A VECTOR-VALUED OPERATIONAL CALCULUS AND ABSTRACT CAUCHY PROBLEMS

A Dissertation

Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in

The Department of Mathematics

by Boris Baeumer Vordiplom, Universität Tübingen, 1991 M.S., Louisiana State University, 1994 August 1997

Acknowledgements

I would like to take this opportunity to express the deepest gratitude to my friends and family; without their love and financial support this work would not have been possible. Among them, a special “thank you” goes to my advisor, Professor Frank Neubrander. His enthusiasm for his work and open heart made me stay at Louisiana State University to pursue this degree. His ceaseless efforts and advice throughout the last five years can not be repaid.

I also would like to thank Professor Rainer Nagel of the University of Tübingen for his advice and guidance during my undergraduate studies and his brilliant suggestion that I should pay a short visit to LSU.

Furthermore, I would like to thank my committee members W. A. Adkins, Y.-S. Chen, J. R. Dorroh, J. A. Goldstein, J. A. Hildebrandt, F. Neubrander and M. M. Tom for their time and help all through my graduate studies.

And finally, I want to thank Kristy Neubrander and Bobby Manuel for proofreading this manuscript and many valuable suggestions.

Prelude

Oliver Heaviside, Electromagnetic Theory, Vol II, §239, London 1895.

“We have now to consider a number of problems which can be solved at once without going to the elaborate theory of Fourier series and integrals. In doing this, we shall have, primarily, to work by instinct, not by rigorous rules. We have to find out first how things go in the mathematics as well as in the physics. When we have learnt the go of it we may be able to see our way to an understanding of the meaning of the processes, and bring them into alignment with other processes. And I must here write a caution. I may have to point out sometimes that my method leads to solutions much more simply than Fourier’s method. I may, therefore, appear to be disparaging and endeavouring to supersede his work. But it is nothing of that sort. In a complete treatise on diffusion Fourier’s and other methods would come side by side – not as antagonists, but as mutual friends helping one another. The limitations of space forbid this, and I must necessarily keep Fourier series and integrals rather in the background. But this is not to be misunderstood in the sense suggested. No one admires Fourier more than I do. It is the only entertaining mathematical work I ever saw. Its lucidity has always been admired. But it was more than lucid. It was luminous. Its light showed a crowd of followers the way to a heap of new physical problems.

The reader who may think that mathematics is all found out, and can be put in a cut-and-dried form like Euclid, in propositions and corollaries, is very much mistaken; and if he expects a similar systematic exposition here he will be disappointed. The virtues of the academical system of rigorous mathematical training are well known. But it has its faults. A very serious one (perhaps a necessary one) is that it checks instead of stimulating any originality the students may possess, by keeping him in regular grooves. Outsiders may find that there are other grooves just as good, and perhaps a great deal better, for their purposes. Now, as my grooves are not the conventional ones, there is no need for any formal treatment. Such would be quite improper for our purpose, and would not be favourable to rapid acquisition and comprehension. For it is in mathematics just as in the real world; you must observe and experiment to find out the go of it. All experimentation is deductive work in a sense, only it is done by trial and error, followed by new deductions and changes of direction to suit circumstances. Only afterwards, when the go of it is known, is any formal expression possible. Nothing could be more fatal to progress than to make fixed rules and conventions at the beginning, and then go by mere deduction. You would be fettered by your own conventions, and be in the same fix as the House of Commons with respect to the dispatch of business, stopped by its own rules.

But the reader may object, ‘Surely the author has got to know the go of it already, and can therefore eliminate the preliminary irregularity and make it logical, not experimental?’ So he has in great measure, but he knows better. It is not the proper way under the circumstances, being an unnatural way. It is ever so much easier to the reader to find the go of it first, and it is the natural way.

The reader may then be able a little later to see the inner meaning of it himself, with a little assistance. To this extend, however, the historical method can be departed from to the reader’s profit. There is no occasion whatever (nor would there be space) to describe the failures which make up the bulk of experimental work. He can be led into successful grooves at once. Of course, I do not write for rigourists (although their attention would be delightful) but for a wider circle of readers who have fewer prejudices, although their mathematical knowledge may be that of the rigourists as a straw to a haystack. It is possible to carry waggonloads of mathematics under your hat, and yet know nothing whatever about the operational solution of physical differential equations.”

Anonymous Fellow of the Royal Society

to Sir Edmund T. Whittaker. See, J.L.B. Cooper [Co].

“There was a sort of tradition that a Fellow of the Royal Society could print almost anything he liked in the Proceedings untroubled by referees: but when Heaviside had published two papers on his symbolic methods, we felt the line had to be drawn somewhere, so we put a stop to it.”

Table of Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Prelude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. Generalized Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

II.1 Basic Concepts and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 II.2 Linear Extensions of Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 II.3 The Convolution on C([0, a]; X). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 II.4 The Vector Space of Generalized Functions . . . . . . . . . . . . . . . . . . . . . . . . . 69

III. The Asymptotic Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 III.1 Asymptotic Laplace Transforms for Functions of Arbitrary Growth . 80 III.2 Asymptotic Laplace Transforms of Generalized Functions . . . . . . . . . . . 95

IV. The Abstract Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 IV.1 The Notion of a Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 IV.2 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Appendix: Relatively Closed Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Abstract

Initial and boundary value problems for linear differential and integro-differential equations are at the heart of mathematical analysis. About 100 years ago, Oliver Heaviside promoted a set of formal, algebraic rules which allow a complete analysis of a large class of such problems. Although Heaviside’s operational calculus was entirely heuristic in nature, it almost always led to correct results. This encouraged many mathematicians to search for a solid mathematical foundation for Heaviside’s method, resulting in two competing mathematical theories:

(a) Laplace transform theory for functions, distributions and other generalized functions,

(b) J. Mikusinski’s field of convolution quotients of continuous functions. In this dissertation we will investigate a unifying approach to Heaviside’s operational calculus which allows us to extend the method to vector-valued functions. The main components are

(a) a new approach to generalized functions, considering them not primarily as functionals on a space of test functions or as convolution quotients in Mikusinski’s quotient field, but as limits of continuous functions in appropriate norms, and

(b) an asymptotic extension of the classical Laplace transform allowing the transform of functions and generalized functions of arbitrary growth at infinity.

The mathematics are based on a careful analysis of the convolution transform f → k ⋆ f. This is done via a new inversion formula for the Laplace transform, which enables us to extend Titchmarsh’s injectivity theorem and Foias’ dense range theorem for the convolution transform to Banach space valued functions. The abstract results are applied to abstract Cauchy problems. We indicate the manner in which the operational methods can be employed to obtain existence and uniqueness results for initial value problems for differential equations in Banach spaces.

I. Introduction

Oliver Heaviside’s 1893 classic “Electromagnetic Theory” [He1-3] proposed formal rules governing manipulations of the differential operator, such that linear differential equations are transformed into algebraic ones; however, his work is not free from debate to say the least. Many mathematicians questioned the platform of his operational calculus. K. Yosida ([Yo], preface) notes, “... the explanation of this operator p (the operator of differentiation) as given by him was difficult to understand and to use, and the range of the validity of his calculus remains unclear still now, although it was widely noticed that his calculus gives correct results in general.”

Though O. Heaviside may deny K. Yosida’s assessment of the rigor of understanding and explaining the calculus, he was unconcerned with remarks on the strength of the foundation of his work (see the prelude). O. Heaviside clearly illustrates his utilitarianism in respect to the calculus in many remarks throughout his work.

In the 1930’s, G. Doetsch, D. V. Widder, and many other mathematicians began to strive for the mathematical foundation of Heaviside’s operational calculus by virtue of the Laplace transform ∫_{0}^{∞} e^{−λt}f(t) dt. However, as K. Yosida ([Yo], preface) remarks, “the use of such integrals naturally confronts restrictions concerning the growth behaviour of the numerical function f.”

Among the goals of this dissertation is the construction of a solid mathematical foundation of Heaviside’s operational calculus. The basis of this foundation lies strictly upon Laplace transform methods and fully extends the operational calculus to Banach space valued functions. J. C. Vignaux’s [Vi] asymptotic version of the Laplace integral ∫ e^{−λt}f(t) dt, which does not require any growth conditions on the locally integrable function f, provides the mathematical mechanism of this work.

For numerical functions, a mathematically sound, algebraic foundation of Heaviside’s operational method was given in 1949, when J. Mikusinski introduced the theory of convolution quotients as a basis for Heaviside’s operational calculus. Due to Titchmarsh’s Theorem ([Ti], Theorem VII), the continuous functions on [0, ∞), with addition and the convolution

k ⋆ f : t ↦ ∫_{0}^{t}k(t − s)f(s) ds

as product, form a ring with no zero divisors, and thus can be extended to a field consisting of the convolution quotients \frac{f}{g}.^{(1)} As anticipated by O. Heaviside, this theory has been successfully applied to Volterra integral equations,

(1) Despite Mikusinski’s contributions, Heaviside’s original operational calculus has all but vanished from modern mathematics. The standard treatment of operational calculus is now almost exclusively based on the Laplace transform. During the last two decades, one of the few proponents of Heaviside’s original operational method was Professor Kôsaku Yosida. H. Komatsu remarks in [Ko]: “ It has always been controversial whether or not Operational Calculus is a mathematics. As far as we know every textbook on Operational Calculus starts with a discussion on this issue. When the Mathematical Society of Japan revised its Encyclopedic Dictionary of Mathematics about ten years ago, the chief editor Professor K. Itô asked opinions of foreign scholars about the then second edition. In his reply a french mathematician wrote ‘Operational Calculus has no value of being mentioned; it is a bad succedaneum of distributions and is very far from being useful.’ Incidentally Professor Yosida was the author of that item. He liked Operational Calculus very much on the contrary. [...] His fondness for Operational Calculus comes probably from his belief that a good mathematics must be not only beautiful but also useful.” integro-differential equations, some nonlinear integral equations, difference equations, differential-difference equations, delay equations, partial differential equations, and naturally to linear ordinary differential equations, in particular to Laplace’s equation

(a_{2}t + b_{2})y′′(t) + (a_{1}(t) + b_{1})y′(t) + (a_{0}(t) + b_{0})y(t) = g(t).  (LE)

The above shown equation contains as special cases the Bessel, Airy, Laguerre, Hermite, Euler, and the confluent hypergeometric differential equation. See, for example, the books “Operational Calculus” by J. Mikusinski ([Mi2]), “Operational Calculus: A Theory of Hyperfunctions” by K. Yosida ([Yo]), or L. Berg’s “Einführung in die Operatorenrechnung” [Ber].

Of course, Mikusinski’s method cannot be extended to functions with values in an arbitrary Banach space since the convolution between two vector-valued functions is in general no longer defined. However, by an extension of Titchmarsh’s theorem to vector-valued functions (see Theorem II.3.12), the space C([0, ∞); X) forms a torsion free^{(2)} module over the convolution ring of scalar functions. We can extend this module to a vector space,^{(3)} considering the set of ordered pairs

M := \{\frac{k ⋆ f}{m} : k, m ∈ C[0, ∞), m ≠ 0, f ∈ C([0, ∞); X)\},

and set \frac{k_{1}⋆f_{1}}{m_{1}} = \frac{k_{2}⋆f_{2}}{m_{2}} if and only if m_{2} ⋆ k_{1} ⋆ f_{1} = m_{1} ⋆ k_{2} ⋆ f_{2}. If one defines (2) Torsion free means that k ⋆ f = 0 implies that f or k is 0.

(3) For an introduction to module theory, see for example W. A. Adkins and S. H. Weintraub [Ad-We]. addition of two elements via

\frac{k_{1} ⋆ f_{1}}{m_{1}} + \frac{k_{2} ⋆ f_{2}}{m_{2}} = \frac{1}{m_{1} ⋆ m_{2}} (m_{2} ⋆ k_{1} ⋆ f_{1} + m_{1} ⋆ k_{2} ⋆ f_{2}) ,

then M becomes a vector space over Mikusinski’s quotient field, a vector space of generalized vector-valued functions.

The abstract nature of the convolution quotients presents a problem in the application of the purely algebraic approach. C. Foias provided an extremely useful analytic result which helps us to understand the nature of such quotients. He showed in 1961 ([Fo]) that Mikusinski’s convolution quotients can be approximated by continuous functions; i.e., for k, m in the Frechét space C[0, ∞) with k(0) = 0 and 0 ∈ supp(m) (i.e., m is not identically zero on [0, ϵ) for all ϵ > 0), there exist h_{n} ∈ C[0, ∞) such that h_{n} ⋆ m → k. Thus, the continuous functions h_{n} “approximate” the convolution quotient \frac{k}{m}. Unfortunately, Foias’ proof of the density of the image of the convolution transform in C_{0}[0, ∞) was done by contradiction, as a result it does not lead to a concrete approximating sequence of a given quotient. The same holds true for the proofs of Foias’ density theorem by J. Mikusinski [Mi3], W. Kierat and K. Skornik [Ki-Sk], or K. Skornik [Sk]. We will give a constructive proof of Foias’ density theorem based on a new inversion formula for the Laplace transform, which holds for the Banach space valued case as well as for the convolution of certain operator families K(t) ∈ L(X)^{(4)} with vector-valued functions.

(4) We denote by L(X, Y) the Banach space of all bounded linear operators between two Banach spaces X and Y. If X = Y, then we write L(X) in lieu of L(X, X). Throughout this dissertation, X will always stand for a Banach space.

Our approach to a mathematical foundation of Heaviside’s operational calculus is not algebraic but entirely analytic in nature. Motivated by Foias’ theorem we view generalized functions primarily as limits of continuous functions in appropriate topologies, rather than defining them as linear functionals on a space of test functions or as convolution quotients. For example, we define Banach spaces of vector-valued generalized functions on the interval [0, 1] to be the completion of C([0, 1]; X) equipped with a norm ||| · ||| which is weaker than the supremums norm ∥ · ∥_{∞}; i.e., |||f||| ≤ ∥f∥_{∞} for all f ∈ C([0, 1]; X). Generalized function spaces are of particular interest in applications to differential equations, where the weaker norm is defined by a convolution operator; i.e.,

|||f||| := ∥K ⋆ f∥_{∞} = sup_{t∈[0,1]} ∥∫_{0}^{t}K(t − s)f(s) ds∥,

where K is a strongly continuous operator family in L(X, Y). In order for |||·||| to be a norm, the convolution operator f → K ⋆ f has to be injective on C([0, 1]; X). We give a proof of Titchmarsh’s injectivity theorem of the convolution transform that holds for Banach space valued functions f ∈ C([0, 1]; X), as well as for certain operator families K(t) ∈ L(X, Y); in particular, it holds for numerical continuous functions k(t) with 0 ∈ supp(k) and strongly continuous semigroups K(t).

Besides generalized functions (defined as limits of continuous functions), a second major aspect of our approach to Heaviside’s operational calculus is an extension of the classical Laplace transform theory, developed by G. Doetsch [Do1-3] and D. V. Widder [Wi1-2], to asymptotic Laplace transforms. Based on Poincare’s method of asymptotic power series (see, for example, R. Remmert [Re], p.294), asymptotic Laplace transforms extend the classical Laplace transform to include functions of arbitrary growth at infinity, while maintaining all essential operational properties of the classical Laplace transform. The Argentinean mathematicians J. C. Vignaux and M. Cotlar ([Vi],[Vi-Co]) first considered asymptotic Laplace transforms in 1939 and 1944. Further contributions were made by W. A. Ditkin (1958) [Di], L. Berg (1962) [Ber], Y. I. Lyubich (1966) [Ly], and M. Deakin (1993) [De]. We follow and extend the approach taken by G. Lumer and F. Neubrander in [Lu-Ne]. Asymptotic Laplace transforms are certain equivalence classes of analytic or meromorphic functions, and as such, they can be multiplied and divided in the obvious manner. The scalar valued asymptotic Laplace transforms form a field, and the vector-valued ones form a vector space over that field. Combining generalized functions and asymptotic Laplace transforms, one can formulate an “operational calculus” for vector-valued functions. Since convolution is transformed by the asymptotic Laplace transform into multiplication, Mikusinski’s quotient field has a one-to-one correspondence to the multiplicative field of asymptotic Laplace transforms. The same holds for the vector space of vector-valued functions over the convolution field of scalar functions and the vector space of vector-valued asymptotic Laplace transforms.

We will demonstrate the use of an operational calculus based on asymptotic Laplace transforms by applying it to Laplace’s differential equation (LE) as well as to abstract linear initial value problems

u′(t) = Au(t), u(0) = x,  (ACP)

where A is a linear operator with domain and range in some Banach space X, where the graph is not necessarily closed in X × X. In particular, we introduce and study the notion of a generalized limit solution of the so-called “Abstract Cauchy Problem” (ACP). The notion of a limit solution resembles the “bonnes solutions” introduced by Ph. Bénilan (see [Ben] or [Ben-Cr-Pa]). They turn out to be helpful in explaining the structure of integrated, k-generalized, C-regularized, distributional, ultradistributional and hyperfunction solutions of (ACP).

Furthermore, we discuss existence, uniqueness and regularity properties of the solutions of (ACP) in terms of the characteristic equation

(λI − A)y(λ) = x + r(λ),  (CE)

where r is a remainder term of exponential decay. The existence, uniqueness and regularity of a solution u = u(t) of (ACP) depends on the existence of local asymptotic resolvents y(λ) solving (CE), their regions of analyticity and the growth therein. This gives us a fine gauge to study (ACP).

This dissertation grew out of the following recent papers and research work by the author, G. Lumer, and F. Neubrander.

• B. Bäumer and F. Neubrander: Laplace transform methods for evolution equations. Conference del Seminario di Matematica dell’ Universitá di Bari 259 (1994), 27–60.

• B. Bäumer and F. Neubrander: Existence and uniqueness of solutions of ordinary linear differential equations in Banach spaces. Submitted.

• G. Lumer and F. Neubrander: Asymptotic Laplace transforms. Preprint.

It is a great pleasure to acknowledge the influence of Prof. G. Lumer’s recent research work on the convolution transform, extending Titchmarsh’s injectivity theorem and Foias’ dense range theorem to vector-valued functions in several variables. Although I know from his results only by hearsay, the knowledge about the existence of such results was a great help and stimulation to find proofs of my own. The proofs of Titchmarsh’s and Foias’ theorem given here were developed independently, with methods disjoint from those used by Professor G. Lumer. We believe that the Laplace transform methods developed (see Section II.3) for the proofs are of independent interest and can be considered as one of the main results of this work.

II. Generalized Functions

II.1 Basic Concepts and Examples

“It has been a standard tactic of the analyst, since the dawn of analysis, that, when forced to deal with a ‘bad’ function, he should try to approximate it with ‘nice’ ones, study the latter and prove that some of the properties in which he happens to be interested, if valid for the approximating nice functions, would carry over to their limit. Of course, the concept of a ‘bad’ function has evolved in time, with the resulting effect that the set of functions considered ‘good’ has steadily increased (but so has the set of functions, or , more generally, of ‘function-like’ objects, considered ‘bad’). We might imagine that Taylor and Mac Laurin felt ill at ease when confronted with analytic functions, and that is why they strove to approximate them by polynomials, whereas for our purposes here, from the local point of view, analytic functions will be regarded as the nicest type of functions (right after polynomials, which retain their supremacy); later on, nondifferentiable continuous functions and the functions which are only measurable would be regarded as bad (they still are), and approximation techniques were devised to deal with them (e.g., approximation by step functions). As we shall see in Part II, functions can become so bad as to stop being functions: they become Dirac’s ‘function’ and measures, and in distribution theory we shall be dealing with derivatives of arbitrary order of measures. In any one of these situations, it will help to have at our disposal approximation techniques, so as to approximate those objects by very smooth functions.”

F. Treves [Tr], p.150.

Usually generalized functions are defined as functionals on a space of test functions.

Starting from this definition, one tries then to shed light on their local structure,

for example, by characterizing them (locally) as (pseudo)-derivatives of continuous

functions. In other words, the local structure theorems for generalized functions

give regularising functions k such that k ⋆ ϕ is a continuous function f;^{(1)} or, at

(1) For example, for any distribution ϕ ∈ D′(Ω), Ω ⊂ IR open, and any relatively compact subset W ⊂ Ω with \overline{W} ⊂ Ω, there exists n ∈ IN such that ϕ ⋆ \frac{t^{n}}{n!} ∈ C(W). See, for example, [Bar], p.70. least formally, that ϕ = \frac{1}{k̂(D)}f, where k̂(λ) := ∫_{0}^{∞} e^{−λt}k(t) dt and D denotes the first derivative operator.

In this section we will explore an alternative approach to generalized functions, taking the local structure theorems as definitions. The following guidelines (see also W. Rudin’s comments in [Ru], p.149) will give us some directions for the definition and construction of particular classes of generalized functions.

(a) Every continuous^{(2)} function should be a generalized function.

(b) Every generalized function should have derivatives which are again generalized functions. For differentiable functions, the new notion of derivative should coincide with the old one.

(c) The usual formal rules of calculus should hold.

(d) There should be a supply of convergence theorems that is adequate for handling the usual limit processes. Our approach to generalized functions is related to a classical approach,

suggested by S. Bochner [Bo] and J. Mikusinski and R. Sikorski ([Mi-Si]), regarding generalized functions as an equivalence class of approximating functions. As mentioned above, this approach is different to the predominant approach via functionals originated by L. Schwartz [Schw] and extended by I. M. Gelfand and G. E. Shilov [Ge-Sh], or the approach taken by J. Mikusinski [Mi], G. Temple [Te], and M. J. Lighthill [Li]. Comparing the three approaches, G. Temple ([Te],

(2) This requirement is rather arbitrary. It would be perfectly justifiable to require (as in the above quote of F. Treves) that C^{∞}-functions, analytic functions or just polynomials are seeds for generalized functions. However, in many of our considerations it will not matter if we start with polynomials or continuous functions since the resulting generalized function spaces will coincide. p.180) remarks: “ […] if any reasonable meaning can be attached to the concept of the derivatives D^{p}f of a continuous function f, then there are at least three different constructive definitions of that concept, namely,

• (Schwartz) D^{p}f is the continuous linear functional T over the space (D) of test functions ϕ, specified by T (ϕ) = (−1)^{p}(fD^{p}ϕ);

• (Mikusinski) D^{p}f is the class of equivalent regular approximations g_{n} such that g_{n}(x) is indefinitely differentiable and (g_{n}ϕ) → (−1)^{p}(fD^{p}ϕ);

• (Bochner) D^{p}f is the class of equivalent regular approximations g_{n} such that g_{n}(x) is indefinitely differentiable, g_{n} = D^{p}f_{n} and f_{n} → f uniformly as n → ∞. […] Each of these three representations has its advantages and disadvantages.

The third definition (Bochner) is the simplest since it does not require the use of the test functions ϕ, but it suffers from an aesthetic disadvantage when the number of independent variables is greater than one […].”

G. Temple clarified and simplified J. Mikusinski’s approach. This was documented by M. J. Lighthill [Li], who remarks in the introduction: “Now, Laurant Schwartz in his Théorie des Distributions has evolved a rigourous theory of these, while Professor Temple has given a version of the theory (generalized functions) which appears to be more readily intelligible to students.”

J. Mikusinski and R. Sikorski also investigated the method suggested by Bochner in their book [Mi-Si]. Their approach, however, as well as the one taken by G. Temple and M. J. Lighthill, had one further drawback: working with defining sequences is not as aesthetic as working with a linear functional, a well defined single mathematical object. For example, E. J. Beltrami in [Bel] was commenting on Mikusinski’s approach by claiming that “the approach of Mikusinski is to work with the concretely defined approximating class {ϕ_{i}} rather than with the more ideal object Φ. It is somewhat like considering irrational numbers only by virtue of their approximation by rationals (which, in fact, is what is done in numerical analysis).”

J. Mikusinski and R. Sikorski noted themselves in their preface of [Mi-Si]: “[...] Notre but est de présenter la théorie des distributions d’une manière simple, accessible également aux physiciens et aux ingénieurs. Afin d’atteindre ce but, nous avons abandonné les méthodes d’analyse fonctionelle et nous avons utilisé le fait que les distributions se laissent approximer par des functions.”

The novelty in our approach is that we bring back the “abandoned functional analytic” aspect by regarding sequences that converge in the above sense of Bochner as elements of a completion of a function space under a new topology. The “unaesthetic” defining sequences vanish in the background of this approach and we can readily consider properties of generalized functions without worrying about the nature of the equivalence class of defining sequences.

In order to define a generalized function space, we adhere to condition (a). Thus, if X is a Banach space and Ω ⊂ IR^{n}, we say that any completion^{(3)} of the space of continuous functions f : Ω → X equipped with a weaker topology (3) The completion of a topological vector space can be obtained by taking the quotient space of all Cauchy sequences modulo sequences converging to zero. See, for example, H. H. Schaefer [Scha] than the one of uniform convergence on compact subsets of Ω is called a space of generalized functions from Ω into X.

To obtain not just a complete topological space, but – more conveniently – a Banach spaces of generalized functions the following construction is useful. Let X be normed vector space, let Y be a Banach space and ν : X → Y be a sublinear operator; i.e., ν is continuous, ν(x) = 0 if and only if x = 0, ∥ν(λx)∥ = ∥λν(x)∥, and ∥ν(x + z)∥ ≤ ∥ν(x)∥ + ∥ν(z)∥ for all x, y, z ∈ X and λ ∈ C. A weaker norm on X is then given by

|||x||| := ∥ν(x)∥.

We call the completion of X equipped with the new norm the ν-extension of X, denoted by \overline{X}^{ν}.^{(4)} Example 1.1 (L^{p}-spaces). Consider X := C[0, 1], the space of continuous functions on [0, 1] equipped with the supremums norm. For 1 ≤ p < ∞ define

ν_{p}f : t ↦ (∫_{0}^{t} |f(s)|^{p} ds)^{1/p} .

Since ν_{p} is a sublinear operator from X into X,

∥f∥_{p} := ∥ν_{p}f∥_{∞} = sup_{t∈[0,1]} |∫_{0}^{t}|f(s)|^{p} ds|^{1/p} = (∫_{0}^{1} |f(s)|^{p} ds)^{1/p}

defines a norm on C[0, 1] and \overline{C[0, 1]}^{ν_{p}} is isometrically isomorphic to L^{p}[0, 1].

It is worthwhile to note that one does not need the definition of the Lebesgue integral nor any notion of measurability to define antiderivatives or definite integrals on intervals of \overline{C[0, 1]}^{ν_{p}}-functions. Let f be a continuous function and If be (4) Since we will not consider closed sets of incomplete spaces, this abuse of notation should not cause any confusion. given by t ↦ ∫_{0}^{t}f(s) ds, where the integral is taken in the Riemann sense. Then

∥If∥_{∞} ≤ ∥f∥_{1} ≤ ∥f∥_{p}

for all f ∈ C[0, 1]. Thus I extends to a continuous linear operator from \overline{C[0, 1]}^{ν_{p}} into X = C[0, 1] (endowed with the supremums norm); i.e., if f ∈ \overline{C[0, 1]}^{ν_{p}} and f_{n} ∈ C[0, 1] with f_{n} → f with respect to the p-norm, then the continuous functions

g_{n} : t ↦ ∫_{0}^{t}f_{n}(s) ds

converge uniformly to the antiderivative If := g ∈ C[0, 1] of f. We can now define the definite integral of f ∈ \overline{C[0, 1]}^{ν_{p}} on an interval (a, b) by

∫_{a}^{b} f(s) ds := g(b) − g(a) = lim g_{n}(b) − g_{n}(a) = lim ∫_{a}^{b} f_{n}(s) ds,

where the latter integral is taken in the Riemann sense. ♢ Clearly, following the above construction, any bounded, injective, linear operator T from C([0, 1]; X) into a Banach space Y will yield a space of generalized functions by setting

∥f∥_{T} := ∥T f∥

and completing C([0, 1]; X) with respect to the norm ∥ · ∥_{T}. In the context of semigroup theory, G. DaPrato and G. Grisvard [DaP-Gr], R. Nagel [Na1-2], and T. Walther [Wa], have used a similar construction to obtain extrapolation spaces which include, for example, the C^{−n} spaces below as a special case. For further references in this direction, see A. Verrusio [Ve].

For us the most important examples for such operators T are the antiderivative operator T : f → ∫_{0}^{(·)} f(s) ds and, more generally, the convolution operators

T_{k} : f ↦ k ⋆ f := ∫_{0}^{(·)} k(· − s)f(s) ds.

In the following example we will give an outline of the main ideas behind our construction of generalized function spaces by studying the antiderivative operator T. For the general convolution case, we refer to the Sections II.2, II.3, and II.4.

Example 1.2 (C_{0}^{−n}-spaces). We consider next Banach spaces consisting of the n^{th}-derivatives of continuous functions. For simplicity we start with n = 1. Let C[0, 1] be equipped with the supremums norm ∥ · ∥_{∞}. Consider the antiderivative operator T f : t ↦ ∫_{0}^{t} f(s) ds. Then T is in L(X) and is one-to-one. Define ∥f∥_{T} := ∥T f∥_{∞} and consider the completion of C[0, 1] under the new norm, in the following denoted as \overline{C[0, 1]}^{T}. Then

\overline{C[0, 1]}^{T} T̃ ; ⟶ ; isom.isom. \overline{Im(T)} = C_{0}[0, 1]
↑ ∪  ∪
C[0, 1] T ; ⟶ ; injective Im(T) =: C_{0}^{1}[0, 1].

We show first that the operator T can be extended to an isometric isomorphism T̃ between \overline{C[0, 1]}^{T} and C_{0}[0, 1]. In order to prove this fact, we start by showing that T can be extended to a bounded linear operator T̃ on \overline{C[0, 1]}^{T}. Let f ∈ \overline{C[0, 1]}^{T}. Thus, by definition of a completion, there exists a sequence (f_{n})_{n∈IN} ⊂ C[0, 1] such that^{(5)}

f =: [f_{n}] = (f_{n})_{n∈IN} + O,

where O = {(h_{n}) ⊂ C[0, 1] : T h_{n} → 0} is the set of zero-sequences, and (f_{n})_{n∈IN} is a Cauchy sequence in the T -norm; i.e., T f_{n} is Cauchy in C[0, 1]. Hence T f_{n} → v for some v ∈ C[0, 1]. Let (g_{n})_{n∈IN} be another sequence with f = T − lim g_{n}. With the same argument, w := lim T g_{n} exists and

∥v − w∥ = lim ∥T f_{n} − T g_{n}∥ = lim ∥f_{n} − g_{n}∥_{T} = ∥f − f∥_{T} = 0.

Thus, T̃f := lim T f_{n} is well defined. Clearly, T̃ is linear and ∥T̃f∥ = lim ∥T f_{n}∥ = lim ∥f_{n}∥_{T} = ∥f∥_{T} for all f ∈ \overline{C[0, 1]}^{T}. Thus T̃ is one-to-one and maps isometrically into C_{0}[0, 1].

To show that T̃ is onto, observe that every function g ∈ C_{0}[0, 1] = \overline{Im(T)} can be approximated by functions g_{n} ∈ C_{0}^{1}[0, 1] = Im(T) and thus g = lim g_{n} = lim T g′_{n}. The sequence (g′_{n})_{n∈IN} is Cauchy with respect to the norm ∥ · ∥_{T} and thus there exists an element g′ ∈ \overline{C[0, 1]}^{T} with

T̃g′ = lim T g′_{n} = lim g_{n} = g.

Hence, the antiderivative operator T̃ is an isometric isomorphism between \overline{C[0, 1]}^{T} and C_{0}[0, 1].

(5) The notation [f_{n}] was introduced by J. Mikusinski and R. Sikorski [Mi-Si] and will be used if we want to stress the approximative nature of the generalized function. Using the natural embedding, we can also say that f = T − lim f_{n}.

We call g′ the generalized derivative of g and denote by

C_{0}^{−1}[0, 1] := \overline{C[0, 1]}^{T}

the space of generalized derivatives of continuous functions in C_{0}[0, 1]. Note that the generalized derivative is so far only defined for continuous functions f with f(0) = 0.

Since the antiderivative operator is an isomorphism, we also have that every generalized function f ∈ C_{0}^{−1} is integrable on intervals; i.e., if f = [f_{n}] with f_{n} ∈ C[0, 1], then T̃f := g ∈ C_{0}[0, 1], T f_{n} := g_{n} ∈ C_{0}[0, 1], and

∫_{a}^{b} f(s) ds := g(b) − g(a) = lim_{n→∞} g_{n}(b) − g_{n}(a) = lim_{n→∞} ∫_{a}^{b} f_{n}(s) ds,

where the latter integral can be taken in the Riemann sense. It is easy to see that this construction also works for functions with values in a Banach space X. Thus, we can formulate the following version of the Fundamental Theorem of Calculus. Theorem 1.3 (Fundamental Theorem of Calculus). Let X be a Banach space and g ∈ C_{0}([0, 1]; X). Then g is differentiable, g′ ∈ C_{0}^{−1}([0, 1]; X) is integrable and

∫_{a}^{b} g′(s) ds = g(b) − g(a)

for all 0 ≤ a ≤ b ≤ 1. Moreover, any f ∈ C_{0}^{−1}([0, 1]; X) is integrable, g(t) := ∫_{0}^{t}f(s) ds ∈ C_{0}([0, 1]; X) and g′ = f. Next we want to shed some light on the nature of the generalized derivatives. Referring to our guidelines (a) − (d) we ask the question, whether this notion of a generalized derivative coincides with the classic notion of a derivative. Clearly, the generalized derivative of a differentiable function f ∈ C_{0}^{1}[0, 1] is its derivative modulo 0-sequences; i.e., sequences (f_{n})_{n∈IN} ⊂ C_{0}[0, 1] with the property that

sup_{t∈[0,1]} |∫_{0}^{t}f_{n}(s) ds| → 0.

A nontrivial example for such a zero-sequence is given by the functions f_{n} : t ↦ e^{int} since T f_{n}(t) = \frac{1}{in} (e^{int} − 1) → 0.

Secondly, note that

L^{p}[0, 1] ⊂ C_{0}^{−1}[0, 1]

for all 1 ≤ p ≤ ∞, since

∥f∥_{−1} = ∥T f∥ ≤ ∥f∥_{L^{1}} ≤ ∥f∥_{L^{p}} ≤ ∥f∥_{∞}.

Moreover, consider the above sequence of functions f_{n} : t ↦ e^{int}. Then f_{n} does not converge in L^{1} since

∫_{0}^{1} |f_{n}(t) − f_{n+1}(t)| dt = ∫_{0}^{1} |e^{int} − e^{i(n+1)t}| dt = ∫_{0}^{1} |1 − e^{it}| dt > 0

for all n ∈ IN. Therefore, the space C_{0}^{−1}[0, 1] is strictly larger than L^{1}[0, 1] and thus strictly larger than L^{p}[0, 1] for all 1 ≤ p ≤ ∞. In the following we identify functions in L^{p}[0, 1] with the corresponding generalized functions in C_{0}^{−1}[0, 1].

Further it is important to notice that the space P [0, 1], the space of all polynomials on [0, 1] (or just the liner span of monomials t ↦ t^{α_{n}} satisfying the Müntz condition ∑_{n=1}^{∞} \frac{1}{α_{n}} = ∞) is dense^{(6)} in \overline{C[0, 1]}^{T}. The reason is that P [0, 1] is (6) Following E. Hille and R. S. Phillips, [Hi-Ph], we say that a set M is dense in a closed set X_{0} if X_{0} = \overline{M ∩ X_{0}}. dense in C_{0}[0, 1] – in the supremums norm and thus in the L^{1}-norm and in the C_{0}^{−1}[0, 1]-norm. The space C_{0}[0, 1] itself is dense in C[0, 1] in the L^{1}-norm and thus in the C_{0}^{−1}[0, 1]-norm. Hence

\overline{P [0, 1]}^{T} = \overline{C_{0}[0, 1]}^{T} = \overline{C[0, 1]}^{T} = \overline{L^{p}[0, 1]}^{T} = C_{0}^{−1}[0, 1].

We employ these embeddings to show that the notion of a generalized derivative coincides also with derivatives of an absolutely continuous function. Let f be a any absolutely continuous function with f(0) = 0. Since C[0, 1] is dense in L^{1}[0, 1] there exists a sequence (g_{n})_{n∈IN} ⊂ C[0, 1] such that g_{n} → f′ in L^{1}-norm, and thus also in ∥ · ∥_{T}. Thus f′ = [g_{n}] can be identified with an element in C_{0}^{−1}[0, 1]. Furthermore, since ∫_{0}^{t}g_{n}(s) ds → f(t) uniformly in t, it is the generalized derivative of f.

Another important class of examples are generalized derivatives of Banach space valued functions. Define f : [0, 1] → L^{∞}[0, 1] =: X via t ↦ χ_{[0,t]}. Then f is not almost separably valued and hence f ∉ L^{1}([0, 1]; X). However, f is Riemannintegrable and ∫_{0}^{t}f(s) ds = (t − ·)χ_{[0,t]} = lim_{n→∞} ∫_{0}^{t}f_{n}(s) ds, where

f_{n}(t) := s ↦ \{ \begin{matrix}1 & if 0 ≤ s ≤ t − \frac{1}{n}, \\ n(t − s) & if t − 1/n ≤ s ≤ t, \\ 0 & else, \end{matrix}

and the limit is uniform in t. Thus f : t ↦ χ_{[0,t]} can be identified with an element in C_{0}^{−1}([0, 1]; X).

As another example, for t ∈ [0, 1], define T (t) : L^{∞}([0, ∞) → L^{∞}[0, ∞) via

T (t)h : s ↦ h(t + s).

Then the operator family (T (t))_{t∈[0,1]} is not strongly Bochner integrable since T (·)h is not separately valued for all discontinuous h. But similar to the previous example, T (·)h can be identified with an element in C_{0}^{−1}([0, 1]; X), and thus (T (t))_{t∈[0,1]} is strongly “integrable.”

Next, we want to comment on item (d) of our guidelines. Clearly, through the approximative nature of the generalized functions we have already an abundance of convergence theorems.^{(7)} On top of them, we will show that the embeddings are compact. Let (f_{n}) be a bounded sequence of continuous functions. Then the sequence (T f_{n}) is equicontinuous and thus, by the theorem of Arzela-Ascoli, there exists a subsequence (f_{n_{k}}), such that T f_{n_{k}} converges; i.e., f_{n_{k}} converges in C_{0}^{−1}[0, 1]. Thus every bounded sequence of continuous functions has a convergent subsequence in C_{0}^{−1}[0, 1], which means that the embedding of C[0, 1] into C_{0}^{−1}[0, 1] is compact. In Section II.2 we will show that any embedding into a generalized function space, that was obtained via a compact operator with a dense image, is compact.

Next, we want to investigate, whether the usual formal rules of calculus hold (see part (c) of our guidelines). In order to talk about the derivative of a product or a composition we have to define products and compositions of generalized functions first.

(7) For example, if C[0, 1] ∋ f_{n} → f in C_{0}^{−1}[0, 1], then f_{n} might not converge pointwise as the example f_{n}(t) := e^{int} shows, but f_{n} converges towards f in the mean; i.e., for all 0 ≤ a < b ≤ 1,

\frac{1}{b − a} ∫_{a}^{b} f_{n}(s) ds → \frac{1}{b − a} ∫_{a}^{b} f(s) ds.

Clearly, we would like to define a product on C_{0}^{−1}[0, 1] that coincides with the product of continuous functions. The naive approach would be to multiply the approximating sequences and hope that they converge in the ∥ · ∥_{T}, but that does not work even for the embedded continuous functions in C_{0}^{−1}[0, 1]. For example, let f_{n} : t ↦ sin(2^{n}πt). Then f_{n} → 0 in C_{0}^{−1}[0, 1]. But with the above definition we would have 0 · 0 = [f_{n}] · [f_{n}] = [f_{n}^{2}] ≠ 0, since

∥[f_{n}^{2}]∥_{T} ≥ ∫_{0}^{1} sin^{2}(2^{n}πs) ds = \frac{1}{2^{n}π} ∫_{0}^{2^{n}π} sin^{2} s ds = \frac{2^{n}}{2^{n}π} ∫_{0}^{π} sin^{2} s ds ↛ 0.

Another hint that defining a meaningful product on all of C_{0}^{−1}[0, 1] might be difficult or impossible, is the fact that even on L^{1}[0, 1] one cannot define a product that stays in L^{1}[0, 1].

The next attempt, mimicking the situation in L^{1}, is that we try to define the product between a continuous function and a generalized function; i.e., for f := [f_{n}] ∈ C_{0}^{−1}[0, 1] and g ∈ C[0, 1] we would like to define

fg := [f_{n}]g := [f_{n}g] = T − lim f_{n}g.  (1)

Unfortunately, as we will see, the limit does not always exist. In particular, there exists a zero sequence (f_{n})_{n∈IN} and a continuous function g such that 0·g = [f_{n}]·g does not converge, which implies that the product defined as above is not well defined. Let h : t ↦= \sqrt{t} sin \frac{1}{t}. Then h′(t) = \frac{1}{2\sqrt{t}} sin \frac{1}{t} − \frac{\sqrt{t}}{t^{2}} cos \frac{1}{t}. Let ξ_{n}, ϵ_{n} be such that \frac{1}{(n+1)π} < ξ_{n} < \frac{1}{nπ}, 0 < ϵ_{n} < ξ_{n}/2 and h′(ξ_{n}) = h′(ϵ_{n}) = 0. Define

f_{n} : t ↦ \{ \begin{matrix}0 & for 0 ≤ t ≤ ϵ_{n}, \\ h′(t) & for ϵ_{n} ≤ t ≤ ξ_{n} \\ 0 & for t ≥ ξ_{n}. \end{matrix}

Then f_{n} → 0 in ∥ · ∥_{T} -norm, since

∥f_{n}∥_{T} ≤ sup_{t∈[ϵ_{n},ξ_{n}]} | ∫_{ϵ_{n}}^{t} h′(s) ds| = sup_{t∈[ϵ_{n},ξ_{n}]} |\sqrt{t} sin \frac{1}{t} − \sqrt{ϵ_{n}} sin \frac{1}{ϵ_{n}} | → 0.

Let g : t ↦ \sqrt{t} cos \frac{1}{t}. However, f_{n}g does not converge since

sup_{t∈[0,1]} |∫_{0}^{t}f_{n}(s)g(s) ds| ≥| ∫_{ϵ_{n}}^{ξ_{n}} f_{n}(s)g(s) ds| = | ∫_{ϵ_{n}}^{ξ_{n}} \frac{1}{2} sin \frac{1}{s} cos \frac{1}{s} − \frac{1}{s} cos^{2} \frac{1}{s} ds|
≥| ∫_{ϵ_{n}}^{ξ_{n}} \frac{1}{s} cos^{2} \frac{1}{s} ds| − | ∫_{ϵ_{n}}^{ξ_{n}} \frac{1}{2} sin \frac{1}{s} cos \frac{1}{s} ds|
= ∫_{1/ξ_{n}}^{1/ϵ_{n}} s cos^{2} s ds − | ∫_{ϵ_{n}}^{ξ_{n}} \frac{1}{2} sin \frac{1}{s} cos \frac{1}{s} ds| → ∞.

However, one can multiply with C^{1}-functions, taking (1) as definition. Suppose g ∈ C^{1}[0, 1], f_{n} ∈ C[0, 1] and f_{n} → f in C_{0}^{−1}[0, 1]. Let F_{n} := T f_{n} and F := T̃f ∈ C_{0}[0, 1]. Then, in C_{0}[0, 1],

\begin{matrix}T (f_{n}g) = & ∫_{0}^{(·)} f_{n}(s)g(s) ds = F_{n}g − ∫_{0}^{(·)} F_{n}(s)g′(s) ds \\ → & F g − ∫_{0}^{(·)} F (s)g′(s) ds = F g − T (F g′). \end{matrix}

Thus f_{n}g → fg := T̃^{−1}(F g) − F g′ in C_{0}^{−1}[0, 1]. Since T̃^{−1} corresponds to differentiation and f = T̃^{−1}F we have proved the following proposition. Proposition 1.4. a) Let f = [f_{n}] ∈ C_{0}^{−1}[0, 1] and g ∈ C^{1}[0, 1]. Then fg := T − lim f_{n}g exists in C_{0}^{−1}[0, 1].

b) Let F ∈ C_{0}[0, 1] and g ∈ C^{1}[0, 1]. Then (F g)′ = F′g + fg′. In a similar fashion, we can define the composition of a C_{0}^{−1}-function f and a C^{2}-function g with g′(t) ≠ 0 for all t ∈ [0, 1]. Define

f(g) := T − lim f_{n}(g).

In order to show that the limit exists, we observe that

F (g) = lim F_{n}(g) = lim T (F_{n}(g))′ = lim T (f_{n}(g)g′)

in C_{0}[0, 1]. Thus T̃^{−1} (F (g)) = T − lim f_{n}(g)g′ ∈ C_{0}^{−1}[0, 1]. Since g′(t) ≠ 0, we have that \frac{1}{g′} ∈ C^{1}[0, 1], and hence, by the fact that we can multiply with differentiable functions, T̃^{−1} (F (g)) \frac{1}{g′} ∈ C_{0}^{−1}[0, 1] and T^{−1} (F (g)) \frac{1}{g′} = T − lim f_{n}(g) =: f(g). This shows that the following proposition holds. Proposition 1.5. a) Let f = [f_{n}] ∈ C_{0}^{−1}[0, 1] and g ∈ C^{2}[0, 1] with g′(t) ≠ 0 for all t ∈ [0, 1]. Then f(g) := T − lim f_{n}(g) exists in C_{0}^{−1}[0, 1].

b) Let F ∈ C_{0}[0, 1] and g ∈ C^{2}[0, 1] with g′(t) ≠ 0 for all t ∈ [0, 1]. Then (F (g))′ = F′(g)g′. Last, we consider item (b) of our guidelines for constructing generalized function spaces. To acquire the differentiability of every generalized function, we will construct a tower of generalized function spaces by considering powers^{(8)} of T. The derivative of a generalized function will then be a generalized function in a generalized function space of higher order. For the antiderivative operator T, the operators T^{2}, T^{n} are also bounded and one-to-one on C[0, 1]. With the same argument as above we can define C^{−n}[0, 1] := \overline{C[0, 1]}^{T^{n}}. If we identify C_{0}[0, 1] with its embedding in C_{0}^{−1}[0, 1], the fact that T̃ is an isometry between C_{0}^{−1}[0, 1] and C_{0}[0, 1]) implies in particular that T̃ ∈ L(C_{0}^{−1}[0, 1]) and that T̃ is one-to-one. We will show in Section II.2 that \overline{C_{0}^{−1}[0, 1]}T̃ = \overline{C[0, 1]}^{T^{2}} = C^{−2}[0, 1]. In particular, we (8) We give here the discrete version. Fractional powers T^{α} of T for α > −1, which are convolutions with \frac{t^{α}}{Γ(α+1)} are discussed in Section II.4. can look at C^{−2}[0, 1] and, inductively at C^{−n}[0, 1], in the following way.

⋮  ⋮

C^{−n}[0, 1] ˜˜T̃ ⟶

isom. isomorph

C^{−n−1}[0, 1]
⋮ ⋮

C^{−2}[0, 1] ˜T̃ ⟶

isom. isomorph

C_{0}^{−1}[0, 1]
↑ ∪  ∪

C_{0}^{−1}[0, 1] T̃ ⟶

isom. isomorph

C_{0}[0, 1]
↑ ∪  ∪

C_{0}[0, 1] T ⟶

isom. isomorph

C_{0,0}^{1}[0, 1]
⋮  ⋮

Here C_{0,0}^{1}[0, 1] := {f ∈ C^{1}[0, 1] : f(0) = f′(0) = 0} with ∥f∥_{1} = ∥f′∥_{∞}.

Clearly, we can construct similar towers for L^{p} spaces, completing them with the new norm

∥f∥_{−n,p} := ∥T^{n}f∥_{p}.

Since C_{0}[0, 1] ↪ L^{p}[0, 1] ↪ C_{0}^{−1}[0, 1], we obtain the embeddings

C_{0}^{−n}[0, 1] ↪ L^{p,−n}[0, 1] ↪ C_{0}^{−(n+1)}[0, 1].

It is worthwhile to note that the spaces C_{0}^{−n}[0, 1] have a partial ordering if we set f ⪯_{−n} g if T^{n}f ≤ T^{n}g. They are in fact Banach lattices, and, even more, AMspaces (see H. H. Schaefer [Scha2] for an introduction to Banach lattices). The positive cones are getting larger if we increase n. For example, t ↦ sin 2πt ≻_{−1} 0 in C^{−1}[0, 1], whereas t ↦ cos 2πt ⊁_{1} 0, but t ↦ cos 2πt ≻_{−2} 0 in C^{−2}[0, 1].

We end the section by giving examples of distributions which are contained in the spaces C_{0}^{−n}[0, 1]. First we consider Dirac’s δ-function. Since the constant 1function is in \overline{C[0, 1]}^{T} = C_{0}^{−1}[0, 1] ⊂ C^{−2}[0, 1], and ˜ T̃ is an isometric isomorphism between C^{−2}[0, 1] and C_{0}^{−1}[0, 1], the generalized derivative ˜ T̃^{−1} of the constant 1function cannot be zero. We can identify the constant 1-function as the generalized derivative of the function F (x) = x. Thus, the generalized derivative of the constant 1-function is the same as the second generalized derivative of F.

Let (f_{n})_{n∈IN} be a sequence in C[0, 1] such that T^{2}f_{n} → F.^{(9)} Then (f_{n})_{n∈IN} is a Cauchy sequence in C^{−2}[0, 1], and f_{n} → F′′ ∈ C^{−2}[0, 1].

The generalized function f := F′′ has the same properties as the functional δ_{0} − δ_{1} on C[0, 1]. To see this, let g ∈ C^{∞}[0, 1] with g′(1) = 0. Then

\begin{aligned}∫_{0}^{1} f(s)g(s) ds & := lim ∫_{0}^{1} f_{n}(s)g(s) ds = lim ∫_{0}^{1} T^{2}(f_{n})(s)g′′(s) ds \\ & = ∫_{0}^{1} sg′′(s) ds = g(0) − g(1). \end{aligned}

Since the functions g ∈ C^{∞}[0, 1] with g′(1) = 0 are dense in C[0, 1] and since the functional L_{f} with L_{f}(g) := ∫_{0}^{1} f(s)g(s) ds = g(0) − g(1) is continuous with respect to the supremums norm on C^{∞}, we can extend L_{f} to C[0, 1] and L_{f}(h) = h(0) − h(1) for all h ∈ C[0, 1].

This result is not too surprising. If we identify functions on [0, 1] with functions on (−∞, ∞) by extending the functions with 0, we may identify the constant (9) For example, let f_{n}(x) := \{ \begin{matrix}2n − 2n^{2}x & for 0 ≤ x ≤ 1/n \\ 0 & else \end{matrix}.

1-function with the Heaviside function H(0) − H(1), whose distributional derivative is δ_{0} − δ_{1}.

In order obtain δ_{0} without δ_{1} we choose the Frechét space C[0, 1). There, the C^{∞}-functions g with lim_{t→1} g(t) = lim_{t→1} g′(t) = 0 are dense, and thus L_{f}(h) = h(0). Thus, the generalized derivative of the constant 1-function coincides with the Dirac functional on the space C[0, 1) and with δ_{0} − δ_{1} as functional on C[0, 1].

As an other example, we locate the function t ↦ \frac{1}{t}, or, in the language of distributions, the finite part of \frac{1}{t} in the tower of spaces C_{0}^{−n}[0, 1]. Since the second derivative of f : t ↦ t ln t−t is \frac{1}{t} for t > 0, and since f ∈ C_{0}[0, 1], we check whether the generalized second derivative of f has the distributional properties of the finite part of \frac{1}{t}H_{0}; i.e., whether ∫_{0}^{1} f′′(t)g(t) dt = ∫_{0}^{1} \frac{g(t)−g(0)}{t} dt for all g ∈ C^{2}[0, 1] with supp(g) ∈ [0, 1) (See, for example, A. H. Zemanian [Ze], p.18 for a discussion of finite parts).

Let f′′ = [h_{n}] be the second generalized derivative of f, i.e., h_{n} ∈ C[0, 1] and T^{2}h_{n} → f. Let g ∈ C^{2}[0, 1] with supp(g) ∈ [0, 1). Then

∫_{0}^{1} h_{n}(t)g(t) dt = ∫_{0}^{1} T^{2}(h_{n}(t))g′′(t) dt → ∫_{0}^{1} f(t)g′′(t) dt.

However, for all 0 < ϵ < 1,

∫_{0}^{1} f(t)g′′(t) dt = ∫_{0}^{ϵ} f(t)g′′(t) dt + ∫_{ϵ}^{1} f(t)g′′(t) dt

= ∫_{0}^{ϵ} f(t)g′′(t) dt − f(ϵ)g′(ϵ) − ∫_{ϵ}^{1} ln tg′(t) dt

= ∫_{0}^{ϵ} f(t)g′′(t) dt − f(ϵ)g′(ϵ) + ln ϵ(g(ϵ) − g(0)) + ∫_{ϵ}^{1} \frac{g(t) − g(0)}{t} dt

= ∫_{0}^{1} \frac{g(t) − g(0)}{t} dt,

since ϵ was chosen arbitrarily and the first three terms converge to 0 as ϵ → 0. Thus, the generalized derivative of t ↦ ln t ∈ L^{1}[0, 1] ⊂ C_{0}^{−1}[0, 1], which is the same as the second generalized derivative of t ↦ t ln t − t ∈ C_{0}[0, 1], corresponds to the finite part distribution \frac{1}{t}H_{0}.

Instead of just looking at the antiderivative operator, we will consider in the following sections the convolution operator T_{k} with T_{k}f := ∫_{0}^{(·)} k(· − s)f(s) ds. E. C. Titchmarsh showed in 1925 ([Ti], Theorem VII) that T_{k} is one-to-one if 0 ∈ supp(k). C. Foias showed in 1961([Fo]), that under the same condition (i.e., 0 ∈ supp(k)), the image of T_{k} is dense in L^{1}[0, 1]. K. Skórnik in [Sk] proved the density of the image in C_{0}[0, 1]. Thus the same construction of generalized function spaces works if one takes the convolution operator T_{k} instead of the antiderivative operator T. Since this is a central point in this dissertation, we will give proofs of vector-valued versions of Titchmarsh’s Injectivity Theorem and Foias’ Dense Range Theorem in Section II.3. The “towers” of generalized function spaces, defined via T_{k}, are investigated in Section II.4. They lead to a “continuous” diagram of spaces which allow, if combined with asymptotic Laplace transforms, an operational calculus as powerful as the one developed by J. Mikusinski in [Mi2] or by K. Yosida in [Yo].

II.2 Linear Extensions of Banach Spaces

In this section we collect some basic properties of linear extensions of Banach spaces. A linear extension of a Banach space X is a completion of X with respect to a new topology which has been obtained via a bounded, linear operator T : X → Y (Y Banach space) that is one-to-one.^{(1)} In other words, the new norm has been obtained by measuring the image of the linear operator with respect to the old norm; i.e.,

∥x∥_{T} := ∥T x∥.

The linear extensions, denoted by \overline{X}^{T}, have the following crucial property which explains why they play an important role in many applications. Theorem 2.1.^{(2)} Let X, Y be Banach spaces and let T ∈ L(X, Y) be one-to-one. On X define a new norm via ∥x∥_{T} := ∥T x∥ and let \overline{X}^{T} denote the completion of the normed vector space X^{T} := (X, ∥ · ∥_{T}). Then

\overline{X}^{T} T̃ ⟶

isom. isomorph

\overline{Im(T)}
↑ ∪  ∪
X  \underrightarrow{T}  Im(T);

i.e., X is continuously embedded in \overline{X}^{T}, and the operator T extends to an isometric isomorphism T̃ between \overline{X}^{T} and \overline{Im(T)}. In particular, (X, ∥ · ∥_{T}) is already a Banach space if and only if Im(T) is closed in Y. Moreover, the operator T is compact if and only if every bounded sequence in X has a convergent subsequence in \overline{X}^{T}.

(1) For the sake of clarity and brevity we confine ourselves to Banach spaces. The whole section also holds for spaces equipped with a separating set of seminorms ∥·∥_{α}. All the theorems can be reformulated replacing “Banach space” with “Frechét space” and ∥ · ∥ with ∥ · ∥_{α}. Moreover, the operator T may depend on α.

(2) See also R. Nagel [Na] and T. Walther [Wa].

Proof. Clearly, X can be identified with a subset of \overline{X}^{T} by identifying x ∈ X with the constant sequence x modulo the set of zero-sequences with respect to ∥ · ∥_{T}. Since

∥x∥_{T} = ∥T x∥ ≤ ∥T ∥ ∥x∥

for all x ∈ X, the identity is a continuous map from the Banach space (X, ∥ · ∥) into \overline{X}^{T}.

Let z ∈ \overline{X}^{T}. Then there exist x_{n} ∈ X such that x_{n} → z in \overline{X}^{T}. This implies that the sequence (x_{n})_{n∈IN} is Cauchy in \overline{X}^{T} or, equivalently, that the sequence (T x_{n})_{n∈IN} is Cauchy in Y. This shows that y := lim_{n→∞} T x_{n} exists. Consider another sequence (v_{n})_{n∈IN} ∈ X such that v_{n} → z in \overline{X}^{T}. Then, as above, w := lim_{n→∞} T v_{n} exists in Y. Now,

∥y − w∥ = lim ∥T x_{n} − T v_{n}∥ = lim ∥x_{n} − v_{n}∥_{T} = ∥z − z∥_{T} = 0.

Thus,

T̃z := lim_{n→∞} T x_{n}

is a well-defined extension of T mapping \overline{X}^{T} into \overline{Im(T)}. Clearly, T̃ is linear and

∥T̃z∥ = lim ∥T x_{n}∥ = lim ∥x_{n}∥_{T} = ∥z∥_{T}

for all z ∈ \overline{X}^{T}. This shows that T̃ ∈ L(\overline{X}^{T} , \overline{Im(T)}) maps isometrically and is one-to-one. To show that T̃ is onto, let y ∈ \overline{Im(T)} and y_{n} ∈ Im(T) with y_{n} → y in Y. Let x_{n} ∈ X with T x_{n} = y_{n}. Since ∥y_{n}∥ = ∥T x_{n}∥ = ∥x_{n}∥_{T} it follows that the sequence (x_{n})_{n∈IN} is a Cauchy sequence in \overline{X}^{T}. Let z ∈ \overline{X}^{T} such that x_{n} → z in \overline{X}^{T}. Then

T̃z = lim T x_{n} = lim y_{n} = y.

Thus T̃ is an isometric isomorphism between \overline{X}^{T} and \overline{Im(T)}.

Suppose X^{T} = (X, ∥ · ∥_{T}) is a Banach space. We show that Im(T) is closed. Let y ∈ \overline{Im(T)}. Then there exist x_{n} ∈ X such that ∥T x_{n} − y∥ → 0. Since the sequence (T x_{n})_{n∈IN} is Cauchy in Y it follows that the sequence (x_{n})_{n∈IN} is Cauchy in X^{T}. Since X^{T} is complete, there exists z ∈ X such that ∥x_{n} − z∥_{T} → 0, or, equivalently, that ∥T x_{n} − T z∥ → 0. Thus, T z = y and therefore Im(T) is closed.

Suppose that Im(T) is closed. Let (x_{n})_{n∈IN} be a Cauchy sequence in X^{T}. Thus (T x_{n})_{n∈IN} is Cauchy and, due to the closedness of Im(T), converges to T x for some x ∈ X. Hence x_{n} → x in X^{T} and thus every Cauchy sequence in X^{T} converges. Thus X^{T} is a Banach space.

The last statement follows from the fact that a map T is compact if and only if for every bounded sequence (x_{n})_{n∈IN} in X, the sequence (T x_{n})_{n∈IN} has a convergent subsequence (T x_{n_{i}})_{i∈IN}; i.e., (x_{n_{i}})_{i∈IN} is convergent in \overline{X}^{T}. ♢

As explained in the previous section, a typical situation we have in mind is when X = Y = C[0, 1], and T is given by the antiderivative operator T : f ↦ ∫_{0}^{(·)} f(s) ds. Then T is injective and \overline{ImT} = C_{0}[0, 1]. Thus, the derivative operator T^{−1} extends to a bounded linear operator from C_{0}[0, 1] onto \overline{C[0, 1]}^{T}. Since we want to do analysis on \overline{C[0, 1]}^{T} we would like to extend typical operations S of analysis (such as integration or convolution) from C[0, 1] to \overline{C[0, 1]}^{T}. This can be achieved via the mechanism described in the following theorem.

Theorem 2.2. Let X be a Banach space, and let T ∈ L(X) be one-to-one. Let S ∈ L(X) with ST = T S. Then the following statements hold.

(a) The operator S has an extension S̃ ∈ L(\overline{X}^{T}) with ∥S̃∥ ≤ ∥S∥ and S̃T̃ = T̃S̃. Moreover, S is one-to-one if and only if S̃ is one-to-one.

(b) Let S be one-to-one. Then \overline{X}^{ST} is isometrically isomorphic to \overline{(\overline{X}^{T})}S̃ (= \overline{(\overline{X}^{S})}T̃).

(c) Suppose there exists V ∈ L(X) with V T = T V, V is one-to-one and \overline{Im(V)} = \overline{Im(T)}. Then \overline{X}^{T} ⊂ \overline{X}^{V} if and only if there exists S ∈ L(X^{T}) with S being one-to-one and ST = T S such that \overline{\overline{X}^{T}}^{S} is isometrically isomorphic to \overline{X}^{V}. Moreover, S can be chosen to be T̃^{−1}V.

(d) If S is compact then S̃ is compact, and the converse holds if the image of T is dense in X.

Proof. (a) Let z ∈ \overline{X}^{T}. Then there exist x_{n} ∈ X such that x_{n} → z in \overline{X}^{T}. Since ∥Sx_{n}−Sx_{m}∥_{T} = ∥T Sx_{n}−T Sx_{m}∥ = ∥ST x_{n}−ST x_{m}∥ ≤ ∥S∥ ∥T x_{n}−T x_{m}∥ = ∥S∥ ∥x_{n} − x_{m}∥_{T}, it follows that there exists y ∈ \overline{X}^{T} such that Sx_{n} → y in \overline{X}^{T}. Consider another sequence (v_{n})_{n∈IN} ⊂ X such that v_{n} → z in \overline{X}^{T}. Then, as above, there exists w ∈ \overline{X}^{T} such that Sv_{n} → w in \overline{X}^{T}. Now,

∥y − w∥_{T} = lim ∥Sx_{n} − Sv_{n}∥_{T} = lim ∥T Sx_{n} − T Sv_{n}∥
≤∥S∥ lim ∥x_{n} − v_{n}∥_{T} = ∥z − z∥_{T} = 0.

Thus, S̃z := T − lim_{n→∞} Sx_{n} is a well-defined extension of S mapping \overline{X}^{T} into \overline{X}^{T}. Clearly, S̃ is linear and

∥S̃z∥_{T} = lim ∥Sx_{n}∥_{T} ≤ ∥S∥ lim ∥x_{n}∥_{T} = ∥S∥ ∥z∥_{T}

for all z ∈ \overline{X}^{T}. This shows that S̃ ∈ L(\overline{X}^{T}) and ∥S̃∥ ≤ ∥S∥. The commutativity of S̃ and T̃ follows from

S̃T̃z = ST̃z (since T̃z ∈ \overline{Im(T)} ⊂ X)
= S(lim T x_{n}) (where the limit is taken in X and x_{n} → z in \overline{X}^{T})
= lim ST x_{n} = lim T Sx_{n} = lim T̃Sx_{n}
= T − lim T̃Sx_{n} (if w = lim w_{n}, then w = T − lim w_{n})
= T̃(T − lim Sx_{n}) = T̃S̃z.

If S̃ is one-to-one, then trivially, S is one-to-one. Suppose that S is one-to-one. If 0 = S̃z, then Sx_{n} → 0 in \overline{X}^{T}, where x_{n} → z in \overline{X}^{T}. Thus T Sx_{n} = ST x_{n} → 0 in X. Since lim T x_{n} exists in X, it follows from the boundedness of S that S(lim T x_{n}) = lim ST x_{n} = 0. Thus T x_{n} → 0 in X. This shows that ∥x_{n}∥_{T} → 0 or, equivalently, z = 0. Hence S̃ is one-to-one.

(b) For x ∈ \overline{X}^{ST} with x = ST − lim x_{n} for some x_{n} ∈ X define

Φ(x) := S̃ − lim_{n→∞} x_{n} in \overline{X}^{T} .

First we show that Φ(x) always exists. This follows from

∥S̃x_{n} − S̃x_{m}∥\overline{X}^{T} = ∥T Sx_{n} − T Sx_{m}∥_{X} = ∥ST x_{n} − ST x_{m}∥_{X} → 0.

Next we show that Φ(x) is well defined. Suppose x = ST − lim x_{n} = ST − lim y_{n}. Then

lim ∥S̃x_{n} − S̃y_{n}∥\overline{X}^{T} = lim ∥ST x_{n} − ST y_{n}∥_{X} = 0.

Thus S̃ − lim x_{n} = S̃ − lim y_{n} in \overline{X}^{T}.

Clearly, Φ is linear. Now we show that Φ is one-to-one. Suppose Φ(x) = 0. Then Φ(x) = S̃ − lim x_{n} = 0 in \overline{X}^{T}. implies that

∥ST x_{n}∥_{X} = ∥T Sx_{n}∥_{X} = ∥S̃x_{n}∥\overline{X}^{T} → 0.

Thus x = ST − lim x_{n} = 0.

The fact that Φ is isometric follows from

∥x∥\overline{X}^{ST} = lim ∥ST x_{n}∥_{X} = lim ∥T Sx_{n}∥_{X} = lim ∥S̃x_{n}∥\overline{X}^{T} = ∥Φ(x)∥\overline{(\overline{X}^{T})}S̃.

It remains to be shown that Φ is onto. Let z ∈ \overline{\overline{X}^{T}} S̃. Then z = S̃ − lim y_{n} for some y_{n} ∈ \overline{X}^{T}. Thus ∥S̃y_{n}−S̃y_{m}∥\overline{X}^{T} → 0 and therefore ∥T̃S̃y_{n}−T̃S̃y_{m}∥ → 0. For each y_{n} choose a sequence (x_{n,k})_{k∈IN} ⊂ X, such that y_{n} = T − lim_{k→∞} x_{n,k} and ∥T̃y_{n}−T x_{n,k}∥ ≤ \frac{1}{k} for all n, k. Thus ∥T̃y_{n}−T x_{n,n}∥ ≤ \frac{1}{n} and thus ∥T̃y_{n}−T x_{n,n}∥ → 0. Combining this with the fact that ∥T̃S̃y_{n} − T̃S̃y_{m}∥ → 0 yields

\begin{matrix}∥ST x_{n,n} & − ST x_{m,m}∥ \\ ≤ & ∥ST x_{n,n} − ST̃y_{n}∥ + ∥ST̃y_{n} − ST̃y_{m}∥ + ∥ST̃y_{m} − ST x_{m,m}∥ \\ ≤ & ∥S∥ ∥T x_{n,n} − T̃y_{n}∥ + ∥T̃S̃y_{n} − T̃S̃y_{m}∥ + ∥S∥ ∥T x_{m,m} − T̃y_{m}∥ → 0. \end{matrix}

Let x = ST − lim x_{n,n}. Then Φ(x) = S̃ − lim x_{n,n} in \overline{X}^{T}. Since

∥S̃x_{n,n} − S̃y_{n}∥\overline{X}^{T} = ∥T̃S̃x_{n,n} − T̃S̃y_{n}∥ ≤ ∥S̃∥ ∥T x_{n,n} − T̃y_{n}∥ → 0

we obtain that z = Φ(x). Hence Φ is onto.

(c) Suppose \overline{X}^{T} ⊂ \overline{X}^{V}. Let S := T̃^{−1}V. Then clearly S ∈ L(\overline{X}^{T}), S is one-to-one and T̃S = ST̃ = V. Since ∥x∥\overline{\overline{X}^{T}} ^{S} = ∥T̃Sx∥ = ∥V x∥ = ∥x∥_{V} for all x ∈ \overline{X}^{T} we obtain that

\overline{\overline{X}^{T}}^{S} = \overline{(\overline{X}^{T} , ∥.∥_{T̃S})} = \overline{(\overline{X}^{T} , ∥.∥_{V})} = \overline{X}^{V} .

Suppose \overline{X}^{V} = \overline{\overline{X}^{T}}^{S}. Then obviously \overline{X}^{T} ⊂ \overline{X}^{V}.

(d) By definition, S̃ is compact if and only if for every \overline{X}^{T} -bounded sequence (z_{n})_{n∈IN} in \overline{X}^{T} there exists a subsequence (z_{n_{i}})_{i∈IN} such that S̃z_{n_{i}} converges in \overline{X}^{T} or, equivalently, T̃S̃z_{n_{i}} converges in X.

Assume S is compact. Let (z_{n})_{n∈IN} be a \overline{X}^{T} -bounded sequence; i.e., ∥T̃z_{n}∥ is bounded. Therefore T̃S̃z_{n} = S̃T̃z_{n} = ST̃z_{n}. Since S is compact and T̃z_{n} is bounded, there exists a subsequence (z_{n_{i}})_{i∈IN} such that ST̃z_{n_{i}} = T̃S̃z_{n_{i}} converges. Thus S̃ is compact.

Suppose S̃ is compact and the image of T is dense in X. Let (x_{n})_{n∈IN} be a bounded sequence in X. Then there exists a sequence (y_{n})_{n∈IN} ⊂ X such that ∥T y_{n} − x_{n}∥ ≤ \frac{1}{n}. Hence y_{n} is \overline{X}^{T} -bounded and therefore S̃y_{n} = Sy_{n} has a \overline{X}^{T} convergent subsequence, i.e., there exists a subsequence (y_{n_{i}})_{i∈IN}, such that T̃Sy_{n_{i}} = ST y_{n_{i}} converges. Thus (Sx_{n_{i}})_{i∈IN} is convergent, and therefore S is compact. ♢ In case that S = T, the last theorem leads to the following observation. Corollary 2.3. Let X be a Banach space. Let T ∈ L(X) be one-to-one, and let X_{0} := \overline{Im(T)}. The isomorphism T̃ : \overline{X}^{T} → X_{0} also defines a bounded, linear operator from \overline{X}^{T} into \overline{X}^{T}. As such, it is one-to-one and ∥T̃∥_{L(\overline{X}^{T})} ≤ ∥T ∥_{L(X)}. Proof. The statement follows immediately from the fact that X_{0} ↪ \overline{X}^{T}. ♢ Using T̃ of the above corollary, we can extend \overline{X}^{T} to \overline{\overline{X}^{T}} T̃. Iterating this procedure leads to the following tower, where, by Theorem 2.2 (b), \overline{\overline{X}^{T}} T̃ = \overline{X}^{T^{2}}. Corollary 2.4 (Tower of linear extensions of X). Let X be a Banach space, and let T ∈ L(X) be one-to-one. Then

⋮  ⋮
↑ ∪  ∪

\overline{X}^{T^{2}} = \overline{\overline{X}^{T}} T̃ ˜T̃ ⟶

isometric isomorphism

\overline{Im(T̃)}T̃ ˆ=Im(˜T̃)

↑ ∪  ∪
\overline{X}^{T} T̃ ⟶

isometric isomorphism

\overline{Im(T)} ˆ=Im(T̃)
↑ ∪  ∪

X T ⟶

isometric isomorphism

(Im(T), ∥ · ∥_{T^{−1}}).

Suppose that the image of T is dense in X; i.e., \overline{Im(T)} = X. Since \overline{Im(T̃)}T̃ = \overline{(\overline{Im(T)})}T̃ = \overline{X}^{T}, the image of T̃ is dense in \overline{X}^{T}. Denoting X_{n} := \overline{X}^{T^{n}} for n ≥ 0 and X_{−n} := Im(T^{n}) equipped with ∥f∥_{−n} := ∥T^{−n}f∥, we obtain the following Sobolev-tower (See also [DaP-Gr], [Na1-2], [Wa]). We identify T with its natural extension or restriction on X_{n}.^{(3)}

(3) In the case that T is the antiderivative operator discussed in Example 1.2, we would take X = C_{0}[0, 1]. Since \overline{C_{0}[0, 1]}^{T} = \overline{C[0, 1]}^{T}, the following corollary coincides with the tower-diagram following Proposition 1.5.

Corollary 2.5 (Abstract Sobolev tower). Let X be a Banach space and let T ∈ L(X) be one-to-one with \overline{Im(T)} = X. Then

⋮  ⋮
↑ ∪  ∪

X_{n} T ⟶

isometric isomorphism

X_{n−1}
⋮ ⋮
↑ ∪  ∪

X_{1} T ⟶

isometric isomorphism

X_{0}
↑ ∪  ∪

X_{0} T ⟶

isometric isomorphism

X_{−1}
↑ ∪  ∪

X_{−1} T ⟶

isometric isomorphism

X_{−2},
⋮  ⋮

In particular, T^{j} is an isometric isomorphism between X_{i} and X_{i+j} for all i, j ∈ ZZ. If T is compact, so are the embeddings.

In Section II.4 we will apply the results of this section to convolution operators T_{k} : f → k ⋆ f acting on generalized function spaces X_{n} = C_{0}^{−n}([0, a]; X). Before doing so, we will study in the following section some properties of the convolution operator on spaces C([0, a]; X).

II.3 The Convolution on C([0,a];X)

The convolution transform T_{K} : f ↦ K ⋆ f, where (K ⋆ f)(t) = ∫_{0}^{t}K(t − s)f(s) ds, holds a central place in modern analysis.^{(1)} In 1925 E. C. Titchmarsh ([Ti], Theorem VII) showed that the convolution T_{k} is one-to-one on L^{1}[0, a] as long as 0 ∈ supp(k) and k ∈ L^{1}[0, a]. Clearly this implies the injectivity of T_{k} as an operator on C[0, 1]. C. Foias showed in 1961 ([Fo]) that with the same condition, the image of the convolution transform is dense in L^{1}[0, a]; K. Skornik [Sk] extended the result to C_{0}[0, 1]. Thus, the convolution operator T_{k} satisfies the conditions of the previous section; i.e., T_{k} is one-to-one and the range of T_{k} is dense in C_{0}[0, 1]. Therefore, we have the following diagram.

\overline{C_{0}[0, 1]}^{T_{k}} T̃_{k} ⟶

isom. isomorph

C_{0}[0, 1] = \overline{Im(T_{k})}
↑ ∪  ∪
C[0, 1]  \underrightarrow{T_{k}}  Im(T_{k}),

where the extension T̃_{k} of the convolution operator is an isometric isomorphism between the generalized function space \overline{C_{0}[0, 1]}^{T_{k}} and C_{0}[0, 1].

(1) W. Kecs noted in the preface of his book “The Convolution Product” ([Ke]) that “ The extension of the convolution product in the distribution space created a natural framework for the extension and enrichment of its properties, and it is due to this fact that the convolution product has become a powerful mathematical tool in symbolic calculus, distribution approximation, Fourier series, and the solution of boundary-value problems. The high effectiveness of the convolution product is especially reflected in its properties with respect to the Fourier and Laplace transforms [...]

We will extend Titchmarsh’s injectivity theorem and Foias’ dense range theorem to convolution operators T_{K} : C([0, a]; X) → C([0, a]; Y), defined by

T_{K}f : t ↦ ∫_{0}^{t}K(t − s)f(s) ds,

where K(t) ∈ L(X, Y) for all t ∈ [0, ∞), t ↦ K(t) is strongly continuous on [0, ∞), 0 ∈ supp(K(·)x) for all x ∈ X,^{(2)} and satisfies a technical condition. These conditions are automatically satisfied for scalar-valued k and uniformly continuous semigroups (K(t))_{t≥0} (see Example 3.11). The technical condition can then be weakened considerably, and we will show that Titchmarsh’s injectivity theorem and Foias’ dense range theorem also hold for strongly continuous semigroups, compositions of strongly continuous operator families that have the injectivity and dense range properties, as well as the composition of such strongly continuous operator families with bounded injective linear operators (see Theorem 3.15).

Contrary to the proofs of Foias’ theorem by C. Foias [Fo], K. Skornik [Sk] and J. Mikusinski [Mi2], our proof is constructive and yields an approximating sequence of continuous functions.

The proofs given here are based on a generalization of the Phragmén-Doetsch inversion formula for the Laplace transform which states that if

f̂(λ) = ∫_{0}^{∞} e^{−λt}f(t) dt = lim_{n} → ∞ ∫_{0}^{n} e^{−}λtf(t) dt

for some exponentially bounded f ∈ L_{loc}^{1}([0, ∞); X), then

∫_{0}^{t}f(s) ds = lim_{k→∞} ∑_{n=1}^{∞}(−1)^{n+1} \frac{1}{n!}e^{tkn}f̂(kn),  (1)

(2) If 0 ∉ suppK(·)x; i.e., K(t)x = 0 for 0 ≤ t ≤ ϵ for some ϵ > 0, then for all f ∈ C[0, a] with f(t) = 0 for 0 ≤ t ≤ a − ϵ, K ⋆ f_{x} = 0, where f_{x}(t) := f(t)x where the limit is uniform for all t ≥ 0 (see [Ba-Ne1], Thm.1.7). The proof of the following extension of this inversion formula was inspired by J. Mikusinski’s proof of a uniqueness theorem for the Laplace transform (see [Mi2], Chapter VII). We obtain Mikusinski’s uniqueness result as a corollary to the inversion procedure to be described below (see Theorem 3.6). The extraordinary aspect of this new inversion formula for the Laplace transform is that it does not require taking infinite sums like in the Phragmén-Doetsch inversion (1).

We say that a sequence (β_{n}) satisfies the Müntz condition (M) if there exists δ > 0, such that for all n ∈ IN,

β_{n} > 0, β_{n+1} − β_{n} > δ > 0, and ∑_{n=1}^{∞} \frac{1}{β_{n}} = ∞.  (M)

Theorem 3.1 (Phragmén-Mikusinski inversion). Let f ∈ L^{1}([0, T]; X) for some T > 0, and q(λ) := ∫_{0}^{T} e^{−λt}f(t) dt.^{(3)} Furthermore, let (β_{n})_{n∈IN} be a sequence satisfying the Müntz condition (M) and N_{k} ∈ IN be such that ∑_{j=1}^{N_{k}} \frac{1}{β_{kj}} ≥ T. Define

α_{k,n} := \frac{1}{e} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N_{k}} \frac{β_{kj}}{β_{kj} − β_{kn}} e^{− \frac{β_{kn}}{β_{kj}}} .

Then |α_{k,n}| ≤ e^{2β_{kn}/kδ} and

∫_{0}^{t}f(s) ds = lim_{k→∞} ∑_{n=1}^{N_{k}} α_{k,n}e^{β_{kn}t}q(β_{kn}),

where the limit is uniform for t ∈ [0, T].

(3) We will extend the result later on to the case that T = ∞; i.e., we will consider in Corollary 3.3 exponentially bounded f ∈ L_{loc}^{1}([0, ∞); X) and q(λ) = ∫_{0}^{∞} e^{−λt}f(t) dt.

Proof. Let N_{k} be such that c_{k} := ∑_{j=1}^{N_{k}} \frac{1}{β_{kj}} ≥ T. Such N_{k} exists since the fact that \frac{k}{β_{kj}} > ∑_{i=1}^{k} \frac{1}{β_{kj+i}} implies that

k ∑_{j=1}^{∞} \frac{1}{β_{kj}} ≥ ∑_{j=1}^{∞} ∑_{i=1}^{k} \frac{1}{β_{kj+i}} = ∑_{j=1}^{∞} \frac{1}{β_{j}} − ∑_{j=1}^{k} \frac{1}{β_{j}} = ∞.

Thus, ∑_{j=1}^{∞} \frac{1}{β_{kj}} = ∞.

The proof of this theorem is built on the fact that the sequence of functions ϕ_{k} : IR → IR^{+} with

ϕ_{k}(t) := \{ \begin{matrix}1 ⋆ β_{k}e^{−β_{k}(·)} ⋆ β_{2k}e^{−β_{2k}(·)} ⋆ … ⋆ β_{N_{k}k}e^{−β_{N_{k}k}(·)}(t + c_{k}) & for t ≥ −c_{k} \\ 0 & else \end{matrix}

converges pointwise towards the Heaviside function H(t) = χ_{(0,∞)} (for all t ≠ 0). We will show first that ϕ_{k}(t) = 1 − ∑_{n=1}^{N_{k}} α_{k,n}e^{−β_{kn}t} for t ≥ −c_{k}. Consider ψ_{k} := 1 ⋆ β_{k}e^{−β_{k}(·)} ⋆ … ⋆ β_{N_{k}k}e^{−β_{N_{k}k}(·)}. Then

\begin{matrix}∫_{0}^{∞} e^{−λt}ψ_{k}(t) dt = & \frac{1}{λ} \frac{β_{k}}{λ + β_{k}} ⋯ \frac{β_{N_{k}k}}{λ + β_{N_{k}k}} \\ = & γ_{k,0} \frac{1}{λ} + γ_{k,1} \frac{1}{λ + β_{k}} + … + γ_{k,N_{k}} \frac{1}{λ + β_{N_{k}k}} \end{matrix}

for some coefficients γ_{k,n}. Multiplying by λ and putting λ = 0, we obtain that γ_{k,0} = 1. Similarly, if we multiply with λ + β_{nk} and put λ = −β_{nk}, we obtain that

γ_{k,n} = − ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N_{k}} \frac{β_{jk}}{β_{jk} − β_{nk}} .

Since the inverse Laplace transform of \frac{1}{λ+β_{nk}} is e^{−β_{nk}t}, we obtain that ψ_{k}(t) = 1 + ∑_{n=1}^{N_{k}} γ_{k,n}e^{−β_{nk}t} for t ≥ 0. Therefore

ϕ_{k}(t) = ψ_{k}(t + c_{k}) = 1 + ∑_{n=1}^{N_{k}} γ_{k,n}e^{−β_{nk}(t+c_{k})}

for all t ≥ −c_{k}. Let

\begin{matrix}α_{k,n} := −γ_{k,n}e^{−β_{nk}c_{k}} = & e^{−β_{nk} ∑_{j=1}^{N_{k}} \frac{1}{β_{jk}}} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N_{k}} \frac{β_{jk}}{β_{jk} − β_{nk}} \\ = & \frac{1}{e} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N_{k}} \frac{β_{jk}}{β_{jk} − β_{nk}} e^{− \frac{β_{nk}}{β_{jk}}} . \end{matrix}

Then ϕ_{k}(t) = 1 − ∑_{n=1}^{N_{k}} α_{k,n}e^{−β_{nk}t} for all t ≥ −c_{k}.

Next we show that |α_{k,n}| ≤ e^{\frac{1+ln 2}{kδ} β_{kn}}. We have that

ln |α_{k,n}| = −β_{nk} ∑_{j=1}^{N_{k}} \frac{1}{β_{jk}} + ∑_{j=1}^{n−1} ln \frac{β_{jk}}{β_{nk} − β_{jk}} + ∑_{j=n+1}^{N_{k}} ln \frac{β_{jk}}{β_{jk} − β_{nk}} =: S_{1} + S_{2} + S_{3}.

We first look at S_{2}. Since β_{jk} < β_{nk} − kδ(n − j) for j < n, and since the function t ↦ \frac{t}{β_{nk}−t} is increasing on (0, β_{nk}), we know that \frac{β_{jk}}{β_{nk}−β_{jk}} < \frac{β_{nk}−kδ(n−j)}{β_{nk}−(β_{nk}−kδ(n−j))} = \frac{β_{nk}−kδ(n−j)}{kδ(n−j)} and thus

S_{2} < ∑_{j=1}^{n−1} ln \frac{β_{nk} − kδ(n − j)}{kδ(n − j)} = ∑_{j=1}^{n−1} ln \frac{β_{nk} − kδj}{kδj} .

The fact that the function t ↦ \frac{β_{nk}−kδt}{kδt} is decreasing for t > 0 yields

S_{2} < ∫_{0}^{n−1} ln \frac{β_{nk} − kδt}{kδt}dt = \frac{β_{nk}}{kδ} ∫_{0}^{kδ(n−1)/β_{kn}} ln (\frac{1}{t} − 1) dt.

Now, ln(1/t − 1) > 0 if t ∈ (0, 1/2) and ln(1/t − 1) < 0, if t ∈ (1/2, 1). Thus

\begin{aligned}S_{2} & < \frac{β_{nk}}{kδ} ∫_{0}^{1/2} ln (\frac{1}{t} − 1) dt = −\frac{β_{nk}}{kδ} ∫_{0}^{1/2} t\frac{t}{1 − t} · \frac{(−t) − (1 − t)}{t^{2}} dt \\ & = \frac{β_{nk}}{kδ} ∫_{0}^{1/2} \frac{1}{1 − t} dt = \frac{β_{nk} ln 2}{kδ}. \end{aligned}

In a similar fashion we find an estimate for S_{1} + S_{3}. The function t ↦ −\frac{β_{nk}}{t} + ln \frac{t}{t−β_{nk}} is positive and decreasing on (β_{nk}, ∞), because its derivative

\frac{β_{nk}}{t^{2}} + \frac{t − β_{nk}}{t} · \frac{t − β_{nk} − t}{(t − β_{nk})^{2}} = \frac{β_{nk}}{t^{2}} − \frac{β_{nk}}{t(t − β_{nk})}

is negative and the limit as t → ∞ is equal to 0. Since β_{nk} + kδ(j − n) < β_{jk} we obtain that

S_{1} + S_{3} = − ∑_{j=1}^{N_{k}} \frac{β_{nk}}{β_{jk}} + ∑_{j=n+1}^{N_{k}} ln \frac{β_{jk}}{β_{jk} − β_{nk}} < ∑_{j=n+1}^{∞} −\frac{β_{nk}}{β_{jk}} + ln \frac{β_{jk}}{β_{jk} − β_{nk}}
< ∑_{j=n+1}^{∞} (−\frac{β_{nk}}{β_{nk} + kδ(j − n)} + ln \frac{β_{nk} + kδ(j − n)}{kδ(j − n)})
= ∑_{j=1}^{∞} (−\frac{β_{nk}}{β_{nk} + kδj} + ln \frac{β_{nk} + kδj}{kδj} )
< ∫_{0}^{∞} (−\frac{β_{nk}}{β_{nk} + kδt} + ln \frac{β_{nk} + kδt}{kδt}) dt
=\frac{β_{nk}}{kδ} ∫_{0}^{∞} (−\frac{1}{1 + t} + ln \frac{1 + t}{t}) dt
= \frac{β_{nk}}{kδ} (−\frac{t}{1 + t} + t ln \frac{1 + t}{t})|_{t=0}^{t=∞}

− \frac{β_{nk}}{kδ} ∫_{0}^{∞}t (\frac{1}{(1 + t)^{2}} + \frac{t}{1 + t} · \frac{t − (1 + t)}{t^{2}}) dt

=\frac{β_{nk}}{kδ} (−1 + 1) + \frac{β_{nk}}{kδ} ∫_{0}^{∞} \frac{1}{(1 + t)^{2}} dt = \frac{β_{nk}}{kδ} .

Hence ln |α_{k,n}| < \frac{β_{nk}(1+ln 2)}{kδ} and thus

|α_{k,n}| < e^{\frac{β_{nk}(1+ln 2)}{kδ}} < e^{\frac{2β_{kn}}{kδ}} .

Next, we show that ϕ_{k}(t) → 1 for all t > 0. This can be seen via the following inequality. Let t > 0, and let k be such that \frac{2}{kδ} < t/2. Then

\begin{matrix}|ϕ_{k}(t) − 1| ≤ & ∑_{n=1}^{N_{k}} |α_{k,n}|e^{−β_{kn}t} ≤ ∑_{n=1}^{∞} e^{\frac{2β_{kn}}{kδ}} e^{−β_{kn}t} \\ ≤ & ∑_{n=1}^{∞} e^{−β_{kn}t/2} ≤ ∑_{n=1}^{∞} e^{−(kn−1)δt/2} \\ = & \frac{e^{−(k−1)δt/2}}{1 − e^{−kδt/2}} , \end{matrix}

since (1 − e^{−kδt/2}) ∑_{n=1}^{∞} e^{−(kn−1)δt/2} = e^{−(k−1)δt/2}. Thus ϕ_{k}(t) → 1 as k → ∞, uniformly for t > ϵ > 0.

Notice from the definition of ϕ_{k}, that ϕ_{k} as convolution of positive functions is positive. Since

ϕ′_{k}(t) := \{ \begin{matrix}β_{k}e^{−β_{k}(·)} ⋆ β_{2k}e^{−β_{2k}(·)} ⋆ … ⋆ β_{N_{k}k}e^{−β_{N_{k}k}(·)}(t + c_{k}) & for t ≥ −c_{k} \\ 0 & else \end{matrix}

is also positive, we know that ϕ_{k} is monotonically increasing. Thus

∫_{0}^{∞} e^{−t}ϕ_{k}(t) dt → 1.

Next, we show that ∫_{−∞}^{∞} e^{−t}ϕ_{k}(t) dt → 1, which implies – again by the positivity and monotonicity of ϕ_{k} – that ϕ_{k}(t) → 0 for all t < 0 and thus uniformly for all t < −ϵ < 0.

Let k be such that β_{kn} ≥ (kn − 1)δ > 1. Since

\frac{β_{kn} + 1}{β_{kn}} = 1 + \frac{1}{β_{kn}} < e^{1/β_{kn}} < \frac{1}{1 − 1/β_{kn}} = \frac{β_{kn}}{β_{kn} − 1},

we know that

1 < \frac{β_{kn}}{β_{kn} + 1}e^{1/β_{kn}} < \frac{β_{kn}^{2}}{β_{kn}^{2} − 1} < 1 + \frac{1}{(kn − 1)^{2}δ^{2} − 1} < e^{1/((kn−1)^{2}δ^{2}−1)}.

Since by the definition of ϕ_{k},

\begin{matrix}∫_{−∞}^{∞} e^{−t}ϕ_{k}(t) dt = & ∫_{−c_{k}}^{∞} e^{−t}ϕ_{k}(t) dt = e^{c_{k}} ∫_{0}^{∞} e^{−t}ϕ_{k}(t − c_{k}) dt \\ = & e^{∑{N_{k} \atop n=1} 1/β_{kn}} ∏_{n=1}^{N_{k}} \frac{β_{kn}}{1 + β_{kn}} = ∏_{n=1}^{N_{k}} \frac{β_{kn}}{1 + β_{kn}} e^{1/β_{kn}}, \end{matrix}

we can conclude that

\begin{matrix}1 ≤ ∫_{−∞}^{∞} e^{−t}ϕ_{k}(t) dt = & ∏_{n=1}^{N_{k}} \frac{β_{kn}}{1 + β_{kn}} e^{1/β_{kn}} ≤ ∏_{n=1}^{∞} e^{1/((kn−1)^{2}δ^{2}−1)} \\ = & e^{∑{∞ \atop n=1} \frac{1}{(kn−1)^{2}δ^{2}−1}} → e^{0} = 1. \end{matrix}

Thus ϕ_{k}(t) → 0 for t < 0, uniformly for t < −ϵ < 0.

Finally, let ϵ > 0 and k_{0} be such that ϕ_{k}(−ϵ) + 1 − ϕ_{k}(ϵ) < ϵ for all k > k_{0}. Then, for t ∈ [0, T],

\begin{matrix}∥∫_{0}^{t} & f(s) ds − ∑_{n=1}^{N_{k}} α_{k,n}e^{β_{kn}t}r(β_{kn})∥ \\ = & ∥∫_{0}^{t}f(s) ds − ∫_{0}^{T} ∑_{n=1}^{N_{k}} α_{k,n}e^{β_{kn}t}e^{−λβ_{kn}}f(λ) dλ∥ \\ = & ∥∫_{0}^{t}f(s) ds − ∫_{0}^{T} (1 − ϕ_{k}(λ − t))f(λ) dλ∥ \\ = & ∥∫_{0}^{t}f(s) − (1 − ϕ_{k}(s − t))f(s) ds − ∫_{t}^{T} (1 − ϕ_{k}(s − t))f(s) ds∥ \\ ≤ & ∥ ∫_{0}^{t−ϵ} ϕ_{k}(s − t)f(s) ds∥ + ∫_{t−ϵ}^{t+ϵ} ∥f(s)∥ ds + ∥ ∫_{t+ϵ}^{T}(1 − ϕ_{k}(s − t))f(s) ds∥ \\ ≤ & ∥f∥_{1}ϕ_{k}(−ϵ) + 2ϵ∥f∥_{1} + ∥f∥_{1}(1 − ϕ_{k}(ϵ)) \\ ≤ & 3ϵ∥f∥_{1}. \end{matrix}

Thus ∑_{n=1}^{N_{k}} α_{k,n}e^{β_{kn}t}r(β_{kn}) converges uniformly to ∫_{0}^{t}f(s) ds. ♢ We show next that the Phragmén-Mikusinski inversion does not register perturbations of exponential decay T; i.e., if for t ∈ [0, T),

∫_{0}^{t}f(s) ds = lim_{k→∞} ∑_{n=1}^{N_{k}} α_{k,n}e^{β_{nk}t}q(β_{nk})

for some function q, then

∫_{0}^{t}f(s) ds = lim_{k→∞} ∑_{n=1}^{N_{k}} α_{k,n}e^{β_{nk}t}q̃(β_{nk})

for all perturbed functions q̃ = q + r, where r is some perturbation of exponential decay T. Corollary 3.2. Let r : (ω, ∞) → X be a function, and let (β_{n})_{n∈IN} ⊂ IR^{+} be a sequence of positive numbers satisfying the Müntz condition (M). Suppose r is of exponential decay T > 0; i.e.,

lim sup_{n→∞} \frac{1}{β_{n}} ln ∥r(β_{n})∥ ≤ −T.

Then

lim_{k→∞} ∑_{n=1}^{N_{k}} α_{k,n}e^{β_{nk}t}r(β_{nk}) = 0

for all 0 ≤ t < T where α_{k,n} and N_{k} are as in the previous theorem, and the limit is uniform for all t ∈ [0, S] and 0 < S < T. Proof. Let t ∈ [0, T). Then −T < −\frac{2T +t}{3}. Thus there exists k_{0} such that ∥r(β_{n})∥ ≤ e^{− \frac{2T +t}{3} β_{n}} for all n ≥ k_{0} and such that 2/k_{0}δ < (T − t)/3.

By the previous theorem we know that |α_{k,n}| < e^{\frac{2}{kδ} β_{kn}} < e^{\frac{T −t}{3} β_{nk}}. Thus

\begin{matrix}∥ ∑_{n=1}^{N_{k}} α_{k,n}e^{β_{nk}t}r(β_{nk})∥ ≤ & ∑_{n=1}^{∞} e^{\frac{T +2t}{3} β_{nk}}e^{− \frac{2T +t}{3} β_{nk}} \\ = & ∑_{n=1}^{∞} e^{− \frac{T −t}{3} β_{nk}} ≤ ∑_{n=1}^{∞} e^{− \frac{T −t}{3} (β_{nk}−β_{1})} \\ ≤ & ∑_{n=1}^{∞} e^{− \frac{T −t}{3} (nk−1)δ} = \frac{e^{− \frac{T −t}{3} (k−1)δ}}{1 − e^{− \frac{T −t}{3} kδ}} → 0 \end{matrix}

as k → ∞, uniformly for all t ∈ [0, S] for all 0 < S < T. ♢ In the following two corollaries we will reformulate Theorem 3.1 in terms of Laplace transforms. The resulting “Phragmén-Mikusinski” inversion formulas for the Laplace transform seem to be new. Corollary 3.3. Let f ∈ L_{loc}^{1}([0, ∞); X) be exponentially bounded and f̂(λ) := ∫_{0}^{∞} e^{−λt}f(t) dt for sufficiently large Re(λ). Let (β_{n})_{n∈IN} be a sequence satisfying the Müntz condition (M). Let 0 < T; let N_{k} ∈ IN be such that ∑_{n=1}^{N_{k}} \frac{1}{β_{nk}} ≥ T, and let α_{k,n} be as in Theorem 3.1. Then, for all t ∈ [0, T),

∫_{0}^{t}f(s) ds = lim_{k→∞} ∑_{n=1}^{N_{k}} α_{k,n}e^{tβ_{kn}} f̂(β_{kn}),

where the limit is uniform for t ∈ [0, S] and all 0 < S < T.^{(4)} Proof. Let q(λ) := ∫_{0}^{T} e^{−λt}f(t) dt, and r(λ) := ∫_{T}^{∞} e^{−λt}f(t) dt. Then f̂ = q + r and

∑_{n=1}^{N_{k}} α_{k,n}e^{tβ_{kn}} f̂(β_{kn}) = ∑_{n=1}^{N_{k}} α_{k,n}e^{tβ_{kn}}q(β_{kn}) + ∑_{n=1}^{N_{k}} α_{k,n}e^{tβ_{kn}}r(β_{kn}).

By Theorem 3.1, the first term converges uniformly on [0, T] to ∫_{0}^{t}f(s) ds. Since ∥f(t)∥ ≤ Me^{ωt} for almost all t ≥ 0 and some positive constants M and ω, it follows that

∥r(λ)∥ ≤ M ∫_{T}^{∞} e^{−λt}e^{ωt} dt ≤ \frac{M}{λ − ω} e^{ωT} e^{−λT}

for all λ > ω. Thus r is of exponential decay T and therefore, by Corollary 3.2, the second term converges uniformly to 0 for all t ∈ [0, S] and 0 < S < T. ♢ Corollary 3.4. Let f ∈ L_{loc}^{1}([0, ∞); X) be exponentially bounded and f̂(λ) = ∫_{0}^{∞} e^{−λt}f(t) dt for sufficiently large Re(λ). Let (β_{n})_{n∈IN} be a sequence satisfying the Müntz condition (M). Let N_{k} be such that ∑_{n=1}^{N_{k}} \frac{1}{β_{nk}} → ∞, as k → ∞, and let α_{k,n} be as in Theorem 3.1. Then for all t ≥ 0,

∫_{0}^{t}f(s) ds = lim_{k→∞} ∑_{n=1}^{N_{k}} α_{k,n}e^{tβ_{kn}} f̂(β_{kn}),

where the limit is uniform on compact sets.

(4) It follows from Corollary 3.2 that one can replace f̂ by any function q, as long as the difference f̂− q is of exponential decay T.

Proof. Since ∑_{n=1}^{∞} \frac{1}{β_{kn}} = ∞ for all k ∈ IN (see the proof of Theorem 3.1), there exists a sequence (N_{k})_{k∈IN} ⊂ IN such that ∑_{n=1}^{N_{k}} \frac{1}{β_{nk}} → ∞. Thus, for all T > 0 there exists k_{0} such that ∑_{n=1}^{N_{k}} \frac{1}{β_{nk}} > T for all k > k_{0}. Hence, by the previous corollary,

∫_{0}^{t}f(s) ds = lim_{k→∞} ∑_{n=1}^{N_{k}} α_{k,n}e^{tβ_{kn}} f̂(β_{kn}),

where the limit is uniform in [0, S] for all 0 < S < T and all T > 0. ♢

If one takes β_{n} = n, then the coefficients α_{k,n} defined in Theorem 3.1 reduce to

\begin{matrix}α_{k,n} = & \frac{1}{e} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N_{k}} \frac{β_{kj}}{β_{kj} − β_{kn}} e^{− \frac{β_{kn}}{β_{kj}}} = e^{− \frac{n}{n}} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N_{k}} \frac{j}{j − n}e^{\frac{−n}{j}} \\ = & \frac{N_{k}!}{n} ∏_{j=1}^{n−1} \frac{1}{j − n} ∏_{j=n+1}^{N_{k}} \frac{1}{j − n}e^{−n ∑_{j=1}^{N_{k}} \frac{1}{j}} = (−1)^{n−1} ( \begin{matrix}N_{k} \\ n \end{matrix} ) e^{−n ∑_{j=1}^{N_{k}} \frac{1}{j}} . \end{matrix}

Thus the previous two corollaries yield the following variant of the Phragmén-Doetsch inversion formula.

Corollary 3.5. Let f ∈ L_{loc}^{1}([0, ∞); X) be exponentially bounded and f̂(λ) = ∫_{0}^{∞} e^{−λt}f(t) dt. Let N_{k} ∈ IN be such that \frac{1}{k} ∑_{n=1}^{N_{k}} \frac{1}{n} ≥ T for all k ∈ IN. Then for all t ∈ [0, T],

∫_{0}^{t}f(s) ds = lim_{k→∞} ∑_{n=1}^{N_{k}}(−1)^{n+1} ( \begin{matrix}N_{k} \\ n \end{matrix} ) e^{−n ∑_{j=1}^{N_{k}} \frac{1}{j}} e^{tkn}q(kn),

where the limit is uniform on [0, S] for all 0 < S < T. If the sequence (N_{k})_{n∈IN} is such that \frac{1}{k} ∑_{n=1}^{N_{k}} \frac{1}{n} → ∞, then the limit exists for all t ≥ 0 and is uniform on compact subsets of IR.

Another consequence of the Phragmén-Mikusinski inversion formula is the following statement characterizing the maximal interval [0, T] on which an exponentially bounded L_{loc}^{1}-function f vanishes in terms of the growth of its Laplace transform λ ↦ f̂(λ) at infinity. This Abelian type theorem will be crucial in the proof of Titchmarsh’s theorem. Theorem 3.6. Let 0 ≤ T and f̂(λ) := ∫_{0}^{∞} e^{−λt}f(t) dt for some exponentially bounded^{(5)} f ∈ L_{loc}^{1}([0, ∞); X). Then the following are equivalent.^{(6)}

(i) Every sequence (β_{n})_{n∈IN} satisfying the Müntz condition (M), satisfies

lim sup_{n→∞} \frac{1}{β_{n}} ln ∥f̂(β_{n})∥ = −T.

(ii) For every sequence (β_{n})_{n∈IN} satisfying the Müntz condition (M), there exists a subsequence (β_{n_{k}})_{k∈IN} satisfying the Müntz condition and

lim_{k→∞} \frac{1}{β_{n_{k}}} ln ∥f̂(β_{n_{k}})∥ = −T.

(iii) There exists a sequence (β_{n})_{n∈IN} satisfying the Müntz condition (M) and

lim sup_{n→∞} \frac{1}{β_{n}} ln ∥f̂(β_{n})∥ = −T.

(iv) f(t) = 0 almost everywhere on [0, T] and T ∈ supp(f).

(v) lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ = −T.

(5) In case that f is not exponentially bounded, the theorem still holds if we replace f with the truncated function f̃(t) := f(t) · χ_{[0,S]} for some S > T.

(6) G. Doetsch ([Do 1], Satz 14.3.1) proved that lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ ≤ −T is equivalent to the statement (iv)′: f = 0 on [0, T] a.e. In fact, it follows from the proof below that statement (iv)′ is equivalent to the statements (i) − (iii) if lim sup_{\frac{1}{βn}} ln ∥f̂(β_{n})∥ = −T is replaced by lim sup_{\frac{1}{βn}} ln ∥f̂(β_{n})∥ ≤ −T.

Proof. We show that first that (i) => (ii) => (iii) => (iv) => (i) and then that (iv) <=> (v). Suppose (i) holds. Let (β_{n})_{n∈IN} be any such sequence. Let (β_{n}^{ϵ})_{n∈IN} be the subsequence that is obtained by dropping the elements of (β_{n})_{n∈IN} for which ∥r(β_{n})∥ ≤ e^{(−T −ϵ)β_{n}}. The dropped subsequence \binom{}{β̃_{n}^{ϵ}} satisfies

lim sup_{n→∞} \frac{1}{β̃_{n}^{ϵ}} ln ∥f̂(β̃_{n}^{ϵ})∥ ≤ −T − ϵ.

Since (i) is assumed to hold, the dropped sequence \binom{}{β̃_{n}^{ϵ}} can not satisfy the Müntz condition. Thus the sum of the reciprocals of the dropped elements is finite, and the sum of the reciprocals of the remaining terms β_{n}^{ϵ} is infinite; therefore, (β_{n}^{ϵ})_{n∈IN} still satisfies the Müntz condition. Now we use a diagonal argument. Let j = 1 and take the first k_{1} elements of (β_{n}^{1})_{n∈IN} such that ∑_{i=1}^{k_{1}} \frac{1}{β_{i}^{1}} > 1. Continue with elements of the sequence (β_{n}^{1/2}), picking consecutive elements until ∑_{j=1}^{k_{1}} \frac{1}{β_{j}^{1}} + ∑_{j=k_{1}}^{k_{2}} _{\frac{1}{β_{j}^{1/2}}} ≥ 2. Continuing this process we clearly end up with a subsequence having the properties stated in (ii).

Clearly (ii) implies (iii). Suppose (iii) holds. In the case that T > 0, combining Corollary 3.3 with Corollary 3.2 we obtain that

∫_{0}^{t} f(s) ds = 0

for all t ∈ [0, T). Hence f = 0 a.e. on [0, T]. Thus (iii) implies that f = 0 a.e. on [0, T] for all T ≥ 0. Now let T ≥ 0 and suppose that f = 0 almost everywhere on [0, T + ϵ]. Then, for λ > 0, we have that

∥ ∫_{0}^{∞} e^{−λt}f(t) dt∥ = ∥ ∫_{T +ϵ}^{∞} e^{−λt}f(t) dt∥ ≤ \frac{M}{λ − ω} e^{ω(T +ϵ)}e^{−λ(T +ϵ)},  (2)

where M, ω are such that ∥f(t)∥ ≤ Me^{ωt} for almost all t ≥ 0. Thus

lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ ≤ −(T + ϵ)

contradicting (iii). Thus (iv) holds.

Suppose (iv) holds. Then by (2), for ϵ = 0, lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ ≤ −T. Thus, for any sequence (β_{n}) satisfying the Müntz condition we have that lim sup_{n→∞} \frac{1}{β_{n}} ln ∥f̂(β_{n})∥ ≤ −T. Suppose lim sup_{n→∞} \frac{1}{β_{n}} ln ∥f̂(β_{n})∥ ≤ −T − ϵ. Then Corollary 3.2 and Corollary 3.3 imply that ∫_{0}^{t}f(s) ds = 0 for t ∈ [0, T + ϵ), contradicting T ∈ supp(f). Thus (i) holds.

The equivalence of (iv) and (v) is proved by virtue of the inequality (2). Suppose (iv) holds. Then, by (2), for ϵ = 0, lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ ≤ −T. Suppose lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ < −T. Then (i) can not hold, and since (iv) implies (i) this contradicts (iv). Thus lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ = −T.

Suppose (v) holds. Then, by Corollaries 3.2 and 3.3, f vanishes on [0, T]. Suppose f vanishes on [0, T + ϵ]. Then (2) implies that lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ ≤ −T − ϵ, contradicting (v). Thus (iv) holds. ♢ For an exponentially bounded, strongly continuous family of linear operators (K(t))_{t≥0} ⊂ L(X, Y)^{(7)} we obtain similar results. We define the Laplace transform of such an operator family to be the family of linear operators (K̂(λ))_{λ>ω}, where

K̂(λ)x := ∫_{0}^{∞} e^{−λt}K(t)x dt.

We often write ∫_{0}^{∞} e^{−λt}K(t) dt instead of K̂(λ).

(7) In case the operator family is only defined on a finite interval [0, T] we identify the operator family with its extension K̃(t) := \{ \begin{matrix}K(t) & for 0 ≤ t ≤ T \\ K(T) & else \end{matrix} onto [0, ∞).

Theorem 3.7. Let (K(t))_{t≥0} be a strongly continuous, exponentially bounded family of linear operators and let T ≥ 0. Then the following are equivalent.

(i) Every sequence (β_{n})_{n∈IN} satisfying the Müntz condition (M), satisfies

lim sup_{n→∞} \frac{1}{β_{n}} ln ∥K̂(β_{n})∥ = −T.

(ii) Every sequence (β_{n})_{n∈IN} satisfying the Müntz condition (M) has a subse-

quence (β_{n_{j}})_{j∈IN} satisfying the Müntz condition and

lim_{j→∞} \frac{1}{β_{n_{j}}} ln ∥K̂(β_{n_{j}})∥ = −T.

(iii) There exists a sequence (β_{n})_{n∈IN} satisfying the Müntz condition (M) and

lim sup_{n→∞} \frac{1}{β_{n}} ln ∥K̂(β_{n})∥ = −T.

(iv) K(t) = 0 almost everywhere on [0, T] and T ∈ supp(K).

(v) lim sup_{λ→∞} \frac{1}{λ} ln ∥K̂(λ)∥ = −T. Proof. With the same proof as in Theorem 3.6 we show that (i) implies (ii) and clearly, (ii) implies (iii). Suppose (iii) holds; i.e.,

lim sup_{n→∞} \frac{1}{β_{n}} ln ∥K̂(β_{n})∥ = −T.

Then lim sup_{β_{n}→∞} \frac{1}{β_{n}} ln ∥K̂(β_{n})x∥ ≤ −T for all x ∈ X. Thus, by the previous corollary, K(t)x = 0 for all 0 ≤ t ≤ T and all x ∈ X. Thus K(t) = 0 on [0, T]. Suppose K(t) = 0 on [0, T + ϵ]. Then, for λ > 0, we have that

∥ ∫_{0}^{∞} e^{−λt}K(t) dt∥ = ∥ ∫_{T +ϵ}^{∞} e^{−λt}K(t) dt∥ ≤ \frac{M}{λ − ω} e^{ω(T +ϵ)}e^{−λ(T +ϵ)}, (3)

where M, ω are such that ∥K(t)∥ ≤ Me^{ωt} for almost all t ≥ 0. Thus

lim sup_{λ→∞} \frac{1}{λ} ln ∥K̂(λ)∥ ≤ −(T + ϵ)

contradicting (iii). Thus (iv) holds.

Suppose (iv) holds. Then, by (3), for ϵ = 0, lim sup_{n→∞} \frac{1}{β_{n}} ln ∥K̂(β_{n})∥ ≤ −T for all sequences (β_{n}) satisfying the Müntz condition. Suppose

lim sup_{n→∞} \frac{1}{β_{n}} ln ∥K̂(β_{n})∥ ≤ −T − ϵ

and thus lim sup_{n→∞} \frac{1}{β_{n}} ln ∥K̂(β_{n})x∥ ≤ −T − ϵ for all x ∈ X. Then the above theorem implies that ∫_{0}^{t}K(s)x ds = 0 for t ∈ [0, T +ϵ) and all x ∈ X, contradicting T ∈ supp(K). Thus (i), and with the same argument, (v) holds.

Suppose (v) holds. Then lim sup_{n→∞} \frac{1}{β_{n}} ln ∥K̂(β_{n})x∥ ≤ −T for all x ∈ X. Then, by Theorem 3.6, K vanishes on [0, T]. Suppose K vanishes on [0, T + ϵ]. Then (3) implies that lim sup_{λ→∞} \frac{1}{λ} ln ∥K̂(λ)∥ ≤ −T − ϵ, contradicting (v). Thus (iv) holds. ♢

The Theorems 3.6 and 3.7 provide us with a powerful tool to prove Titchmarsh’s theorem. But first we want to establish the some facts about convoluting with a strongly continuous operator family.

Lemma 3.8. Let (K(t))_{t∈[0,∞)} ⊂ L(X, Y) be a strongly continuous family and f ∈ C([0, ∞); X). Then s ↦ K(t − s)f(s) is continuous on [0, t] for all t > 0 and the convolution operator T_{K}f : t ↦ ∫_{0}^{t}K(t − s)f(s) ds is a bounded linear operator from C([0, T]; X) to C([0, T]; Y) for all T > 0. Furthermore, if K and f are exponentially bounded, then for all sufficiently large λ,

∫_{0}^{∞} e^{−λt}(K ⋆ f)(t) dt = ∫_{0}^{∞} e^{−λt}K(t) ∫_{0}^{∞} e^{−λs}f(s) ds dt = K̂(λ)f̂(λ).

Proof. Let M_{t} := sup_{s∈[0,t]} ∥K(s)∥.^{(8)} The function s ↦ K(t − s)f(s) is continuous in [0, t] since

∥K(t − (s + h))f(s + h) − K(t − s)f(s)∥
≤∥K(t − (s + h))f(s + h) − K(t − (s + h))f(s)∥
+ ∥K(t − (s + h))f(s) − K(t − s)f(s)∥
≤M_{t}∥f(s + h) − f(s)∥ + ∥K(t − (s + h))f(s) − K(t − s)f(s)∥ → 0

as h → 0. The convolution operator T_{K} : C([0, T]; X) → C([0, T]; Y) is bounded since

∥T_{K}f∥ = sup_{t∈[0,T]} ∥∫_{0}^{t}K(t − s)f(s) ds∥ ≤ sup_{t∈[0,T]} M_{T} ∫_{0}^{t}∥f(s)∥ ds ≤ M_{T} T ∥f∥.

The second statement of the theorem holds since

\begin{matrix}K̂ ⋆ f(λ) := & ∫_{0}^{∞} e^{−λt}∫_{0}^{t}K(t − s)f(s) ds dt = ∫_{0}^{∞} ∫_{s}^{∞} e^{−λt}K(t − s)f(s) dt ds \\ = & ∫_{0}^{∞}∫_{0}^{∞}e^{−λ(t+s)}K(t)f(s) dt ds = ∫_{0}^{∞}∫_{0}^{∞}e^{−λt}K(t)e^{−λs}f(s) ds dt \\ = & K̂(λ)f̂(λ). \end{matrix}

♢

As remarked earlier, we can not expect the convolution operator to be injective for a strongly continuous operator families K, assuming that 0 ∈ supp(K). A necessary condition is that 0 ∈ suppK(·)x for all x ∈ X. Otherwise, if K(t)x = 0 on [0, ϵ], the convolution K ⋆ f_{x} = 0 on [0, T] for every scalar valued function f with f(t) = 0 on [0, T − ϵ] and f_{x}(t) := f(t)x. We do not know at this point,

(8) Since K is strongly continuous, there exists for all x ∈ X and all t > 0 a constant M_{x} such that ∥K(s)x∥ ≤ M_{x} for all s ∈ [0, t]. By the Principle of Uniform Boundedness, there exists a constant M_{t} such that ∥K(s)∥ ≤ M_{t} for all ∈ [0, t]. whether this condition is also sufficient. We give a technical condition, which combined with the requirement that 0 ∈ supp(K), is a sufficient condition. Once we proved Titchmarsh injectivity theorem and Foias’ density theorem for operator families satisfying this technical condition, we give ways to extend Titchmarsh’s and Foias’ Theorems to a wider class of operator families including, for example, strongly continuous semigroups. For the following definition we recall that

K_{T} (t) := \{ \begin{matrix}K(t) & for 0 ≤ t ≤ T ≤ ∞ \\ 0 & else. \end{matrix}

Definition 3.9. Let (K(t))_{t≥0} ⊂ L(X, Y) be a strongly continuous operator family with K(T_{0}) ≠ 0 for some T_{0} > 0. Let T > T_{0}. We say that (K(t))_{t≥0} satisfies condition (A_{T}) if there exists a Müntz sequence (β_{n})_{n∈IN}, such that for all ϵ > 0 there exists a constant N_{ϵ} with

∥K̂_{T} (β_{n})∥ ∥x∥ ≤ e^{β_{n}ϵ}∥K̂_{T} (β_{n})x∥  (A_{T})

for all n > N_{ϵ} and all x ∈ X. We say that (K(t))_{t≥0} satisfies condition (A) if there exists a T > T_{0} such that (K(t))_{t≥0} satisfies condition (A_{T}). Proposition 3.10. Let (K(t))_{t≥0} ⊂ L(X, Y) be a strongly continuous operator family satisfying condition (A_{T_{0}}) for some T_{0} > 0. Then (K(t))_{t≥0} satisfies condition (A_{T}) for all T > T_{0}. If, in addition, (K(t))_{t≥0} is exponentially bounded, then there exists a sequence (β_{n})_{n∈IN} satisfying the Müntz condition (M), such that for all ϵ > 0 there exists a constant N_{ϵ} with

∥K̂(β_{n})∥ ∥x∥ ≤ e^{β_{n}ϵ}∥K̂(β_{n})x∥

for all n > N_{ϵ} and all x ∈ X.

Proof. Let T > T_{0}, and suppose (K(t))_{t≥0} satisfies condition (A_{T_{0}}). Thus there exists t ∈ [0, T_{0}) such that K(t) ≠ 0. Let (β_{n}) be a Müntz sequence such that for all ϵ > 0 there exists a constant N_{ϵ} > 0, and such that

∥K̂_{T_{0}}(β_{n})∥ ∥x∥ ≤ e^{β_{n}ϵ}∥K̂_{T_{0}}(β_{n})x∥  (4)

for all n > N_{ϵ} and all x ∈ X. Without loss of generality, it suffices to prove the assertion for all 0 < ϵ < \frac{T_{0}−t}{4}.

We show first that there exists a subsequence (β_{n_{j}})_{j∈IN} such that for all 0 < ϵ < \frac{T_{0}−t}{4} there exists a constant J_{ϵ} such that

1 − \frac{e^{−β_{n_{j}} (T_{0}−2ϵ)}}{∥K̂_{T} (β_{n_{j}})∥} ≥ e^{−β_{n_{j}} ϵ/2}

for all j > J_{ϵ}. By Theorem 3.7 there exists a subsequence of (β_{n}) such that

lim_{j→∞} \frac{1}{β_{n_{j}}} ln ∥K̂_{T} (β_{n_{j}})∥ ≥ −t.

Let 0 < ϵ < \frac{T_{0}−t}{4}. Hence, there exists J̃_{ϵ} such that

∥K̂_{T} (β_{n_{j}})∥ ≥ e^{−(t+ϵ)β_{n_{j}}} ≥ e^{−(T_{0}−3ϵ)β_{n_{j}}}

for all j > J̃_{ϵ}. Thus

1 − \frac{e^{−β_{n_{j}} (T_{0}−2ϵ)}}{∥K̂_{T} (β_{n_{j}})∥} ≥ 1 − e^{−β_{n_{j}} ϵ} ≥ e^{−β_{n_{j}} ϵ/2}

for β_{n_{j}}ϵ/2 > 1. Let J_{ϵ} be such that β_{n_{j}}ϵ/2 > 1 for all j > J_{ϵ}, and such that J_{ϵ} ≥ J̃_{ϵ}. Then (β_{n_{j}})_{n∈IN} satisfies the above inequality for all j > J_{ϵ}.

We now look for estimates of (4) on both sides. We know that

\begin{matrix}∥K̂_{T_{0}}(β_{n})x∥ = & ∥ ∫_{0}^{T_{0}} e^{−β_{n}t}K(t)x dt∥ \\ ≤ & ∥ ∫_{0}^{T} e^{−β_{n}t}K(t)x dt∥ + ∥ ∫_{T_{0}}^{T} e^{−β_{n}t}K(t) dt∥ ∥x∥ \\ ≤ & ∥K̂_{T} (β_{n})x∥ + C_{1}|e^{−β_{n}T_{0}} − e^{−β_{n}T} |∥x∥ \end{matrix}

for all n > N_{ϵ} and some constant C_{1} > 0. On the other hand,

∥K̂_{T_{0}}(β_{n})∥ = ∥ ∫_{0}^{T} e^{−β_{n}t}K(t) dt − ∫_{T_{0}}^{T} e^{−β_{n}t}K(t) dt∥

≥ ∥K̂_{T} (β_{n})∥ − ∥ ∫_{T_{0}}^{T} e^{−β_{n}t}K(t) dt∥ ≥ ∥K̂_{T} (β_{n})∥ − C_{2}|e^{−β_{n}T_{0}} − e^{−β_{n}T} |

for all n > N_{ϵ} and some constant C_{2} > 0. Thus

\begin{matrix}(∥K̂_{T} (β_{n})∥− & C_{2}|e^{−β_{n}T_{0}} − e^{−β_{n}T} | − C_{1}e^{β_{n}ϵ}|e^{−β_{n}T_{0}} − e^{−β_{n}T} |)∥x∥ \\ ≤ e^{β_{n}ϵ}∥K̂_{T} (β_{n})x∥ \end{matrix}  (5)

for all x ∈ X and for all n > N_{ϵ}.

Let N_{0} > N_{ϵ} be such that

C_{2}|e^{−β_{n}T_{0}} − e^{−β_{n}T} | + C_{1}e^{β_{n}ϵ}|e^{−β_{n}T_{0}} − e^{−β_{n}T} | ≤ e^{−β_{n}(T_{0}−2ϵ)}

for all n > N_{0}. Thus (5) reduces to

∥K̂_{T} (β_{n})∥(1 − \frac{e^{−β_{n}(T_{0}−2ϵ)}}{∥K̂_{T} (β_{n})∥})∥x∥ ≤ e^{β_{n}ϵ}∥K̂_{T} (β_{n})x∥

for all n ≥ N_{0}. Let N_{1} > max{N_{0}, J_{ϵ}}. Then

e^{−β_{n_{j}} ϵ/2}∥K̂_{T} (β_{n_{j}})∥∥x∥ ≤ e^{β_{n_{j}} ϵ}∥K̂_{T} (β_{n_{j}})x∥,

and thus

∥K̂_{T} (β_{n_{j}})∥∥x∥ ≤ e^{β_{n_{j}} 3ϵ/2}∥K̂_{T} (β_{n_{j}})x∥

for all x ∈ X and all j > N_{1}.

Let K is an exponentially bounded operator family. Replacing T by ∞ and K̂_{T} by K̂ in the above argument, we obtain the desired properties for K. ♢ Condition (A) is obviously satisfied by continuous scalar valued functions k ≠ 0, if we identify k(t) with the linear operator K(t) : x ↦ k(t)x. Another important class of strongly continuous operator families that satisfy condition (A) is the class of uniformly continuous semigroups K(t) = e^{tB} with bounded generator B.^{(9)} Example 3.11. Let (K(t))_{t≥0} be a uniformly continuous semigroup. Then (K(t))_{t≥0} satisfies condition (A). Proof. For a uniformly continuous semigroup (K(t))_{t≥0} we know that

∫_{0}^{∞} e^{−λt}K(t) dt = R(λ, B) := (λI − B)^{−1}

for some bounded linear operator B and all λ > ω for some ω > 0. Since

∥λx − Bx∥ ≤ λ∥x∥ + ∥B∥∥x∥

we obtain, for x = R(λ, B)y, that

∥y∥ ≤ (λ + ∥B∥)∥R(λ, B)y∥,

and thus

∥K̂(λ)∥∥y∥ ≤ (λ + ∥B∥)∥K̂(λ)∥∥R(λ, B)y∥.

Since ∥R(λ, B)y∥ = ∥K̂(λ)y∥ and ∥K̂(λ)∥ ≤ \frac{M}{λ−ω} for some constants M, ω > 0, we obtain that

∥K̂(λ)∥∥y∥ ≤ e^{λϵ}∥K̂(λ)y∥

for λ large enough. Hence (K(t)) satisfies condition (A). ♢ (9) See, for example, A. Pazy [Pa] or J. Goldstein [Go] for an introduction to semigroup theory.

Theorem 3.12 (Titchmarsh). Let (K(t))_{t≥0} ⊂ L(X, Y) be a strongly continuous operator family satisfying condition (A). Let T ≥ 0, let f ∈ C([0, T]; X). Then the following are equivalent:

(i) K ⋆ f = 0 on [0, T]. (ii) There exist constants 0 ≤ t_{1}, t_{2} ≤ T with t_{1} + t_{2} ≥ T such that K = 0 on

[0, t_{1}] and f = 0 on [0, t_{2}]. Proof. By Proposition 3.10 it suffices to show that the theorem holds for exponentially bounded, strongly continuous operator families, since any operator family on [0, T] satisfying condition (A) can be extended to an exponentially bounded operator family satisfying condition (A). Suppose K ⋆ f = 0 on [0, T]. Then, identifying f with its zero extension on [0, ∞),

lim sup_{λ→∞} \frac{1}{λ} ln ∥K̂ ⋆ f(λ)∥ = lim sup_{λ→∞} \frac{1}{λ} ln ∥K̂(λ)f̂(λ)∥ ≤ −T.

By Proposition 3.10, for all ϵ > 0 there exists a Müntz sequence (β_{n})_{n∈IN} and N_{ϵ} such that

e^{−β_{n}(T −2ϵ)} ≥ e^{β_{n}ϵ}∥K̂(β_{n})f̂(β_{n})∥ ≥ ∥K̂(β_{n})∥ ∥f̂(β_{n})∥

for all n > N_{ϵ}. The first inequality follows from the estimate above and the second from condition (A). Let

t_{1} := lim sup_{λ→∞} \frac{1}{λ} ln ∥K̂(λ)∥ and t_{2} := lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥.

Then, by Theorems 3.6 and 3.7, K = 0 on [0, t_{1}] and f = 0 on [0, t_{2}]. By Theorem 3.7 there exists a subsequence of (β_{n})_{n∈IN} satisfying the Müntz condition (M) such that t_{1} = lim_{k→∞} \frac{1}{β_{n_{k}}} ln ∥K̂(β_{n_{k}})∥; by Theorem 3.6 there exists a further subsequence, (β_{n_{k_{j}}} ) such that t_{2} = lim_{j→∞} _{\frac{1}{β_{n_{k_{j}}}}} ln ∥f̂(β_{n_{k_{j}}})∥. Thus

−(t_{1} + t_{2}) = lim_{j→∞} \frac{1}{β_{n_{k_{j}}}} ln ∥K̂(β_{n_{k_{j}}})∥∥f̂(β_{n_{k_{j}}})∥ ≤ −T + 2ϵ

for all ϵ > 0. Hence t_{1} + t_{2} ≥ T.

Suppose (ii) holds. By Theorems 3.6 and 3.7, we know that

lim sup_{λ→∞} \frac{1}{λ} ln ∥K̂(λ)∥ ≤ −t_{1} and lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ ≤ −t_{2}.

Thus,

lim sup_{λ→∞} \frac{1}{λ} ln ∥K̂ ⋆ f(λ)∥ ≤ lim sup_{λ→∞} \frac{1}{λ} ln ∥K̂(λ)∥∥f̂(λ)∥
≤ −(t_{1} + t_{2}) ≤ −T.

Therefore, by Theorem 3.6, K ⋆ f ≡ 0 on [0, T]. ♢

This theorem yields the following corollary.

Corollary 3.13 (Injectivity of the convolution transform). Let (K(t))_{t≥0} be a strongly continuous family of linear operators with 0 ∈ supp(K)^{(10)} that satisfies condition (A), and let f ∈ C([0, T]; X) for some T > 0. Then K ⋆ f = 0 on [0, T] implies that f = 0 on [0, T].

Proof. Suppose K ⋆ f = 0 on [0, T]. Then, by the previous theorem, K = 0 on [0, t_{1}] and f = 0 on [0, T − t_{1}] for some t_{1} ≥ 0. Since 0 ∈ supp(K) we obtain that t_{1} = 0 and thus f = 0 on [0, T]. ♢ (10) Recall that 0 ∈ supp(K) if there exist sequences t_{n} ≥ 0 and x_{n} ∈ X such that t_{n} → 0 and K(t_{n})x_{n} ≠ 0.

After showing the injectivity of the convolution transform, we consider the density of its image. This is known in the scalar-valued case as Foias’ dense range theorem (see [Fo]). The proof of Foias is based on the Hahn-Banach theorem and the Riesz representation theorem of the dual of L^{1}[0, T]. We give a different proof which is constructive and yields an approximating sequence of continuous functions in the image of the convolution transform. Moreover, we extend Foias’ dense range theorem to the Banach space valued setting.

Of course, we can not expect a dense image if we do not have a condition on the range of K(t). Suppose (K(t))_{t≥0} with 0 ∈ supp(K) is exponentially bounded and satisfies condition (A); i.e., there exists a sequence (β_{n})_{n∈IN} satisfying the Müntz condition (M) and ∥K̂(β_{n})∥∥x∥ ≤ e^{β_{n}ϵ}∥K̂(β_{n})x∥ for all ϵ > 0, x ∈ X, and n > N_{ϵ}. Let ϵ > 0. Since 0 ∈ supp(K), by Theorem 3.7 there exists a subsequence of (β_{n})_{n∈IN} satisfying the Müntz condition (M) with ∥K̂(β_{n_{k}})∥ ≥ e^{−β_{n_{k}} ϵ}. Thus, by condition (A) there exists a constant N_{ϵ} such that

∥x∥e^{−β_{n_{k}} ϵ} ≤ e^{β_{n_{k}} ϵ}∥K̂(β_{n_{k}})x∥

for all x ∈ X and all k > Ñ_{ϵ}. Thus K̂(β_{n_{k}}) is one-to-one and for y ∈ Im(K̂(β_{nk})),

∥K̂(β_{n_{k}})^{−1}y∥ ≤ e^{2β_{n_{k}} ϵ}∥y∥.  (6)

Hence, K̂(β_{n_{k}})^{−1} is a bounded linear operator and Im(K̂(β_{n_{k}})) = D(K̂(β_{n_{k}})^{−1}) is a closed subspace of Y. We denote this subspace as Y_{β_{n_{k}}} and the intersection of the ranges with Ỹ := ⋂_{k≥Ñ_{ϵ}} Y_{β_{n_{k}}}, with (β_{n_{k}}) as a defining Müntz sequence.

Theorem 3.14 (Foias’ Dense Range Theorem). Let (K(t))_{t≥0} ⊂ L(X, Y) be a strongly continuous operator family with 0 ∈ supp(K) satisfying condition (A). Let K̃ be an exponentially bounded strongly continuous operator family that coincides with K on [0, T]. Let β_{n} be a Müntz sequence defining Ỹ. Then for all f ∈ C_{0}([0, T]; Ỹ) there exist g_{n} ∈ C([0, T]; X) such that K ⋆ g_{n} → f.^{(11)}

Proof. It suffices to show that the image is dense for K̃, since by Proposition 3.10, K̃ also satisfies condition (A) and, by definition, coincides with K on [0, T]. Let f ∈ C_{0}([0, T]; Ỹ). Let ϵ > 0 and Let

f_{ϵ}(t) := \{ \begin{matrix}0 & for 0 ≤ t ≤ ϵ \\ \frac{1}{ϵ} (∫_{t−ϵ}^{t} f(s) ds − ∫_{0}^{ϵ} f(s) ds) & else. \end{matrix}

Then

∥f − f_{ϵ}∥ ≤ sup_{t∈[0,ϵ]} ∥f(t)∥ + sup_{t∈[ϵ,T]} \frac{1}{ϵ} ∫_{t−ϵ}^{t} ∥f(t) − f(s)∥ ds.

Since f is uniformly continuous we have that f_{ϵ} → f. Thus, it suffices to show that we can approximate each f_{ϵ}. Let

h(t) := \{ \begin{matrix}\frac{f(t+ϵ)−f(t)}{ϵ} & for 0 ≤ t ≤ T − ϵ \\ \frac{f(T)−f(T −ϵ)}{ϵ} & else. \end{matrix}

Then h ∈ C([0, T]; Ỹ) and we identify h with its zero extension on C([0, ∞); Ỹ). Then, in particular, ĥ(β_{n}) ∈ Ỹ for all n ∈ IN.

(11) If f ∈ C_{0}([0, ∞); Ỹ), then the functions g̃_{n} ∈ C([0, ∞); X) that satisfy

sup_{t∈[0,n]} ∥(K ⋆ g̃_{n})(t) − f(t)∥ ≤ \frac{1}{n}

approximate f in the Frechét topology of C_{0}([0, ∞); Ỹ).

Since (β_{n})_{n∈IN} is a defining Müntz sequence there exists a constant N_{ϵ} such that ∥ ˆK̃(β_{n})^{−1}y∥ ≤ e^{β_{n}2ϵ/3}∥y∥ for all n > N_{ϵ} and y ∈ Ỹ.

Let

g_{k}(t) := ∑_{n=1}^{N_{k}} α_{k,n}e^{(t−ϵ)β_{kn}} ˆK̃(β_{kn})^{−1}ĥ(β_{kn}),

where α_{k,n} and N_{k} are as in Theorem 3.1. Then

K ⋆ g_{k}(t) = ∑_{n=1}^{N_{k}} α_{k,n}∫_{0}^{t}K(s)e^{(t−s−ϵ)β_{kn}} ˆK̃(β_{kn})^{−1}ĥ(β_{kn}) ds
= ∑_{n=1}^{N_{k}} α_{k,n}e^{(t−ϵ)β_{kn}}∫_{0}^{t}e^{−sβ_{kn}} K̃(s) ˆK̃(β_{kn})^{−1}ĥ(β_{kn}) ds
= ∑_{n=1}^{N_{k}} α_{k,n}e^{(t−ϵ)β_{kn}} ∫_{0}^{∞} e^{−sβ_{kn}} K̃(s) ˆK̃(β_{kn})^{−1}ĥ(β_{kn}) ds

− ∑_{n=1}^{N_{k}} α_{k,n}e^{(t−ϵ)β_{kn}} ∫_{t}^{∞} e^{−sβ_{kn}} K̃(s) ˆK̃(β_{kn})^{−1}ĥ(β_{kn}) ds

= ∑_{n=1}^{N_{k}} α_{k,n}e^{(t−ϵ)β_{kn}} ĥ(β_{kn})

− ∑_{n=1}^{N_{k}} α_{k,n}e^{(t−ϵ)β_{kn}} ∫_{t}^{∞} e^{−sβ_{kn}} K̃(s) ˆK̃(β_{kn})^{−1}ĥ(β_{kn}) ds.

Since

e^{−ϵβ_{kn}} ĥ(β_{kn}) = ∫_{0}^{∞} e^{−β_{kn}(t+ϵ)}h(t) dt = ∫_{ϵ}^{∞} e^{−β_{kn}t}h(t − ϵ) dt,

by Theorem 3.1, the first term converges uniformly for T ∈ [0, T] to

\{ \begin{matrix}0 & for 0 ≤ t ≤ ϵ \\ ∫_{ϵ}^{t} h(s − ϵ) ds & else. \end{matrix}

But

\begin{aligned}∫_{ϵ}^{t} h(s − ϵ) ds & = \frac{1}{ϵ} ∫_{ϵ}^{t} f(s) − f(s − ϵ) ds = \frac{1}{ϵ} (∫_{ϵ}^{t} f(s) − ∫_{0}^{t−ϵ} f(s) ds) \\ & = \frac{1}{ϵ} ∫_{t−ϵ}^{t} f(s) ds − ∫_{0}^{ϵ} f(s) ds = f_{ϵ}(t). \end{aligned}

Thus the first term converges to f_{ϵ}, uniformly on [0, T].

We conclude the proof by showing that the second term converges to zero. Choose k such that 2/kδ < ϵ/4. Recall that ∥β_{kn}∥ ≥ (kn − 1)δ. Then

\begin{matrix}∥ ∑_{n=1}^{N_{k}} & α_{k,n}e^{(t−ϵ)β_{kn}} ∫_{t}^{∞} e^{−sβ_{kn}} K̃(s) ˆK̃(β_{kn})^{−1}ĥ(β_{kn}) ds∥ \\ ≤ & ∑_{n=1}^{∞} e^{2β_{kn}/kδ}e^{(t−ϵ)β_{kn}} ∫_{t}^{∞} e^{−sβ_{kn}}∥K̃(s) ˆK̃(β_{kn})^{−1}ĥ(β_{kn})∥ ds \\ ≤ & ∑_{n=1}^{∞} e^{2β_{kn}/kδ+(t−ϵ)β_{kn}} ∫_{t}^{∞} e^{−sβ_{kn}}∥K̃(s)∥e^{β_{kn}2ϵ/3}∥ĥ(β_{kn})∥ ds \\ ≤ & \frac{Me^{ωt}∥ĥ(β_{kn})∥}{β_{kn} − ω} ∑_{n=1}^{∞} e^{2β_{kn}/kδ+(t−ϵ)β_{kn}}e^{−β_{kn}t}e^{β_{kn}2ϵ/3} \\ ≤ & C ∑_{n=1}^{∞} e^{β_{kn}(2/kδ−ϵ/3)} ≤ C ∑_{n=1}^{∞} e^{−(kn−1)δϵ/12} \\ = & C \frac{e^{−(k−1)δϵ/12}}{1 − e^{−kδϵ/12}} → 0. \end{matrix}

♢

As seen in inequality (6), the condition (A) implies that ˆ K̃(β_{n})^{−1} ∈ L(Ỹ , X) for some Müntz sequence (β_{n})_{n∈IN}. This shows that a strongly continuous semigroup satisfies condition (A) if and only if K is uniformly continuous. We will now develop tools to extend the class of strongly continuous operator families that yield an injective convolution product or a convolution product with a dense range.

Suppose (K(t)) with 0 ∈ supp(K) satisfies condition (A) and thus, by Corollary 3.13, yields an injective convolution operator T_{K}. Let C be any bounded, injective linear operator. Then the operator family (CK(t)) does not, in general, satisfy condition (A). However, T_{CK} is also injective, since CK ⋆ f = 0 implies that K⋆f = 0. Furthermore, if T_{K_{1}} and T_{K_{2}} are injective, then T_{K_{1}⋆K_{2}} is injective.

Theorem 3.15. Let (K_{1}(t))_{t∈[0,∞)} ⊂ L(X, Y) and (K_{2}(t))_{t∈[0,∞)} ⊂ L(Y, Z) be strongly continuous operator families and let C ∈ L(Y, Z) be a bounded linear operator with a dense range. Suppose the convolution transforms T_{K_{1}} : C([0, T]; X) → C([0, T]; Y) and T_{K_{2}} : C([0, T]; Y) → C([0, T]; Z) have dense ranges. Then the convolution operators T_{CK}, T_{K_{2}⋆K_{1}} : C([0, T]; X) → C([0, T]; Z) have dense ranges in C([0, T]; Z). Proof. First we show that for all f ∈ C([0, T]; Z) and ϵ > 0 there exists g ∈ C([0, T]; Y) such that sup_{t∈[0,T]} ∥f(t) − Cg(t)∥_{Z} < ϵ. Let {t_{j}} be a partition of [0, T] such that sup_{t∈[t_{j},t_{j+1}]} ∥f(t) − f(t_{j})∥_{Z} < ϵ for all j. Since the range of C is dense, we can find y_{j} ∈ Y such that ∥Cy_{j} − f(t_{j})∥ < ϵ. For t ∈ [t_{j}, t_{j+1}] define

g(t) := (1 − \frac{t − t_{j}}{t_{j+1} − t_{j}})y_{j} + \frac{t − t_{j}}{t_{j+1} − t_{j}})y_{j+1}.

Then

sup_{t∈[0,T]}∥Cg(t) − f(t)∥ = sup_{j} sup_{t∈[t_{j},t_{j+1}]} ∥Cg(t) − f(t)∥
≤ sup_{j} sup_{t∈[t_{j},t_{j+1}]} ∥Cg(t) − Cy_{j}∥ + ∥Cy_{j} − f(t_{j})∥ + ∥f(t_{j}) − f(t)∥
≤\frac{t − t_{j}}{t_{j+1} − t_{j}} ∥Cy_{j+1} − Cy_{j}∥ + 2ϵ
≤2ϵ + sup_{j} sup_{t∈[t_{j},t_{j+1}]} ∥Cy_{j+1} − f(t_{j+1})∥
+ ∥f(t_{j+1}) − f(t_{j})∥ + ∥f(t_{j}) − Cy_{j}∥ ≤ 5ϵ.

Let h ∈ C([0, T]; X) be such that ∥T_{K_{1}}h − g∥ ≤ ϵ. Then

\begin{matrix}∥T_{CK_{1}}h − f∥ = & sup_{t∈[0,T]} ∥∫_{0}^{t}CK_{1}(t − s)h(s) ds − f(t)∥ \\ ≤ & sup_{t∈[0,T]} ∥C∫_{0}^{t}K_{1}(t − s)h(s) ds − Cg(t)∥ + ∥Cg(t) − f(t)∥ \\ ≤ & ∥C∥ϵ + 5ϵ = (∥C∥ + 5)ϵ. \end{matrix}

Thus the image of T_{CK_{1}} is dense.

In order to show that the range of T_{K_{2}⋆K_{1}} is dense, let f ∈ C([0, T]; Z). Then there exists g ∈ C([0, T]; Y) such that ∥K_{2} ⋆ g∥ ≤ ϵ. Pick h ∈ C([0, T]; Z) such that ∥K_{1} ⋆ h − g∥ ≤ ϵ. Then

∥T_{K_{2}⋆K_{1}}h − f∥ = sup_{t∈[0,T]} ∥∫_{0}^{t}K_{2}(t − s)K_{1} ⋆ h(s) ds − f(t)
≤ sup_{t∈[0,T]} ∥∫_{0}^{t}K_{2}(t − s)K_{1} ⋆ h(s) ds∫_{0}^{t}K_{2}(t − s)g(s) ds∥
+ ∥∫_{0}^{t}K_{2}(t − s)g(s) ds − f(t)∥
≤∫_{0}^{t}∥K_{2}(t − s)∥ ∥K_{1} ⋆ h(s) − g(s)∥ ds + ϵ ≤ T Mϵ + ϵ,

where M = sup_{t∈[0,T]} ∥K_{2}(t)∥. Thus the range of T_{K_{2}⋆K_{1}} is dense in C([0, T]; Z) ♢ As an application of the previous theorem we show that for strongly continuous semigroups the image of the convolution transform is dense. Proposition 3.16. Let (K(t))_{t≥0} be a strongly continuous semigroup on a Banach space X and T > 0. Then the convolution operator T_{K} : C([0, T]; X) → C([0, T]; X) defined by T_{K}f := ∫_{0}^{t}K(t − s)f(s) ds is one-to-one and has dense range.^{(12)} Proof. Let A be the generator of (K(t))_{t≥0} and let λ_{0} be in the resolvent set of A. We show first that

(F (t))_{t≥0} := ((λ_{0} − A)∫_{0}^{t}K(s) ds)_{t≥0}

(12) Since for all g ∈ C([0, ∞); X), and for all T > 0, ϵ > 0 there exist f_{T} ∈ C([0, ∞); X) such that sup_{t∈[0,T]} ∥∫_{0}^{t}K(t−s)f_{T} (s) ds−g(t)∥ < ϵ, the image is also dense in the Frechét space C([0, ∞); X). satisfies condition (A). By definition, F (t) = λ_{0}(1 ⋆ K(t)) − K(t) + Id. Thus F (t) is strongly continuous. Furthermore, F̂(λ) = ∫_{0}^{∞} e^{−λt}F (t) dt = \frac{λ_{0}−A}{λ}R(λ, A) and hence F̂(λ)^{−1} = λ(λ − A)R(λ_{0}, A). Therefore

\begin{matrix}∥x∥ = & ∥F̂(λ)(λ(λ − A)R(λ_{0}, A)x)∥ = ∥(λ(λ − A)R(λ_{0}, A)F̂(λ)x∥ \\ ≤ & λ∥(λ − λ_{0})R(λ_{0}, A) + Id∥∥F̂(λ)x∥ \\ ≤ & λ(λ∥R(λ_{0}, A)∥ + λ_{0}∥R(λ_{0}, A)∥ + 1)∥F̂(λ)x∥ ≤ λ^{2}C∥F̂(λ)x∥ \end{matrix}

for some constant C > 0 and all λ > ω for some ω > 0. Since

∥F̂(λ)∥ ≤ λ_{0}(\frac{1}{λ}∥R(λ, A)∥ + ∥R(λ, A)∥ + 1) ≤ \frac{M̃}{λ}

we obtain ∥x∥∥F̂(λ)∥ ≤ λMM̃∥F̂(λ)x∥. Hence, the operator family (F (t))_{t∈[0,T]} satisfies condition (A) and it is easy to see that 0 ∈ supp(F). Thus T_{F} is one-to-one (by Titchmarsh’s theorem), and the image of T_{F} is dense in C([0, T]; X) for all T > 0 (by Foias’ theorem). By the previous theorem, the operator R(λ_{0}, A)F = 1⋆K also has these properties. But if (1⋆K)⋆f_{n} converges to f, so does K⋆(1⋆f_{n}), and if (1 ⋆ K) ⋆ f = 0, then f = 0. Thus the convolution product with a strongly continuous semigroup has a dense range and is one-to-one.

Clearly, these statements can also be proved directly using semigroup methods without referring to Titchmarsh’s and Foias’ theorems. We will give the direct proofs next.

To see that T_{K}f is one-to-one, observe that T̂_{K}f = R(λ, A)f̂(λ), where we identify f with its zero continuation onto [0, ∞). Thus, if T_{K}f = 0, then f̂ = 0 for all sufficiently large λ, and thus f = 0.

For the semigroup proof of the density of the range of T_{K}f in C_{0}([0, T]; X) we show first that C_{0}^{1}([0, T]; [D(A)]) is dense in C_{0}([0, T]; X), where [D(A)] denotes the Banach space consisting of the domain D(A) of the generator A endowed with the graph norm ∥x∥_{A} := ∥x∥+∥Ax∥. It follows from the Hille-Yosida theorem that the operators λR(λ, A) are uniformly bounded (λ > λ_{0}) and that λR(λ, A)x = R(λ, A)Ax + x → x for all x ∈ D(A) as λ → ∞. Thus by the density of the domain D(A), it follows from the Banach-Steinhaus theorem that λR(λ, A)x → x for all x ∈ X, where the convergence is uniform for x in compact subsets of X. Let g ∈ C_{0}[0, T]; X). Then

g_{n} : t ↦ \frac{1}{h} ∫_{t}^{t+h} g(s) ds − \frac{1}{h} ∫_{0}^{h} g(s) ds ∈ C_{0}^{1}([0, T]; X)

and g_{n} → g as h → ∞ uniformly on [0, T]. Let

g_{h,λ} : t ↦ λR(λ, A)g_{h}(t) ∈ C_{0}^{1}([0, T]; [D(A)]).

Since C_{h} := {g_{h}(t) : t ∈ [0, T]} is a compact subset of X, it follows that g_{h,λ} → g_{h} as λ → ∞. Now for f_{h,λ} ∈ C([0, T]; X), we have that T_{K}f_{h,λ} = g_{h,λ} if and only if T̂_{K} f̂_{h,λ} or f̂_{h,λ}(μ) = (μ − A)ĝ_{h,λ}(μ) = μĝ_{h,λ}(μ) − Âg_{h,λ}(μ). Thus, f_{h,λ} : t ↦ g′_{h,λ}(t) − Ag_{h,λ}(t) ∈ C([0, T]; X) satisfies T_{K}f_{h,λ} → g. Thus, the range of T_{K} is dense in C_{0}([0, T]; X). ♢

Consider the inhomogeneous abstract Cauchy problem

u′(t) = Au(t) + f(t), u(0) = 0,  (ICP)

where A is the generator of a strongly continuous semigroup K(t) on a Banach space X, and where f ∈ C([0, ∞); X) is a forcing term. Then the unique mild solution of (ICP) is given by

u(t) := ∫_{0}^{t}K(t − s)f(s) ds = (T_{K}f)(t)

(see, for example A. Pazy [Pa], or J. Goldstein [Go]). By the previous proposition, the range of T_{K}f is dense in C_{0}([0, T]; X). Thus for any given target function z ∈ C_{0}([0, T]; X) and any ϵ > 0, there exists a forcing term f_{ϵ} ∈ C([0, T]; X) such that the solution u_{ϵ} = T_{K}f_{ϵ} of the evolutionary system described by (ICP) satisfies

∥T_{K}f_{ϵ} − z∥ = sup_{t∈[0,T]} ∥u_{ϵ}(t) − z(t)∥ < ϵ.

Therefore any evolutionary system governed by a strongly continuous semigroup can be steered by appropriate forcing terms f_{ϵ} ∈ C([0, T]; X) to any preassigned target orbit z ∈ C_{0}([0, T]; X).

Furthermore, since ∥T_{K}f_{ϵ} − z∥ = ∥u_{ϵ} − z∥ < ϵ, it follows that there exists a generalized forcing term f_{0} ∈ \overline{C([0, T]; X)}^{T_{K}} such that f_{ϵ} → f_{0} ∈ \overline{C([0, T]; X)}^{T_{K}} and

z(t) = (T̃_{K}f_{0})(t) = lim_{ϵ→0}(T_{K}f_{ϵ})(t) = lim_{ϵ→0} u_{ϵ},

uniformly in t ∈ [0, T], where u_{ϵ} is the solution of

u′_{ϵ}(t) = Au_{ϵ}(t) + f_{ϵ}, u_{ϵ}(0) = 0.

In this sense, any target z ∈ C_{0}([0, T]; X) is a “limit solution” or generalized solution of the inhomogeneous problem

u′(t) = Au(t) + f_{0}(t), u(0) = 0,

for a unique forcing term f_{0} ∈ \overline{C([0, T]; X)}^{T_{K}}.

Having Titchmarsh’s and Foias’ theorems to our disposal, we do expect similar results for cosine families, solutions of Volterra integral equations and other abstract integro-differential equations that can be treated with the Laplace transform. However, the scope of that investigation is going beyond the range of topics treated in this dissertation. We will return to the concept of “limit solution” in Chapter IV. There we will show how this concept arises naturally in the study of abstract Cauchy problems

u′(t) = Au(t); u(0) = x,

where the operator A generates an integrated or k-generalized semigroup on a given Banach space X.

II.4 The Vector Space of Generalized Functions

In this section we will show that the generalized functions deduced from the convolution product with a scalar valued function form, in the scalar-valued case a field, and in the vector-valued case a vector space over that field. Extending the domain of the generalized function spaces and considering all k-generalized vector-valued functions on [α, ∞) for some α ∈ IR, we obtain a class of generalized functions that includes the abstract quotient field developed by J. Mikusinski.

But first, we want to shed some light on the structure and interplay between different generalized function spaces that were obtained by the convolution product. In Section II.1 we introduced the notion of a generalized derivative of continuous functions. This led to the following tower of generalized function spaces, where T denotes the antiderivative operator T f : t ↦ ∫_{0}^{t}f(s) ds.

⋮  ⋮

C^{−n}[0, 1] T ⟶

isom. isomorph

C^{−n−1}[0, 1]
⋮ ⋮

C^{−2}[0, 1] T ⟶

isom. isomorph

C^{−1}[0, 1]
↑ ∪  ∪

C^{−1}[0, 1] T ⟶

isom. isomorph

C_{0}[0, 1]
↑ ∪  ∪

C_{0}[0, 1] T ⟶

isom. isomorph

C_{0}^{1}[0, 1]

Since integrating is the same as convoluting with the constant one function, we will consider now towers of generalized function spaces generated by convolution operators T_{k}. In the last section we showed that the convolution operator T_{k} is one to one and has a dense range for k ∈ C[0, T] with 0 ∈ supp(k). In the case that k ∈ L^{1}[0, 1] with 0 ∈ supp(k), the resulting convolution operator T_{k} : C([0, 1]; X) → C_{0}([0, 1]; X) is also injective and has a dense range in C_{0}[0, 1], since 1 ⋆ k is continuous and 0 ∈ supp(1 ⋆ k). Thus T_{1⋆k} is an injective operator with a dense range. But since k ⋆ f = 0 implies that 1 ⋆ k ⋆ f = 0, and hence f = 0, we obtain that T_{k} is injective and since 1 ⋆ k ⋆ f_{n} → g implies that k ⋆ (1 ⋆ f_{n}) → g we have that the range of T_{k} is dense. Thus the results of Section II.2 apply to k ∈ L^{1}[0, 1] as well. In particular, Theorem 2.1 yields the following diagram for any k ∈ L^{1}[0, 1] with 0 ∈ supp(k).

\overline{C[0, 1]}^{T_{k}} T̃_{k} ⟶

isom. isomorph

C_{0}[0, 1]

↑ ∪  ∪

C[0, 1]  \underrightarrow{T_{k}}  Im(T_{k}).

To simplify notation, from now on we will identify the operator T̃_{k} with T_{k}.

As special cases, consider k_{α} : t ↦ \frac{t^{α−1}}{Γ(α)} for α > 0. Then k_{α} ∈ L^{1}[0, 1] and we

denote with T_{α} the convolution operator f ↦ k_{α} ⋆ f. Then T_{α}T_{β} = T_{α+β}. Note

that in following diagram we consider T_{α} as an operator from C_{0}[0, 1] into itself.

By Theorem 2.2, \overline{C_{0}[0, 1]}^{T_{β}} can be obtained by extending \overline{C_{0}[0, 1]}^{T_{α}} with T_{β−α}.

Let 0 < α ≤ β < n. Then

⋮  ⋮

C^{−n}[0, 1] T_{n} ⟶

isom. isomorph

C_{0}[0, 1]

↑ ∪  | |

\overline{C_{0}[0, 1]}^{T_{β}} T_{β} ⟶

isom. isomorph

C_{0}[0, 1]

↑ ∪  | |

\overline{C_{0}[0, 1]}^{T_{α}} T^{α} ⟶

isom. isomorph

C_{0}[0, 1]

↑ ∪  ∪

C_{0}[0, 1] \underrightarrow{T^{α}} ImT^{α}

Let ϕ : t ↦ e^{− \frac{1}{t}}. Then ϕ^{(n)} := (\frac{d}{dt})^{n} ϕ ∈ C_{0}[0, 1], ϕ^{(n)}(0) = 0, and hence T_{n}ϕ^{(n)} = \frac{t^{n−1}}{n−1!} ⋆ ϕ^{(n)} = ϕ for all n ∈ IN. Hence, by Theorem 2.2 (c), all the C_{0}^{−n}[0, 1]-spaces are contained in the generalized function space \overline{C[0, 1]}^{T_{ϕ}}. Thus

C_{0}[0, 1] ↪ \overline{C[0, 1]}^{T_{α}} ↪ \overline{C[0, 1]}^{T_{β}} ↪ C^{−n}[0, 1] ↪ \overline{C[0, 1]}^{T_{ϕ}}

for all 0 < α < β ≤ n. Since distributions are, at least locally, distributional derivatives of some order n, all distributions with support in (0, 1) are contained in some C_{0}^{−n}[0, 1] space. The generalized function T_{ϕ}^{−1}(t ↦ t) cannot be a distribution, since that would imply that \frac{t^{n}}{n!} ⋆ T_{ϕ}^{−1}(t ↦ t) is a continuous function f for some n ∈ IN, and therefore that \frac{t^{n+2}}{n+2!} = e^{− \frac{1}{t}} ⋆ f. Hence

1 = (\frac{d}{dt})^{n+2} (e^{− \frac{1}{t}} ⋆ f) = ((\frac{d}{dt})^{n+2} e^{− \frac{1}{t}} ) ⋆ f ∈ C_{0}[0, 1].

This is a contradiction. Therefore the space \overline{C[0, 1]}^{T_{ϕ}} contains more objects than just distributions.

If we look at arbitrary generalized function spaces deduced from two convolution operators T_{k_{1}} and T_{k_{2}}, with k_{1}, k_{2} ∈ L^{1}[0, 1], Theorem 2.2 yields the following structure.

\overline{C[0, 1]}^{T_{k_{1}⋆k_{2}}} ↩ \overline{C[0, 1]}^{T_{k_{2}}} \overline{C[0, 1]}^{T_{k_{1}⋆k_{2}}} T̃_{k_{1}} ⟵

isom. isomorph

\overline{C[0, 1]}^{T_{k_{2}}}
↑ ∪  ↑ ∪  T̃_{k_{2}} ↑ isom.isom.  isom.isom. ↑ T_{k_{2}}
\overline{C[0, 1]}^{T_{k_{1}}} ↩ C_{0}[0, 1] \overline{C[0, 1]}^{T_{k_{1}}} T_{k_{1}} ⟵

isom. isomorph

C_{0}[0, 1].

For elements f ∈ \overline{C[0, 1]}^{T_{k_{1}}} we have that T_{k_{2}}T_{k_{1}}f = T_{k_{2}⋆k_{1}}f, whether f is considered as a generalized function in \overline{C[0, 1]}^{T_{k_{1}}} or as its embedding in \overline{C[0, 1]}^{T_{k_{1}⋆k_{2}}} .

Thus we identify an element f ∈ \overline{C[0, 1]}^{T_{k_{1}}} with an element f̃ ∈ \overline{C[0, 1]}^{T_{k_{2}}} , if their embeddings in \overline{C[0, 1]}^{T_{k_{1}⋆k_{2}}} are the same; i.e., if

k_{2} ⋆ k_{1} ⋆ f = k_{1} ⋆ k_{2} ⋆ f̃.

This identification enables us to define the convolution between two generalized functions.

Definition 4.1. Let k_{1}, k_{2} ∈ L^{1}[0, 1] with 0 ∈ supp(k_{1}) ∩ supp(k_{2}). Let f ∈ \overline{C[0, 1]}^{k_{1}} and g ∈ \overline{C[0, 1]}^{k_{2}}. Then we define the convolution between the two generalized functions via

f ⋆ g := T_{k_{1}⋆k_{2}}^{−1} (k_{1} ⋆ f ⋆ k_{2} ⋆ g) .

Clearly, f ⋆ g ∈ \overline{C[0, 1]}^{k_{1}⋆k_{2}} is again a generalized function. Next, we show that this convolution is well defined: we show that for f ∈ \overline{C[0, 1]}^{k_{1}} ∩ \overline{C[0, 1]}^{k_{2}} the convolution f ⋆g is independent of the representative for f; i.e., it does not matter whether f is considered as an element of \overline{C[0, 1]}^{k_{1}} or as an element of \overline{C[0, 1]}^{k_{2}}. We know that

f = T_{k_{1}⋆k_{2}}^{−1} (k_{1} ⋆ k_{2} ⋆ f) ∈ \overline{C[0, 1]}^{k_{1}⋆k_{2}}.

Thus, for g ∈ \overline{C[0, 1]}^{k} for some k ∈ L^{1}[0, 1] with 0 ∈ supp(k) we obtain that

f ⋆ g = T_{k_{1}⋆k_{2}⋆k}^{−1} (k_{1} ⋆ k_{2} ⋆ f ⋆ k ⋆ g) ,

which is independent to the choice of the space of f.

Furthermore, let k_{1}, k_{2} be as above and let h := T_{k_{1}⋆k_{1}⋆1}^{−1}(1 ⋆ k_{1}). Then for all f ∈ \overline{C[0, 1]}^{T_{k_{2}}} , k_{1} ⋆ f ∈ \overline{C[0, 1]}^{T_{k_{2}}} and

h ⋆ (k_{1} ⋆ f) = T_{k_{1}⋆k_{1}⋆1⋆k_{2}}^{−1} (1 ⋆ k_{1} ⋆ k_{1} ⋆ k_{2} ⋆ f) = f.  (1)

Thus T_{h} = T_{k_{1}}^{−1}. This shows that the inverse convolution transform of functions k with 0 ∈ supp(k) is again a convolution, a convolution with a particular generalized function.

Through the additive component of the generalized function spaces, we will always have to deal with generalized functions that do not have zero in their support.^{(1)} Thus in order to obtain a field with respect to addition and convolution we have to take a closer look at convolutions of functions that are zero on some interval [0, a].

Let k ∈ C[0, ∞) with k = 0 on [0, a] for some a > 0. Then

\begin{matrix}k ⋆ f(t) = & ∫_{0}^{t}f(t − s)k(s) ds = ∫_{a}^{t} f(t − s)k(s) ds \\ = & ∫_{0}^{t−a} f(t − a − s)k(s + a) ds = (k_{−a} ⋆ f)(t − a), \end{matrix}

where k_{−a}(t) := k(t + a); i.e., the graph of k_{−a} is the graph of k shifted by a to the left. Thus, convoluting with a function k that is zero on some interval [0, a] is the same as convoluting with the shifted function k_{−a} and then shifting back the result.

Therefore if the “shift”-operations were invertible, we could define the inverse of the convolution transform with respect to functions that do not have zero in (1) Let f ∈ \overline{C([0, 1]}^{T_{k}}. Then supp(f) := supp(y), where T_{k}f = k ⋆ f = y ∈ C_{0}[0, 1]. their support. Unfortunately, only the right-shift is invertible on [0, ∞) and neither shift is invertible on [0, 1]. On (−∞, ∞), however, they are invertible. Thus, if we regard functions f in C_{0}[0, ∞) as functions f_{∞} ∈ C(−∞, ∞) with

f_{∞}(t) := \{ \begin{matrix}f(t) & for 0 ≤ t \\ 0 & else, \end{matrix}

the shift-operator S_{α} with S_{α}f(t) = f(t − α) for some α ∈ IR is an invertible operator. The resulting functions can be identified with functions in C_{0}[α, ∞) for some α ∈ IR. With this embedding of the continuous functions C_{0}[α, ∞) into C(−∞, ∞) we have to adapt the definition of the convolution operator in order to keep it one-to-one.

For k ∈ [a, ∞) and f ∈ C([b, ∞); X) for some constants a, b ∈ IR, define

k ⋆ f : t ↦ ∫_{−∞}^{∞} k_{∞}(t − s)f_{∞}(s) ds,

where k_{∞} and f_{∞} are the functions obtained by extending k and f with zero to the left. With this interpretation we follow J. Mikusinski’s interpretation of the shift operator in the context of his operational calculus ([Mi2] §66). If a = b = 0, then obviously this convolution coincides with the convolution discussed so far. We cannot use the space C(−∞, ∞) as a seed for our generalized function spaces, since it does not have the property that the convolution transform is one-to-one. For example

1 ⋆ \frac{x}{1 + x^{2}} := ∫_{−∞}^{∞} \frac{t}{1 + t^{2}} dt = 0.

For functions f, g ∈ C(−∞, ∞) with f = 0 on (−∞, α] and g = 0 on (−∞, β] we have that

\begin{matrix}f ⋆ g(t) = & ∫_{−∞}^{∞} f(t − s)g(s) ds = ∫_{β}^{t−α} f(t − s)g(s) ds \\ = & ∫_{0}^{t−α−β} f(t − β − s)g(s + β) ds \\ = & ∫_{0}^{t−α−β} f(t − α − β − s + α)g(s + β) ds \\ = & ∫_{0}^{t−α−β} f_{−α}(t − α − β − s)g_{−β}(s) ds \\ = & f_{−α} ⋆ g_{−β}(t − α − β) = S_{α+β} (S_{−α}f ⋆ S_{−β}g) (t). \end{matrix}

Since the right shift S_{−α}f and S_{−β}g are zero on (−∞, 0) and the convolution transform is one-to-one on C[0, ∞), we obtain that the convolution product on the space of continuous functions that are zero on the interval (−∞, α] for some α ∈ IR has no zero divisors. Furthermore, if f = 0 on (−∞, α] and g = 0 on (−∞, β], then f ⋆ g = 0 on (−∞, α + β].

We can extend function spaces C_{0}[a, ∞) to \overline{C_{0}[a, ∞)}^{T_{k}} for T_{k} : C_{0}[a, ∞) → C_{0}[a, ∞) with

T_{k}f : t ↦ ∫_{a}^{t} k(t − s)f(s) ds

for k ∈ C[0, ∞) by taking the completion of the space C[a, ∞) with respect to the seminorms ∥f∥_{α,T_{k}} := sup_{t∈[a,α]} ∥T_{k}f(t)∥. Any generalized function f ∈ \overline{C_{0}[b, ∞)}^{T_{k}} is naturally embedded in \overline{C_{0}[a, ∞)}^{T_{k}} for a ≤ b by identifying k ⋆ f with its zero-extension to [a, ∞) and then taking T_{k}^{−1}. Recall that we identify f ∈ \overline{C_{0}[a, ∞)}^{T_{k_{1}}} with g ∈ \overline{C_{0}[a, ∞)}^{T_{k_{2}}} , if k_{2} ⋆ k_{1} ⋆ f = k_{1} ⋆ k_{2} ⋆ g. Thus we consider two generalized functions f_{1} ∈ \overline{C_{0}[a_{1}, ∞)}^{T_{k_{1}}} and f_{2} ∈ \overline{C_{0}[a_{2}, ∞)}^{T_{k_{2}}} to be the same if

k_{2} ⋆ k_{1} ⋆ f_{1} = k_{1} ⋆ k_{2} ⋆ f_{2},

where the functions k_{2} ⋆ k_{1} ⋆ f_{1} and k_{1} ⋆ k_{2} ⋆ f_{2} are considered as functions in C(−∞, ∞) by extending them with zero to the left. Theorem 4.2 (The field of generalized scalar functions). Let

F := \{f ∈ \overline{C_{0}[a, ∞)}^{T_{k}} : a ∈ IR, k ∈ C[0, ∞) with 0 ∈ supp(k)\},

and define

f ⋆ g := T_{k_{1}⋆k_{2}}^{−1}(k_{1} ⋆ f ⋆ k_{2} ⋆ g)

for f ∈ \overline{C_{0}[a, ∞)}^{T_{k_{1}}} and g ∈ \overline{C_{0}[a, ∞)}^{T_{k_{2}}} . Then F is a field with respect to addition and convolution. Proof. Clearly, with the above identification, F is an additive group. By the natural embedding of functions with different domains, the convolution is defined for all f, g ∈ F. Furthermore, f ⋆ g = 0 implies that k_{1} ⋆ f ⋆ k_{2} ⋆ g = 0. Since the convolution transform is one-to-one, we obtain that either k_{1} ⋆ f or k_{2} ⋆ g = 0 and hence f = 0 or g = 0. Let f ∈ \overline{C_{0}[a, ∞)}^{T_{k}} with a ∈ supp(k ⋆ f). With the above identification, such a constant a always exists for f ≠ 0. Then S_{−a}(f ⋆ k) := h ∈ C[0, ∞) with 0 ∈ supp(h). Let f^{−1} := T_{h}^{−1}(S_{−a}k). Then for all g ∈ F we have that

f ⋆f^{−1} ⋆g = T_{k⋆h⋆k_{1}}^{−1}(k⋆f ⋆h⋆T_{h}^{−1}S_{−a}k⋆k_{1} ⋆g) = T_{k⋆h⋆k_{1}}^{−1}(S_{−a}h⋆S_{−a}k⋆k_{1} ⋆g) = g.

Clearly, f ⋆ (g + h) = f ⋆ g + f ⋆ h, and hence, F is a field. ♢ Since every element of the field is a generalized function and is therefore equal to T_{k}^{−1}f for some continuous f and k, the field is the same as the one obtained via a field extension by J. Mikusinski.

So far we have only considered scalar-valued functions. Clearly, the vectorvalued generalized functions cannot form a field. The convolution between two vector-valued functions is not defined if the vector space has no algebra structure. However, we can consider the spaces \overline{C_{0}([(a, ∞); X)}^{T_{k}}. As above, we can define the convolution of a generalized scalar-valued function with a vector-valued function. The vector-valued generalized functions form a vector space over the field of generalized functions. Thus integration (convoluting with the Heaviside function H_{0}), for example, and differentiation (convoluting with H_{0}^{−1} = δ′_{0}) are “scalar” multiplications in this vector space. Corollary 4.3 (The vector space of generalized functions). Let X be a Banach space and let F be the field of generalized functions. Then

V := \{f ∈ \overline{C_{0}([a, ∞); X)}^{T_{k}} : a ∈ IR, k ∈ C[0, ∞) with 0 ∈ supp(k)\}

is a vector space over F where scalar multiplication of a vector f ∈ V with some h ∈ F is defined by

h ⋆ f := T_{k_{1}⋆k_{2}}^{−1}(k_{1} ⋆ h ⋆ k_{2} ⋆ f),

where h ∈ \overline{C_{0}[a, ∞]}_{k_{1}}^{T} and f ∈ \overline{C_{0}([a_{2}, ∞); X)}_{k_{2}}^{T}.

III. The Asymptotic Laplace Transform

“The modern Laplace transform is relatively recent. It was first used by Bateman in 1910, explored and codified by Doetsch in the 1920s and was first the subject of a textbook as late as 1937. In the 1920s and 1930s it was seen as a topic of front-line research; the applications that call upon it today were then treated by an older technique – the Heaviside operational calculus. This, however, was rapidly displaced by the Laplace transform and by 1950 the exchange was virtually complete. No other recent development in mathematics has achieved such ready popularisation and acceptance among the users of mathematics and the designers of undergraduate curricula.”

M. A. B. Deakin [De2].

The asymptotic Laplace transform is an extension of the classical Laplace transform. It was introduced in 1939 by J. C. Vignaux [Vi] and further investigated by J. C. Vignaux and M. Cotlar [Vi-Co], [Vi2] in 1944. It allows one to transform functions and generalized functions of arbitrary growth at infinity, while preserving all operative features of the classical Laplace transform. It fills the gap between the operational calculus developed by J. Mikusinski and the classical Laplace transform method. J. Mikusinski noted in Part V, Chapter II, §11 in [Mi2]: “ In spite of the formal resemblance of the Laplace transform method and the direct method, the two methods are not equivalent. [...] This (the Laplace transform) method restricts the range of applicability of the operational calculus to a class of functions for which the integral

∫_{0}^{∞}e^{−ts}f(t) dt

is convergent. [...] The Laplace transform method does not provide the full solution of the problem since it is necessary to assume during the calculation that the function sought does not increase too fast, i.e., to be more exact, that it is transformable. Consequently we do not know if the solution obtained is unique.”

It was observed by W. A. Ditkin and P. L. Kusnezow [Di-Ku], and L. Berg [Ber], that the multiplicative field obtained in the image space of the asymptotic Laplace transform corresponds to the quotient field of J. Mikusinski. The advantage of the transform approach, especially in the vector valued case, is that the emphasis is kept on the approximative nature of the generalized functions and that one associates via the asymptotic Laplace transform a class of analytic, vectorvalued functions with any element of Mikusinski’s abstract quotient space. This perception allows the study of vector-valued differential and integral equations involving unbounded linear operators.

The basic approach to asymptotic Laplace transforms presented and extended here is due to G. Lumer and F. Neubrander [Lu-Ne]. The main difference to their work is that we consider functions f ∈ L_{loc}^{1}[0, T); X) for 0 < T ≤ ∞, whereas G. Lumer and F. Neubrander consider only functions in the spaces L^{1}[0, T]; X) and L_{loc}^{1}([0, ∞); X). Moreover, we restrict the asymptotic Laplace transform from the beginning to equivalence classes of analytic functions, whereas G. Lumer and F. Neubrander allow arbitrary functions.

III.1 Asymptotic Laplace Transforms for Functions of Arbitrary Growth

The goal of this section is to extend the Laplace transform f̂(λ) := ∫_{0}^{∞} e^{−λt}f(t) dt to functions of arbitrary growth at infinity in such a way that the main operative features of the Laplace transform are still valid. The asymptotic Laplace transform L is a set-valued operator that assigns to every f ∈ L_{loc}^{1}([0, ∞); X) an equivalence class of certain analytic functions with the following properties.

(a) f̂ ∈ L(f), if f is Laplace transformable.

(b) L is linear.

(c) L(f) ∩ L(g) ≠ ∅ if and only if f = g.

(d) L(f ⋆ g) = L(f) · L(g) + L(0).

(e) L(f′) = λL(f) − f(0).

(f) L(−tf) = (L(f))′ + L(0). The asymptotic Laplace transform does not only exist for functions of arbi-

trary growth on [0, ∞), but also for functions that have a finite time blowup; i.e., for functions f ∈ L_{loc}^{1}([0, T); X) with arbitrary growth in [0, T). The asymptotic Laplace transform of such functions will be denoted by L_{T} (f) and has the same operational features than L. As mentioned above, the main difference between the Laplace transform f ↦ f̂ and its asymptotic versions f ↦ L(f) or f ↦ L_{T} (f), is that the latter are set-valued. They consist of equivalence classes of analytic functions q : Ω → X, where the complex domain Ω is a post-sectorial region containing some real halfline (ω, ∞).

Definition 1.1. Let Ω ⊂ C be an open set. We say that Ω is a post-sectorial region if for all −π/2 < α < π_{2} there exists a constant λ_{α} > 0, such that

{λe^{iβ} : −|α| ≤ β ≤ |α|, λ > λ_{α}} ⊂ Ω.

We denote with A(P; X) the vector space of all analytic functions defined in some postsector Ω with values in a Banach space X.

On the space A(P; X) we define an equivalence class via the equivalence relation ≈_{T}, where r ≈_{T} q if r − q and its derivatives are of exponential decay T; i.e., for T ∈ IR we define

r ≈_{T} q if lim sup_{λ→∞} \frac{1}{λ} ln ∥r^{(n)}(λ) − q^{(n)}(λ)∥ ≤ −T

for all n ∈ IN.

Proposition 1.2. The relation r ≈_{T} q is an equivalence relation on A(P; X). The relation r ≈_{T} 0 defines a linear subspace {0}_{T} of A(P, X), and the quotient space Â_{T} (P; X)/{0}_{T} is a vector space. Furthermore, if q ∈ A(P; C) with q ≈_{S} 0 and r ∈ A(P; X) with r ≈_{T} 0, then qr ∈ A(P; X) with qr ≈_{T +S} 0.

Proof. Clearly, “≈_{T} ” is an equivalence relation, since 1) r ≈_{T} r, 2) r ≈_{T} q if and only if q ≈_{T} r, and 3) p ≈_{T} q and q ≈_{T} r implies that for all ϵ > 0 there exists λ_{ϵ} > 0 such that

∥p^{(n)}(λ) − r^{(n)}(λ)∥ ≤ ∥p^{(n)}(λ) − q^{(n)}(λ)∥ + ∥q^{(n)}(λ) − r^{(n)}(λ)∥ ≤ 2e^{ϵλ}

for all λ > λ_{ϵ}. Hence p ≈_{T} r.

Clearly, r ≈_{T} 0, implies that αr ≈_{T} 0. Let r ≈_{T} 0 and q ≈_{T} 0. Then −q ≈_{T} 0 and thus r ≈_{T} −q which implies, by definition, that r + q ≈_{T} 0. Thus r ≈_{T} 0 defines a linear subspace in A(P; X). The last statement follows from the fact that

\begin{matrix}∥ (q(λ)r(λ))^{(n)} ∥ ≤ & ∑_{j=0}^{n} ( \begin{matrix}n \\ j \end{matrix} ) ∥q(λ)^{(j)}∥ ∥r(λ)^{(n−j)}∥ ≤ 2^{n}e^{−λ(S−ϵ)}e^{−λ(T −ϵ)} \\ = & 2^{n}e^{−λ(T +S−2ϵ)} \end{matrix}

for λ large enough. ♢ Definition 1.3. Let 0 < T ≤ ∞, let f ∈ L_{loc}^{1}([0, T); X). A function q ∈ A(P; X) is an asymptotic Laplace transform of f, i.e., q ∈ L_{T} (f), if and only if

q(λ) ≈_{t} ∫_{0}^{t} e^{−λs}f(s) ds

for all t ∈ [0, T).^{(1)} We first show that for all f ∈ L_{loc}^{1}([0, T); X) the asymptotic Laplace transform L_{T} exists. Theorem 1.4 (Existence). If f ∈ L_{loc}^{1}([0, T); X), then

λ ↦ q(λ) := λ ∫_{0}^{T} e^{−λt}(1 − e^{− \frac{λ ln λ}{G(t)}} )F (t) dt ∈ L_{T} (f),

where F (t) := ∫_{0}^{t}f(s) ds, and G(t) := max{∥F (t)∥, 1}. Proof. Notice that

|1 − e^{−μ}| = |μ ∫_{0}^{1} e^{−μt} dt| ≤ |μ|

for all μ ∈ C with Re(μ) > 0. Let

λ ∈ Ω := {λ ∈ C : Re(λ) ≥ 0 and Re(λ ln λ) > 0}.

(1) To keep notation as simple as possible, we sometimes write r(λ) ≈_{t} q(λ) instead of (λ ↦ r(λ)) ≈_{t} (λ ↦ q(λ)).

Then

∫_{0}^{T} ∥e^{−λs}(1 − e^{− \frac{λ ln λ}{G(s)}} )F (s)∥ ds = ∫_{0}^{T} e^{−λs}|1 − e^{− \frac{λ ln λ}{G(s)}} |∥F (s)∥ ds
≤ ∫_{0}^{T} e^{−Re(λ)s} \frac{|λ ln λ|}{G(s)}∥F (s)∥ ds
≤|λ ln λ| ∫_{0}^{T} e^{−Re(λ)s} ds ≤ \frac{|λ ln λ|}{Re(λ)}.

Thus q(λ) = λ ∫_{0}^{T} e^{−λs}(1 − e^{− \frac{λ ln λ}{G(s)}} )F (s) ds exists for all λ ∈ Ω > 0. It is easy to see that q is an analytic function on Ω. Moreover, Ω is a post-sector since

Re(λ ln λ) = Re(re^{iα}(ln r + iα)) = r ln r cos α − αr sin α > 0

if and only if r > e^{α tan α}, where λ =: re^{iα}.

Now we show that q ∈ L_{T} (f). For all λ ∈ Ω and all n ∈ IN_{0}, we have that

∥ (\frac{d}{dλ})^{n} ∫_{0}^{t} e^{−λs}f(s) ds − (\frac{d}{dλ})^{n} q(λ)∥
=∥ (\frac{d}{dλ})^{n} (e^{−λt}F (t) + λ ∫_{0}^{t} e^{−λs}F (s) ds − λ ∫_{0}^{T} e^{−λs}(1 − e^{− \frac{λ ln λ}{G(s)}} )F (s) ds) ∥
≤∥ (\frac{d}{dλ})^{n} (λ ∫_{0}^{t} e^{−λs}e^{− \frac{λ ln λ}{G(s)}} F (s) ds − λ ∫_{t}^{T} e^{−λs}(1 − e^{− \frac{λ ln λ}{G(s)}} )F (s) ds) ∥

+ ∥ (\frac{d}{dλ})^{n} e^{−λt}F (t)∥

≤∥ (\frac{d}{dλ})^{n} ∫_{0}^{t} λe^{− \frac{λ ln λ}{G(s)} −λs}F (s) ds∥ + ∥ (\frac{d}{dλ})^{n} λ ∫_{t}^{T} e^{−λs}(1 − e^{− \frac{λ ln λ}{G(s)}} )F (s) ds∥

+ ∥ (\frac{d}{dλ})^{n} e^{−λt}F (t)∥.

We want to show that the right hand side is of exponential decay t. Clearly, the third term decays at that rate for any n ∈ IN_{0}. For the first term, there exists a function p with |p(λ, s)| ≤ C(s^{n}λ^{n+1} +1) for all s ∈ [0, t], λ > 1 and some constant C such that

\begin{aligned}∥ (\frac{d}{dλ})^{n} λ ∫_{0}^{t} e^{− \frac{λ ln λ}{G(s)} −λs}F (s) ds∥ & = ∥ ∫_{0}^{t} p(λ, s)e^{− \frac{λ ln λ}{G(s)} −λs}F (s) ds∥ \\ & ≤ e^{− \frac{λ ln λ}{M}} ∫_{0}^{t} e^{−λs}M|p(λ, s)| ds, \end{aligned}

where M = sup_{s∈[0,t]} ∥F (s)∥. Thus, the first term is of exponential decay of order ∞. For the second term, there exist functions q_{j} and p which satisfy |q_{j}(λ, s)| + |p(λ, s)| ≤ C_{j}(s^{n}λ^{n+1} + 1) for all s ∈ [t, T), λ > 1 and some constant C_{j} such that

∫_{t}^{T} ∥ (\frac{d}{dλ})^{n} λe^{−λs}(1 − e^{− \frac{λ ln λ}{G(s)}} )F (s)∥ ds
= ∫_{t}^{T} ∥p(λ, s)e^{−λs}(1 − e^{− \frac{λ ln λ}{G(s)}} )F (s) + ∑_{j=1}^{n} \frac{q_{j}(λ, s)e^{− \frac{λ ln λ}{G(s)} −λs}}{(G(s))^{j}} F (s)∥ ds
≤ ∫_{t}^{T} |p(λ, s)|e^{−λs} \frac{|λ ln λ|}{G(s)}∥F (s)∥ ds + ∫_{t}^{T} ∑_{j=1}^{n} \frac{|q_{j}(λ, s)|e^{− \frac{λ ln λ}{G(s)} −λs}}{(G(s))^{j}} ∥F (s)∥ ds
≤|λ ln λ| ∫_{t}^{T} p(λ, s)e^{−λs} ds + ∫_{t}^{T} ∑_{j=1}^{n} |q_{j}(λ, s)|e^{−λs} ds.

Hence the third term is of exponential decay t and therefore ∫_{0}^{t} e^{−λs}f(s) ds − q(λ) ≈_{t} 0 for all 0 < t < T and hence q ∈ L_{T} (f). ♢ We proved the existence of an analytic element of the asymptotic Laplace transform defined on a post-sectorial region. In the above proof, in case that T < ∞ we can replace ln λ by T. Then q(λ) := λ ∫_{0}^{T} e^{−λt}(1 − e^{\frac{−λT}{G(t)}} )F (t) dt is analytic in the right halfplane and q ∈ L_{T} (f). J. C. Vignaux and M. Cotlar ([Vi-Co]) proved the existence of an element of the asymptotic Laplace transform which is analytic in the right halfplane also for the case T = ∞. Since their proof is not constructive and since for our purposes the existence in a post-sectorial region is fully sufficient, we will not go into details.

Now that we know that the asymptotic Laplace transform always exists, we will check the operative rules (a) − (f) mentioned above. Proposition 1.5. Let f ∈ L_{loc}^{1}([0, ∞); X). If f̂ exists, then f̂ ∈ L(f). Proof. We recall the following facts from Laplace-transform theory (see [Ba-Ne1]): If f̂:= lim_{T →∞} ∫_{0}^{T} e^{−λt}f(t) dt exists for some λ_{0} ∈ C, then f̂(λ) exists for all λ ∈ C with Re(λ) > Reλ_{0}. Moreover, f̂ exists if and only if the antiderivative F (t) := ∫_{0}^{t}f(s) ds of f is exponentially bounded. Since

f̂(λ) = ∫_{0}^{t}e^{−λs}f(s) ds + r_{t}(λ),

where

\begin{matrix}r_{t}(λ) := & ∫_{t}^{∞} e^{−λs}f(s) ds = −e^{−λt}F (t) + λ ∫_{t}^{∞} e^{−λs}F (s) ds \\ = & e^{−λt} (−F (t) + λ∫_{0}^{∞} e^{−λt}F (s + t) ds) , \end{matrix}

it follows that r_{t} ≈_{t} 0. Thus f̂ ∈ L(f). ♢ Proposition 1.6 (Linearity). L_{T} : L_{loc}^{1}([0, T); X) → A(P, X)/{0}_{T} is a linear operator. Proof. Let f ∈ L_{loc}^{1}([0, T); X) and p, q ∈ L_{T} (f). Then p ≈_{t} q for all t ∈ [0, T); i.e., lim sup_{λ→∞} \frac{1}{λ} ln ∥p^{(n)}(λ) − q^{(n)}(λ)∥ ≤ −t for all 0 ≤ t < T. Hence p ≈_{T} q. Thus, the asymptotic Laplace transform L_{T} is a singlevalued operator from L_{loc}^{1}([0, T); X) into A(P, X). It is easy to see that L_{T} is linear; i.e., that αL_{T} (f) + L_{T} (g) = L_{T} (αf + g) for all f, g ∈ L_{loc}^{1}[0, T); X) and all α ∈ C. ♢ It is often more convenient to work in Banach spaces L^{1}([0, t]; X) for 0 < t < T instead of L_{loc}^{1}([0, T); X). The following proposition provides the framework that allows the deduction from properties of the Laplace transform on L^{1}[0, t] for 0 < t < T to properties of the asymptotic Laplace transforms on L_{loc}^{1}([0, T); X). Proposition 1.7. Let f ∈ L_{loc}^{1}([0, T); X). Then

L_{T} (f) = ⋂_{0≤t<T} L_{t}(f).

Proof. Let f ∈ L_{loc}^{1}([0, T); X) and q ∈ L_{T} (f). Then clearly, q ∈ ⋂_{0≤t<T} L_{t}(f). By definition, for 0 ≤ ϵ < T − t, any function p ∈ L_{t+ϵ}(f) satisfies p ≈_{t} ∫_{0}^{t}e^{−λs}f(s) ds. Thus for p ∈ ⋂_{0≤t<T} L_{t}(f) we know that p ≈_{t} ∫_{0}^{t}e^{−λs}f(s) ds for all 0 ≤ t < T. Hence, p ∈ L_{T} (f). ♢ The uniqueness property “L_{T} (f) ∩ L_{T} (g) ≠ ∅ <=> f = g” was already shown by J. C. Vignaux and M. Cotlar [Vi-Co], L. Berg [Ber], and Y. Lyubich [Ly]. We give a new proof and add a new inversion formula. Theorem 1.8 (Uniqueness and Inversion). Let f, g ∈ L_{loc}^{1}([0, T); X) for some 0 < T ≤ ∞. Then L_{T} (f) ∩ L_{T} (g) ≠ ∅ if and only if f = g. Furthermore, for any Müntz sequence sequence (β_{n})_{n∈IN} and q ∈ L_{T} (f), we have that

∫_{0}^{t}f(s) ds = lim_{k→∞} ∑_{n=1}^{N_{k}} α_{k,n}e^{tβ_{kn}}q(β_{kn}),

converging uniformly on compact subsets of [0, T), where the constants α_{k,n} and N_{k} are the same as in Theorem II.3.1. Proof. The proof is based on the Phragmén-Mikusinski inversion formula for Laplace transforms (see Theorem II.3.1). Clearly, f = g implies that L_{T} (f) = L_{T} (g).

Let q ∈ L(f) ∩ L(g). Then for all 0 < t < T there exists ω > 0 and r_{f}, r_{g} ∈ A(P, X) with r_{f}, r_{g} ∈ {0}_{t} such that

q(λ) = ∫_{0}^{t}e^{−λs}f(s) ds + r_{f}(λ) = ∫_{0}^{t}e^{−λs}g(s) ds + r_{g}(λ).

By Corollary II.3.2, the Phragmén-Mikusinski inversion formula applied to r_{f} and r_{g} yields zero on [0, t), and thus, for all 0 < s < t,

lim_{k→∞} ∑_{n=1}^{N_{k}} α_{k,n}e^{sβ_{kn}}q(β_{kj}) = ∫_{0}^{s} f(u) du = ∫_{0}^{s} g(s) ds.

Since t was chosen arbitrarily, the above equation holds for all 0 ≤ s < T and hence f = g almost everywhere on [0, T). ♢ In applications to differential and integral equations, the most essential operational property of the Laplace transform is that it transforms convolution (in particular, integration and differentiation) into multiplication. It therefore transforms linear differential and integral equations into algebraic equations. We will show next that this crucial feature extends to the asymptotic Laplace transform; i.e., the convolution k ⋆ f between an operator-valued function K and a vector-valued function f is mapped by L_{T} to the application of L_{T} (K) to L_{T} (f).

By the uniform boundedness principle, any strongly continuous operator family (K(t))_{t∈[0,T)} ⊂ L(X, Y) is uniformly bounded on compact subsets of [0, T). Thus K̂_{t}(λ) := ∫_{0}^{t}e^{−λs}K(s) ds is a bounded linear operator and λ ↦ K̂_{t}(λ) is an entire function. As in the vector-valued case we define

L_{T} (K) := {Q ∈ A(P; L(X, Y)) : Q(λ) ≈_{t} ∫_{0}^{t}e^{−(·)s}K(s) ds for all t ∈ [0, T)}. The proofs of Theorem 1.4 and Theorem 1.8 are easily extendable to asymptotic Laplace transforms of strongly continuous operator families.

Theorem 1.9 (Transform of convolutions). Let X, Y be Banach spaces, 0 < T ≤ ∞, let f ∈ C([0, T); X), and let (K(t))_{t∈[0,T)} ⊂ L(X, Y) be a strongly continuous operator family. Then

L_{T} (K ⋆ f) = L_{T} (K)L_{T} (f) + {0}_{T} .

Proof. Let 0 < t < T and let Q(λ) := ∫_{0}^{t}e^{−λs}K(s) ds + R(λ) ∈ L_{T} (K) and q(λ) := ∫_{0}^{t}e^{−λs}f(s) ds + r(λ) ∈ L_{T} (f), where R ≈_{t} 0 and r ≈_{t} 0. Define

m(λ) := R(λ)∫_{0}^{t}e^{−λs}f(s) ds + ∫_{0}^{t}e^{−λs}K(s)r(λ) ds + R(λ)r(λ).

Then

\begin{matrix}Q(λ) & q(λ) = ∫_{0}^{t}∫_{0}^{t}e^{−λ(s+u)}K(s)f(u) ds du + m(λ) \\ = & ∫_{0}^{t} ∫_{u}^{t+u} e^{−λs}K(s − u)f(u) ds du + m(λ) \\ = & ∫_{0}^{t} ∫_{0}^{s} e^{−λs}K(s − u)f(u) du ds + ∫_{t}^{2t} ∫_{s−t}^{t} e^{−λs}K(s − u)f(u) du ds + m(λ) \\ = & ∫_{0}^{t}e^{−λs}(K ⋆ f)(s) ds + e^{−λt}∫_{0}^{t}e^{−λs} ∫_{s}^{t} K(s + t − u)f(u) du ds + m(λ) \\ =: & ∫_{0}^{t}e^{−λs}(K ⋆ f)(s) ds + c(λ), \end{matrix}

where c(λ) = e^{−λt}∫_{0}^{t}e^{−λs} ∫_{s}^{t} K(s + t − u)f(u) du ds + m(λ). Since c ≈_{t} 0 it follows that L_{T} (K)L_{T} (f) ⊂ L_{T} (K ⋆ f).

Let g ∈ L_{T} (K ⋆ f). Then, for all 0 < t < T there exists d ≈_{t} 0 such that g(λ) = ∫_{0}^{t}e^{−λs}(K ⋆ f)(s) ds + d(λ). Let Q ∈ L_{T} (K), and let q ∈ L_{T} (f). As shown above,

∫_{0}^{t}e^{−λs}(K ⋆ f)(s) ds = Q(λ)q(λ) − c(λ)

for some c ≈_{t} 0. Thus g(λ) ≈_{t} Q(λ)q(λ) for all t ∈ [0, T), and therefore g(λ) ≈_{T} Q(λ)q(λ). Hence g(λ) ∈ L_{T} (K)L_{T} (f) + {0}_{T}. ♢ Corollary 1.10. Let f ∈ L_{loc}^{1}([0, T); X) be such that f′ exists a.e. and f′ ∈ L_{loc}^{1}([0, T); X). Then L_{T} (f′) = λL_{T} (f) − f(0). Proof. By Proposition 1.2 we know that \frac{1}{λ}\{0\}_{T} = \{0\}_{T}. Since (1 ⋆ f′)(t) = ∫_{0}^{t}f′(s) ds = f(t) − f(0), we obtain by the previous theorem and Proposition 1.3 that

\begin{matrix}\frac{1}{λ}L_{T} (f′) = & (\frac{1}{λ} + \{0\}_{T})L_{T} (f′) + \{0\}_{T} = L_{T} (1)L_{T} (f′) + \{0\}_{T} \\ = & L_{T} (1 ⋆ f′) = L_{T} (f − f(0)) = L_{T} (f) − \frac{f(0)}{λ} + \{0\}_{T} = L_{T} (f) − \frac{f(0)}{λ}. \end{matrix}

Hence, L_{T} (f′) = λL_{T} (f) − f(0). ♢ The main difference in our definition of asymptotic Laplace transforms to the literature is that we restrict the range of L_{T} to analytic functions on post-sectors. This yields the following operational property, corresponding to Mikusinski’s algebraic derivative (see [Mi2] pp.294), which is crucial, for example, in dealing with Laplace’s equation

(a_{2}t + b_{2})y′′(t) + (a_{1}t + b_{1})y′(t) + (a_{0}t + b_{0})y(t) = g(t).

Theorem 1.11. Let f ∈ L_{loc}^{1}([0, T); X). Then L(−tf) = (L(f))′. Proof. Let q ∈ L_{T} (−tf). Then, for all 0 < t < T, there exists r_{t} ≈_{t} 0 such that

\begin{matrix}q(λ) = & ∫_{0}^{t}e^{−λs}(−s)f(s) ds + r_{t}(λ) = \frac{d}{dλ} (∫_{0}^{t}e^{−λs}f(s) ds) + r_{t}(λ) \\ = & \frac{d}{dλ} (∫_{0}^{t}e^{−λs}f(s) ds + R_{t}(λ)) , \end{matrix}

where R_{t}(λ) := − ∫_{−λ}^{∞} r_{t}(μ) dμ satisfies R_{t} ≈_{t} 0. Thus q ∈ (L_{T} (f))′.

Now let p ∈ L_{T} (f). Then, for all 0 < t < T, there exists r_{t} ≈_{t} 0 such that p(λ) = ∫_{0}^{t}e^{−λs}f(s) ds + r_{t}(λ). Thus p′(λ) = ∫_{0}^{t}e^{−λs}(−s)f(s) ds + r′_{t}(λ). Since r′ ≈_{t} 0 it follows that p′ ∈ L(−tf). ♢ In Theorem 1.4 we have seen that the asymptotic Laplace transform exists for all f ∈ L_{loc}^{1}([0, T); X) and maps to f to analytic functions q : Ω → X, where Ω is some post-sectorial region in C. Now we ask the converse question. For which analytic functions q : Ω → X, defined on a post-sectorial region Ω, does there exist T > 0, and f ∈ L_{loc}^{1}([0, T); X) such that q ∈ L_{T} (f)? The following theorem, which is a slightly reformulated version of a result due to G. Lumer and F. Neubrander [Lu-Ne], gives a partial answer to this important representation problem. Theorem 1.12 (Complex Representation). Let q : Ω → X be analytic in a region Ω containing

Ω_{Ψ} := {λ : Re(λ) ≥ β > 0, |Im(λ)| ≤ Ψ(Re(λ))},

where Ψ is a positive, strictly increasing C^{1}-function with Ψ(r) → ∞ as r → ∞ and sup_{r≥β} \frac{Ψ′(r)}{r^{α}Ψ(r)} < ∞ for some α ≥ 0. Then the following statements hold.

(a) If there exists T > 0 such that

∥q(λ)∥ ≤ \frac{e^{−T (Ψ^{−1}(|λ|))}}{|λ|}

for all λ ∈ Ω_{Ψ}, then there exists f ∈ C([0, T); X) such that r ∈ L_{T} (f).

(b) If there exist constants c > 0, d > 1 such that

∥q(λ)∥ ≤ \frac{e^{−c(Ψ^{−1}(|λ|))^{d}}}{|λ|}

for all λ ∈ Ω_{Ψ}, then there exists f ∈ C([0, ∞); X) such that r ∈ L(f).

Proof. (a) Let Γ be the oriented boundary of the region Ω_{Ψ}; i.e., Γ = Γ_{−} ∪ Γ_{β} ∪ Γ_{+}, where

Γ_{±} := {r ± iΨ(r); β ≤ r < ∞} and Γ_{β} := {β + ir; −Ψ(β) ≤ r ≤ Ψ(r)}.

Let C_{0}, C_{1} > 0 be such that

|\frac{1 ± iΨ′(r)}{r ± iΨ(r)}| ≤ C_{0} + C_{1}r^{α}

for all λ ∈ Ω_{Ψ} and all r ≥ β. Since Ψ^{−1} is increasing it follows that

Ψ^{−1}(|r ± iΨ(r)|) ≥ Ψ^{−1}(|Ψ(r)|) = r

for all r ≥ β. This implies for λ = r ± iΨ(r) ∈ Γ_{±} the estimate

\begin{aligned}∥e^{λt}q(λ) dλ∥ & ≤ e^{rt}e^{−T Ψ^{−1}(|r±iΨ(r)|)} |\frac{1 ± iΨ′(r)}{r ± iΨ(r)}| dr \\ & ≤ (C_{0} + C_{1}r^{α})e^{−r(T −t)}dr. \end{aligned}

It follows that

f(t) := \frac{1}{2πi} ∫_{Γ} e^{λt}q(λ) dλ

is well defined and continuous for 0 ≤ t < T. Next it will be shown that

∫_{0}^{t} e^{−λs}f(s) ds = q(λ) − a_{t}(λ)

for all 0 < t < T and λ in the interior Ω_{Ψ}^{0} of Ω_{Ψ}, where

a_{t}(λ) := \frac{e^{−λt}}{2πi} ∫_{Γ} \frac{1}{μ − λ}e^{μt}q(μ) dμ.

To see this, let Γ(n) := Γ ∩ {Reλ ≤ n} and Π_{n} := {n + ir; −Ψ(n) ≤ r ≤ Ψ(r)}. Then, for all λ ∈ Ω_{Ψ}^{0},

∫_{0}^{t} e^{−λs}f(s) ds = ∫_{0}^{t} e^{−λs} \frac{1}{2πi} ∫_{Γ} e^{μs}q(μ) dμds = \frac{1}{2πi} ∫_{Γ} ∫_{0}^{t} e^{(μ−λ)s} dsq(μ) dμ

= \frac{1}{2πi} ∫_{Γ} \frac{1}{μ − λ} [e^{(μ−λ)t} − 1] q(μ) dμ

= lim_{n→∞} \frac{1}{2πi} [∫_{Γ(n)} \frac{1}{μ − λ}e^{(μ−λ)t}q(μ) dμ − ∫_{Γ(n)} \frac{1}{μ − λ}q(μ) dμ]

= lim_{n→∞} \frac{1}{2πi}[ ∫_{Γ(n)} \frac{1}{μ − λ}e^{(μ−λ)t}q(μ) dμ

+ (− ∫_{Γ(n)} + ∫_{Π_{n}} − ∫_{Π_{n}}) \frac{1}{μ − λ}q(μ) dμ]

= q(λ) + lim_{n→∞} \frac{1}{2πi}[ ∫_{Γ(n)} \frac{1}{μ − λ}e^{(μ−λ)t}q(μ) dμ

− ∫_{Π_{n}} \frac{1}{μ − λ}q(μ) dμ]

= q(λ) + \frac{1}{2πi} ∫_{Γ} \frac{1}{μ − λ}e^{(μ−λ)t}q(μ) dμ,

since, by the residue theorem,

\frac{1}{2πi} (− ∫_{Γ(n)} + ∫_{Π_{n}}) \frac{1}{μ − λ}q(μ) dμ = q(λ)

for n > Reλ, and lim_{n→∞} \frac{1}{2πi} ∫_{Π_{n}} \frac{1}{μ−λ}q(μ) dμ = 0. The last statement follows from the estimate

\begin{aligned}∥ ∫_{Π_{n}} \frac{1}{μ − λ}q(μ) dμ∥ & ≤ ∫_{−Ψ(n)}^{Ψ(n)} \frac{1}{|n + ir − λ|}e^{−T Ψ^{−1}(|n+ir|)} \frac{1}{|n + ir|} dr \\ & ≤ e^{−T Ψ^{−1}(n)} ∫_{−∞}^{∞} \frac{1}{|n + ir − λ|^{2}} dr. \end{aligned}

To show that a_{t} ≈_{t} 0, it suffices to prove the boundedness of

(\frac{d}{dλ})^{n} ∫_{Γ} \frac{1}{μ − λ}e^{μt}q(μ) dμ

for λ → ∞. Since ∫_{Γ} \frac{n!}{(μ−λ)^{n}} e^{μt}q(μ) dμ exists, we have that

(\frac{d}{dλ})^{n} ∫_{Γ_{β}} \frac{1}{μ − λ}e^{μt}q(μ) dμ = ∫_{Γ_{β}} \frac{n!}{(μ − λ)^{n}} e^{μt}q(μ) dμ → 0

as λ → ∞. Moreover,

∥ (\frac{d}{dλ})^{n} ∫_{Γ_{±}} \frac{1}{μ − λ}e^{μt}q(μ) dμ∥ = ∥ ∫_{Γ_{±}} \frac{n!}{(μ − λ)^{n}} e^{μt}q(μ) dμ∥

≤ ∫_{β}^{∞} \frac{n!}{dist(λ, Γ_{±})^{n}} e^{rt}e^{−T Ψ^{−1}(|r±iΨ(r)|)} |\frac{1 ± iΨ′(r)}{r ± Ψ(r)}| dr

≤ \frac{n!}{dist(λ, Γ_{±})^{n}} ∫_{β}^{∞}(C_{0} + C_{1}r^{α})e^{−r(T −t)} dr.

This shows that a_{t} ≈_{t} 0. Thus, q ∈ L_{T} (f).

(b) Let the path Γ and the constants C_{0}, C_{1} be as above. Then, for λ = r ± iΨ(r) ∈ Γ_{±},

\begin{aligned}∥e^{λt}q(λ) dλ∥ & ≤ e^{rt}e^{−c(Ψ^{−1}(|r±iΨ(r)|))^{d}} |\frac{1 ± iΨ′(r)}{r ± iΨ(r)}| dr \\ & ≤ (C_{0} + C_{1}r^{α})e^{rt−cr^{d}}dr. \end{aligned}

It follows that

f(t) := \frac{1}{2πi} ∫_{Γ} e^{λt}q(λ) dλ

is well defined and continuous for t ≥ 0. As in part (a) one shows that

∫_{0}^{t} e^{−λs}f(s) ds = q(λ) + \frac{e^{−λt}}{2πi} ∫_{Γ} \frac{1}{μ − λ}e^{μt}q(μ) dμ.

Since ∫_{Γ_{β}} \frac{n!}{(μ−λ)^{n}} e^{μt}q(μ) dμ → 0 as λ → ∞ and

∥ ∫_{Γ_{±}} \frac{n!}{(μ − λ)^{n}} e^{μt}q(μ) dμ∥

≤ ∫_{β}^{∞} \frac{n!}{dist(λ, Γ_{±})^{n}} e^{rt}e^{−c(Ψ^{−1}(|r±iΨ(r)|))^{d}} |\frac{1 ± iΨ′(r)}{r ± Ψ(r)}| dr

≤ \frac{n!}{dist(λ, Γ_{±})^{n}} ∫_{β}^{∞}(C_{0} + C_{1}r^{α})e^{rt−cr^{d}} dr,

it follows that q ∈ L(f). ♢ In essence, the previous theorem tells that an analytic function q defined on some post-sectorial region has an asymptotic Laplace representation q = L_{T} f for some continuous function f, if q decays sufficiently fast at infinity. Analytic functions q defined on postsectors which grow at infinity can not have a Laplace representation q = L_{T} (f) for some f ∈ L_{loc}^{1}([0, T); X). In fact, if q ∈ L_{T} (f) for some f ∈ L_{loc}^{1}([0, T); X), then there exists r_{t} ≈_{t} 0 such that q(λ) = ∫_{0}^{t}e^{−λs}f(s) ds + r_{t}(λ) = e^{−λt}∫_{0}^{t}f(s) ds + λ∫_{0}^{t}e^{−λs} ∫_{0}^{u} f(u) du ds + r_{t}(λ). Thus ∥q(λ)∥ ≤ M for all λ ∈ IR sufficiently large. As we will see in the following section, such functions have an asymptotic Laplace representation q ∈ L_{T} (f) for some generalized function f, if lim sup_{λ→∞} \frac{1}{λ} ln ∥q(λ)∥ < ∞.

III.2 Asymptotic Laplace Transforms of Generalized Functions As it was shown in Section II.4, generalized functions f ∈ \overline{C_{0}[a, ∞)}^{T_{k}} for some constant a ∈ IR and some k ∈ C[0, ∞) with 0 ∈ supp(k) form a field with respect to convolution and addition, and their vector-valued counterparts f ∈ \overline{C_{0}([a, ∞); X)}^{T_{k}} form a vector space over that field. In the last section we gave a method that assigns to any function f ∈ C[a, ∞), a ≥ 0, an equivalence class of analytic functions defined on a post-sector, namely, the asymptotic Laplace transform. Moreover, the asymptotic Laplace transform converts convolution into multiplication. It is not surprising that we can extend the asymptotic Laplace transform to all elements of the convolution field, taking the multiplicative structure as a basis for the extension. It is easy to see that for f ∈ L_{loc}^{1}[0, ∞), the asymptotic Laplace transform of the shifted function f_{α} : t ↦ \{ \begin{matrix}0 & for 0 ≤ t ≤ α \\ f(t − α) & else \end{matrix}, satisfies

L(f_{α}) = e^{−λα}L(f).

Thus the asymptotic Laplace transform of the generalized function representing the right shift operator, denoted by δ_{α}, must be L(δ_{α}) = e^{−λα}+{0}. That suggests the following definition. Definition 2.1 Let k ∈ C[0, ∞) with 0 ∈ supp(k) and f ∈ \overline{C_{0}([a, ∞); X)}^{T_{k}} for some a ∈ IR. Then k ⋆ f_{−a} ∈ C_{0}([0, ∞; X) and we define the asymptotic Laplace transform of f to be

L(f) := e^{−λa} \frac{L(k ⋆ f_{−a})}{L(k)}.

For functions f ∈ \overline{C[0, T); X)}^{T_{k}} and k ∈ C[0, T) with 0 ∈ supp(k), define

L_{T} (f) := \frac{L_{T} (k ⋆ f)}{L_{T} (k)}.

The resulting functions q ∈ L_{T} (f) are now meromorphic function defined in a postsectorial region (except at the poles). We denote the vector space of meromorphic functions with M(P; X). Notice that this vector space is a vector space over C as well as over M(P; C).

We will show first that this extension of the asymptotic Laplace transform is well defined and that the operational properties of the asymptotic Laplace transform extend to the generalized function case. Proposition 2.2 The asymptotic Laplace transform of generalized functions is well defined; i.e., generalized functions that are identified via natural embeddings yield the same asymptotic Laplace transform.

Proof. Recall from Section II.4, that we identify two generalized functions f ∈ \overline{C_{0}[a, ∞); X)}^{T_{k_{1}}} and g ∈ \overline{C_{0}[b, ∞); X)}^{T_{k_{2}}} if

k_{2} ⋆ k_{1} ⋆ f = k_{1} ⋆ k_{2} ⋆ g,

considered as functions on C(−∞, ∞) by extending them with zero to the left.

Suppose that k_{2} ⋆ k_{1} ⋆ f = k_{1} ⋆ k_{2} ⋆ g for f ∈ \overline{C_{0}[a, ∞); X)}^{T_{k_{1}}} and g ∈ \overline{C_{0}[b, ∞); X)}^{T_{k_{2}}} . Then

\begin{matrix}L(f) = & e^{−λa} \frac{L((k_{1} ⋆ f)_{−a})}{L(k_{1})} = e^{−λa} \frac{L((k_{1} ⋆ f)_{−a} ⋆ k_{2})}{L(k_{1} ⋆ k_{2})} \\ = & e^{−λ(a+b)} \frac{L((k_{1} ⋆ f ⋆ k_{2})_{−a−b})}{L(k_{1} ⋆ k_{2})} = e^{−λb} \frac{L(k_{1} ⋆ (g ⋆ k_{2})_{−b})}{L(k_{1} ⋆ k_{2})} \\ = & e^{−λb} \frac{(g ⋆ k_{2})_{−b})}{L(k_{2})} = L(g). \end{matrix}

Clearly, the same argument holds for generalized functions f ∈ \overline{C[0, T); X}^{T_{k_{1}}} and g ∈ \overline{C[0, T); X}^{T_{k_{2}}} with 0 ∈ supp(k_{1}) ∩ supp(k_{2}). ♢

Next, we show that the operative features of the asymptotic Laplace transform extend to asymptotic Laplace transforms of generalized functions. Denoting with L : V_{X}(+, ⋆) the vector space of generalized functions with values in a Banach space X as described in Section II.4, we show that L : V_{X}(+, ⋆) → M(P; X)/{0} is linear; i.e.,

(a) L(f + g) = L(f) + L(g)

(b) L(f ⋆ g) = L(f)L(g) + {0},

Proposition 2.3. The map L : V_{X}(+, ⋆) → A(P; X)/{0} is linear; i.e., L(f + g) = L(f) + L(g) and L(f ⋆ g) = L(f)L(g).

98

Proof. Let f, g ∈ V_{X}. Then f ∈ \overline{C_{0}[a, ∞); X)}^{T_{k_{1}}} and g ∈ \overline{C_{0}[b, ∞); X)}^{T_{k_{2}}} for some constants a ≤ b ∈ IR and functions k_{1}, k_{2} ∈ C[0, ∞) with 0 ∈ supp(k_{1}) ∩ supp(k_{2}). Thus, by definition, f + g = T_{k_{1}⋆k_{2}}^{−1}(k_{2} ⋆ k_{1} ⋆ f + k_{1} ⋆ k_{2} ⋆ g). Hence,

\begin{matrix}L(f + g) = & e^{−λa} \frac{L((k_{2} ⋆ k_{1} ⋆ f + k_{1} ⋆ k_{2} ⋆ g)_{a})}{L(k_{1} ⋆ k_{2})} \\ = & e^{−λa} \frac{L(k_{2})L((k_{1} ⋆ f)_{a}) + L(k_{1})L((k_{2} ⋆ g)_{a})}{L(k_{1})L(k_{2})} \\ = & e^{−λa} \frac{L((k_{1} ⋆ f)_{a})}{L(k_{1})} + e^{−λb} \frac{L((k_{2} ⋆ g)_{b})}{L(k_{2})} = L(f) + L(g). \end{matrix}

Furthermore,

\begin{matrix}L(f ⋆ g) = & e^{−λ(a+b)} \frac{L((k_{2} ⋆ k_{1} ⋆ f ⋆ k_{1} ⋆ k_{2} ⋆ g)_{a+b})}{L(k_{1} ⋆ k_{2})} \\ = & e^{−λa} \frac{L(k_{2})L((k_{1} ⋆ f)_{a})}{L(k_{2})L(k_{1})}e^{−λb} \frac{L(k_{1})L((k_{2} ⋆ g)_{b})}{L(k_{1})L(k_{2})} = L(f)L(g). \end{matrix}

♢

Since integrating is the same as convoluting with the Heaviside function χ_{[0,∞)} (which we continue to denote by 1), and since L(1 ⋆ f) = \frac{1}{λ}L(f), the inverse function is differentiation and thus

L(f′) = λL(f).

Notice the difference of the above equality to item (e) of Section III.1, where we show that L(f′) = λL(f) − f(0). At first glance this looks like a discrepancy; however, the generalized derivative of a constant function in C[0, ∞) is not zero, but a multiple of the δ-function (see Section II.1). This is not the case if we just look at the usual derivative in C^{1}[0, 1], and hence this is corrected by the term −f(0).

Thus, multiplying λ with the Laplace transform of a generalized function f ∈ \overline{C_{0}[0, ∞); X)}^{T_{k}} corresponds to differentiating, while dividing by λ is equivalent to integrating the generalized function. Example 2.4 (Laplace transform of Dirac’s δ-function). We showed in Section I.1 that the δ-function can be obtained as the second generalized derivative of f, where f(t) = t on [0, ∞). Thus, δ ⋆ 1 ⋆ 1 = f or

δ̂(λ) · \frac{1}{λ} \frac{1}{λ} = \frac{1}{λ^{2}} .

Thus δ̂(λ) = 1. Moreover, L(δ′) = λ. Recall from Section II.1 that the definition of a product between a generalized function and a continuous function was somewhat tricky; for generalized functions in C^{−1}[0, 1] it was only well defined for differentiable functions. However, we will show that for a given generalized function f, the product tf := [tf_{n}] for f = [f_{n}] is always well defined and

L(f)′ = L(−tf).

Since

(L(f))′ = (\frac{L(k ⋆ f)}{L(k)})′ = \frac{L(k ⋆ f)′L(k) − L(k ⋆ f)L(k)′}{L(k)^{2}} ,

we do not expect the sequence tf_{n} to converge in the T_{k}-topology, but in the T_{k⋆k} topology. Theorem 2.5 Let f = k − lim f_{n} ∈ \overline{C([a, ∞); X)}^{T_{k}} for some k ∈ C[0, ∞) with 0 ∈ supp(k). Then the generalized function

tf := T_{k⋆k} − lim tf_{n} ∈ \overline{C([a, ∞); X)}^{T_{k⋆k}} and L(tf) = −L(−tf).

Proof. Suppose k ⋆ f_{n} converges in C_{0}([a, ∞); X). Then, identifying f_{n} and k with their zero extensions,

\begin{matrix}(k ⋆ k ⋆ (tf_{n})) = & k ⋆ (t ↦ ∫_{−∞}^{∞}(t − s)f_{n}(t − s)k(s) ds) \\ = & k ⋆ (t ↦ t ∫_{−∞}^{∞} f_{n}(t − s)k(s) ds − ∫_{−∞}^{∞} f_{n}(t − s)sk(s) ds) \\ = & k ⋆ (t(f_{n} ⋆ k)) − k ⋆ f_{n} ⋆ (tk) \\ → & k ⋆ (t(f ⋆ k)) − k ⋆ f ⋆ (tk) ∈ C_{0}([a, ∞); X). \end{matrix}

Hence, by Theorem 1.9 and Theorem 1.11,

\begin{matrix}Ltf_{n} = & \frac{L(k ⋆ (t(f ⋆ k)) − k ⋆ f ⋆ (tk)}{L(k ⋆ k)} = \frac{−L(k)L(f ⋆ k)′ + L(k ⋆ f)L(k)′}{L(k)^{2}} \\ = & − L(f)′. \end{matrix}

♢

Equipped with this operational property we consider the following example. Example 2.6 (Singular ordinary differential equation). To demonstrate the operational method developed in the last sections we consider the problem

ty′′(t) + (t + 3)y′(t) + y(t) = 0; y(0) = 0  (1)

and determine all ω ∈ F (the field of generalized functions) that satisfy (1). Suppose there exist k ∈ C[0, ∞) with 0 ∈ supp(k) and T > 0 such that ω ∈ \overline{C[0, T)}^{T_{k}} satisfies (1); i.e., z := k ⋆ ω ∈ C_{0}[0, T) is a classical solution of the convoluted problem

k ⋆ (ty′′(t) + (t + 3)y′(t) + y(t)) = 0.

From the operational rules (a) L_{T} (ω)′ = L_{T} (−tω) and

(b) L_{T} (ω′) = λL_{T} (ω), we deduce the following familiar rules from Laplace transform theory:

(c) L_{T} (ω) + λL_{T} (ω)′ = L_{T} (ω′)′ = L_{T} (−tω′)

(d) L_{T} (ω′′) = λL_{T} (ω′) = λ^{2}L_{T} (ω)

(e) 2λL_{T} (ω) + λ^{2}L(ω)′ = L(ω′′)′ = L(−tω′′). Thus, if 0 = tω′′(t) + (t + 3)ω′(t) + ω(t), then

\begin{matrix}\{0\}_{T} = & L(tω′′(t)) + L(tω′(t) + 3L(ω′(t)) + L(ω(t)) \\ = & − 2λ(L_{T} (ω) − λ^{2}L_{T} (ω)′ − L_{T} (ω) − λL(ω)′ + 3λL_{T} (ω)′ + λL_{T} (ω) \\ = & (−λ^{2} − λ)L_{T} (ω)′ + λL_{T} (ω) \end{matrix} ;

or equivalently, L_{T} (ω)′ − \frac{1}{λ+1}L_{T} (ω) = \{0\}_{T}. Thus, the asymptotic Laplace transform reduces the second order problem to a first order problem which can be solved explicitly. Let q ∈ L_{T} (ω). Then

q′(λ) − \frac{1}{λ + 1}q(λ) = r(λ),

where r ≈_{T} 0; so (\frac{1}{λ}q(λ))′ = \frac{1}{λ+1}r(λ) =: r̃(λ), where r̃ ≈_{T} 0. This shows that there exists a constant c such that \frac{1}{λ}q(λ) = c + R̃(λ), where

R̃(λ) = − ∫_{λ}^{∞} r̃(μ) dμ ≈_{T} 0.

Then

L_{T} (ω) = c(λ + 1) + {0}_{T}

is the operational solution of (1), and ω = cL^{−1}(λ + 1) = c(δ′ + δ) for c ∈ C is the unique class of solutions in F that solves (1).

The next proposition follows immediately from the definition.

Proposition 2.7. Let q ∈ A(P; X), let k ∈ C[0, ∞) with 0 ∈ supp(k). Then the following are equivalent.

(i) There exists a generalized function f ∈ \overline{C([0, ∞); X)}^{T_{k}} such that q ∈ L(f). (ii) k̃ · q ∈ L(g) for some g ∈ C_{0}([0, ∞); X) and k̃ ∈ L(k).

Proposition 2.7 sets the stage for several representation theorems, characterizing (ii) in different ways. G. Lumer and F. Neubrander in [Lu-Ne] proved the following sufficient conditions (recall from Section III.1 that for all f ∈ C([0, a); X) there exists f̂ ∈ L_{A}(f) such that f̂ is analytic in a postsectorial region):

Theorem 2.8 (Complex Representation. Let r ∈ A(P; X) be a function. Suppose there exists k ∈ C[0, ∞) with 0 ∈ supp(k) and q ∈ A(P; X) such that q = r · k̃ ∈ A(P; X) for k̃ ∈ L(k). Suppose the post sector on which q is analytic contains

Ω_{Ψ} := {λ : Re(λ) ≥ β > 0, |Im(λ)| ≤ Ψ(Re(λ))},

where Ψ is a positive, strictly increasing C^{1}-function with Ψ(λ) → ∞ as λ → ∞, and sup_{λ≥β} \frac{Ψ′(λ)}{λ^{α}Ψ(λ)} < ∞ for some α ≥ 0. Then the following hold:

(a) If there exists T > 0 such that

∥q(λ)∥ ≤ \frac{e^{−T (Ψ^{−1}(|λ|))}}{|λ|}

for all λ ∈ Ω_{Ψ}, then there exists f ∈ \overline{C([0, T); X)}^{k} such that r ∈ L_{T} (f).

(b) If there exist c > 0, d > 1 such that

∥q(λ)∥ ≤ \frac{e^{−c(Ψ^{−1}(|λ|))^{d}}}{|λ|}

for all λ ∈ Ω_{Ψ} and for some c > 0, d > 1, then there exists f ∈ \overline{C([0, ∞); X)}^{k} such that r ∈ L(f).

Proof. The proof follows immediately from Theorem 1.6 and the definition of asymptotic Laplace transforms of generalized functions. ♢ The above results give us a gauge to measure the regularity of f. The less growth in a post-sectorial region, the more regularity we obtain for f. Especially, if f̂ grows polynomially we can multiply by \frac{1}{λ^{n}} in order to satisfy the growth requirement. But multiplying by 1/λ^{n} corresponds to integrating n-times, thus f would be in C^{−n}([0, ∞); X).

The philosophy that the growth determines the regularity is the idea behind the next theorem. J. Prüss in [Pr] has a similar characterization, although we do not need any derivatives of the analytic function. Theorem 2.9 (Complex Representation Theorem for differentiable functions). Let q : {Re(λ) > ω ≥ 0} → X be analytic in a right halfplane, let 0 < β ≤ 1, let n ∈ IN_{0}, and assume there exists a polynomial p(λ) := ∑_{i=0}^{n−1} λ^{i}x_{i} such that λ^{β+1}(λ^{n}q(λ) − p(λ)) is bounded for all λ with Re(λ) > ω. Then there exists a function f such that q(λ) = ∫_{0}^{∞} e^{−λt}f(t) dt and f ∈ C^{n,α}([0, ∞); X) for all 0 ≤ α < β.^{(1)}

(1) A function f is locally α-Hölder continuous, or f ∈ C^{0,α}([0, ∞); X), if for all Proof. Let r(λ) := λ^{n}q(λ) − p(λ). Then λ^{β+1}r(λ) = λλ^{β−α}λ^{α}r(λ) and hence, by the Complex Representation Theorem, for every α there exists a function g_{α} ∈ C_{0}([0, ∞); X) such that λ^{(β−α)}λ^{α}r(λ) = λ^{β−α}∫_{0}^{∞} e^{−λt}g_{α}(t) dt and thus

r(λ) = \frac{1}{λ^{α}} ∫_{0}^{∞} e^{−λt}g_{α}(t) dt = ∫_{0}^{∞} e^{−λt} \frac{t^{α−1}}{Γ(α)} ⋆ g_{α}(t) dt =: ∫_{0}^{∞} e^{−λt}h(t) dt

for 0 < α < β. By the uniqueness of the Laplace transform, h is independent of α; and hence h ∈ C_{0}^{0,α}([0, ∞); X) for all 0 ≤ α < β. Since

\begin{matrix}q(λ) = & \frac{1}{λ^{n}} (r(λ) + p(λ)) \\ = & ∫_{0}^{∞} e^{−λt}((\frac{t^{n−1}}{n − 1!} ⋆ h)(t) + ∑_{i=0}^{n−1} \frac{t^{i}}{i!} x_{n−1−i})dt =: ∫_{0}^{∞} e^{−λt}f(t) dt \end{matrix}

we obtain that f ∈ C^{n,α}([0, ∞); X). ♢

N there exists a constant M such that sup_{t,s∈[0,N]} ∥f(t) − f(s)∥ ≤ M|t − s|^{α}. A function f is said to be in the Hölder space C^{n,α}([0, ∞); X) if f ∈ C^{n}([0, ∞); X) and f^{(n)} ∈ C^{0,α}([0, ∞); X).

IV. The Abstract Cauchy Problem

J. Hadamard [Ha], Chapter I.

“A differential equation – whether ordinary or partial – admits of an infinite number of solutions. The older and classic point of view, concerning its integration, consisted in finding the so-called ‘general integral,’ i.e. a solution of the equation containing as many arbitrary elements (arbitrary parameters or arbitrary functions) as are necessary to represent any solution, save some exceptional ones. But, in more recent research, especially as concerns partial differential equations, this point of view had to be given up, not only because of the difficulty or impossibility of obtaining this ‘general integral,’ but, above all, because the question does not by any means consist merely in its determination. The question, as set by most applications, does not consist in finding any solution u of the differential equation, but in choosing, amongst all those possible solutions, a particular one defined by properly given accessory conditions.[...]

The true questions which actually lie before us are, therefore, the ‘boundary problems,’ each of which consists in determining an unknown function u so as to satisfy:

(1) an ‘indefinite’ partial differential equation;

(2) some ‘definite’ boundary conditions. Such a problem will be ‘correctly set’ if those accessory conditions are such as to determine one and only one solution of the indefinite equation. The simplest of boundary problems is Cauchy’s problem. It represents, for partial differential equations, the exact analog of the well-known fundamental problem in ordinary differential equations. The theory of the latter was founded by Cauchy on the following theorem: Given an ordinary differential equation, say of the second order, [...] a solution of this equation is (under proper hypotheses) determined if, for x = 0, we know the numerical values y_{0}, y′_{0} of y and \frac{dy}{dx}.[...]

Strictly, mathematically speaking, we have seen (this is Holmgren’s theorem) that one set of Cauchy’s data u_{0}, u_{1} corresponds (at most) to one solution u of \frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂y^{2}} = 0, so that, if these quantities u_{0}, u_{1} were ‘known,’ u would be determined without any possible ambiguity.

But, in any concrete application, ‘known,’ of course, signifies ‘known with a certain approximation,’ all kinds of errors being possible, provided their magnitude remains smaller than a certain quantity; and, on the other hand, we have seen that the mere replacing of the value zero for u_{1} by the (however small) value changes the solution not by very small but by very great quantities. Everything takes place, physically speaking, as if the knowledge of Cauchy’s data would not determine the unknown function. This shows how very differently things behave in this case and in those which correspond to physical questions. If a physical phenomenon were to be dependant on such an analytical problem as Cauchy’s for ∇^{2}u = 0, it would appear to us as being governed by pure chance (which, since Poincaré, has been known to consist precisely in such a discontinuity in determinism) and not obeying any law whatever.”

One of the applications of the theory of Banach space valued generalized functions and their asymptotic Laplace transforms is the study of the abstract Cauchy problem

u′(t) = Au(t); u(0) = x; t ∈ [0, a)

where x ∈ X for some Banach space X, A is a linear operator on X, and a ∈ IR^{+} ∪ {+∞}. We will show the equivalence of the solvability of the above problem with the solvability of the asymptotic characteristic equation

(λI − A)y(λ) = x + r_{a}(λ)

for some remainder term r_{a}(λ) of exponential decay a. The growth in the region of analyticity of the asymptotic local resolvent y will determine the regularity and growth of the (generalized) solution u.

IV.1 The Notion of a Generalized Solution

Recall from Chapter II, that the generalized derivative coincides with the derivative of a differentiable function u, as long as u(0) = 0, and that the generalized derivative of a constant function is the Dirac δ-function at 0. This means that we have to modify slightly the above notion of the abstract Cauchy problem in order to fit the setup of generalized derivation. We obtain an equivalent formulation of the abstract Cauchy problem if we write

(u(t) − x)′ = Au(t); u(0) = x; t ∈ [0, a).

But now the derivative symbol on the left can be understood as a generalized derivative and solving the abstract Cauchy problem in a generalized sense will mean that both sides represent the same generalized function. Convoluting both sides with 1 leads to an equivalent equation without derivation, namely

u − x = 1 ⋆ Au.  (ACP)

In order to claim that a generalized function^{(1)} u ∈ \overline{C([0, a); X)}^{new} solves (ACP), we face the problem that u(t) is no longer defined for a specific t, nor in an “almost everywhere” sense. Hence, what is Au(t)? This problem can be resolved by emphasizing the limiting processes involved. Definition 1.1 (Generalized solutions). We say that u ∈ \overline{C([0, a); X)}^{new} is a generalized solution of (ACP), if and only if there exists a sequence u_{n} ∈ C([0, a); X) such that

(a) u_{n}(t) ∈ D(A) for all t ∈ [0, a) and Au_{n} ∈ C([0, a); X),

(b) u = new − lim u_{n},

(c) u − x = new − lim 1 ⋆ Au_{n}. If A is closed and if we can choose u_{n} ∈ C([0, a); X) satisfying (a) − (c) such that u_{n} → u and Au_{n} → Au in C([0, a); X), then clearly u is a classical solution to the (1) Clearly, \overline{C([0, a); X)}^{new} is the completion of the Frechét space C([0, a); X) equipped with a new set of seminorms. abstract Cauchy problem. If we omit the requirement that Au_{n} → Au, we obtain a mild solution to the abstract Cauchy problem since 1⋆Au_{n} = A(1⋆u_{n}) → A(1⋆u) and hence u − x = A(1 ⋆ u).

Next we want to investigate how this notion of solution correlates to the notions of integrated, k-regularized and C-regularized solutions to the abstract Cauchy problem. Recall that v ∈ C([0, a); X) is a k-generalized solution of the abstract Cauchy problem if ∫_{0}^{t}v(s) ds ∈ D(A) and

v(t) = A∫_{0}^{t}v(s) ds + (k ⋆ x)(t)  (ACP_{k})

for all t ∈ [0, a). In the case that k = \frac{t^{n+1}}{n+1!} we speak of an n-times integrated solution. If we replace k(t) with a bounded operator C ∈ L(X) we obtain the notion of C-regularized solutions to the abstract Cauchy problem. See, for example, R. deLaubenfels [deL].

The following proposition is crucial in the interplay between generalized limit solutions and regularized solutions. We say that a linear operator A on a Banach space X is (X_{A} ↪ X)-closed, if there exists an auxiliary Banach space X_{A} such that D(A) ⊂ X_{A} ↪ X and the graph of A is closed in X_{A} × X (see Appendix). We denote with [D(A)] the Banach space

(D(A) : ∥x∥_{[D(A)]} := ∥x∥_{X_{A}} + ∥Ax∥_{X}) .

Proposition 1.2. Let A be a (X_{A} ↪ X)-closed, linear operator on a Banach space X and let a > 0. Then for all v ∈ C([0, a); X) with 1⋆v ∈ C_{0}([0, a); [D(A)]), there exists v_{n} ∈ C([0, a); [D(A)]) such that v_{n} → v in X and 1 ⋆ v_{n} → 1 ⋆ v in [D(A)].

Proof. Let v ∈ C([0, a); X_{A}) with 1 ⋆ v ∈ C_{0}([0, a); [D(A)]). For 0 < h < a/2, define

v_{h}(t) := \{ \begin{matrix}\frac{1}{h} ∫_{t}^{t+h} v(s) ds & for 0 ≤ t < a − 2h \\ \frac{1}{h} ∫_{a−2h}^{a−h} v(s) ds & else. \end{matrix}

Then, for 0 ≤ t ≤ a − 2h, we have that v_{h}(t) = \frac{1⋆v(t+h)−1⋆v(t)}{h} and thus v_{h} ∈ C([0, a); [D(A)]). Clearly, v_{h} → v in C([0, a); X) as h → 0. Since

(1 ⋆ v_{h})(t) = \frac{1}{h}∫_{0}^{t} ∫_{s}^{s+h} v(r) dr ds = \frac{1}{h}∫_{0}^{t} ∫_{0}^{h} v(r + s) dr ds

= \frac{1}{h} ∫_{0}^{h} ∫_{0}^{t}v(r + s) ds dr = \frac{1}{h} ∫_{t}^{t+h} ∫_{0}^{t}v(r − t + s) ds dr

= \frac{1}{h} ∫_{t}^{t+h} ∫_{r−t}^{r} v(s) ds dr = \frac{1}{h} ∫_{t}^{t+h}(1 ⋆ v)(r) − (1 ⋆ v)(r − t) dr

for 0 ≤ t < a − 2h, we obtain that 1 ⋆ v_{h} → 1 ⋆ v in C([0, a); [D(A)]). ♢ Recall from Chapter II the following definition. For an injective linear operator T ∈ L(C([0, a); X)), we define \overline{C([0, a); X)}^{T} to be the completion of the space C([0, a); X) equipped with the seminorms

∥f∥_{t,T} := ∥T f∥_{t} := sup_{s∈[0,t]} ∥T f(s)∥

for t ∈ (0, a). Theorem 1.3. Let A be an (X_{A} ↪ X)-closed linear operator on a Banach space X and x ∈ X. Let a > 0 and let T ∈ L(C([0, a); X)) be injective with T (1 ⋆ v) = 1 ⋆ T (v). Suppose that for all f ∈ C([0, a); [D(A)]) we have that T f ∈ C([0, a); [D(A)]) and AT f = T Af. Suppose that the image of T as an operator on C([0, a); [D(A)]) is dense in C([0, a); [D(A)]).^{(2)} Let u ∈ \overline{C([0, a); X)}^{T} with 1 ⋆ u ∈ \overline{C([0, a); X_{A})}^{T}. Then the following are equivalent:

(2) Clearly, the theorem also holds for v ∈ C_{0}([0, a); X) with the image of T being dense in C_{0}([0, a); X).

(i) u is a generalized limit solution of (ACP), (ii) T u =: v is a continuous solution of

v(t) = A ∫_{0}^{t} v(s) ds + T (1x)(t).

Proof. Let [u_{n}] := u ∈ \overline{C([0, a); X)}^{T} be a generalized limit solution of u − x = 1 ⋆ Au with u_{n} ∈ C([0, a); [D(A)]). Then

T u − T (1x) =T (T − lim 1 ⋆ Au_{n}) = T − lim T (1 ⋆ Au_{n})
= lim 1 ⋆ A(T (u_{n})) = A(1 ⋆ T (u)).

Thus, for v = T u, we obtain that v(t) = A ∫_{0}^{t} v(s) ds + T (1x)(t).

Suppose there exists v ∈ C([0, a); X) with 1 ⋆ v ∈ C([0, a); X_{A}) and

v(t) = A∫_{0}^{t}v(s) ds + T (1x)(t).

Then 1 ⋆ v ∈ C_{0}([0, a); [D(A)]). By Proposition 2.2 we know that there exist v_{n} ∈ C([0, a); [D(A)]) such that v_{n} → v in C([0, a); X) and 1 ⋆ v_{n} → 1 ⋆ v in C_{0}([0, a); [D(A)]). Since the image of T as an operator on C([0, a); [D(A)]) is dense in C([0, a); [D(A)]), there exist u_{n} ∈ C([0, a); [D(A)]) with T u_{n} − v_{n} → 0 in C([0, a); [D(A)]). Thus T u_{n} → v in C([0, a); X) and 1 ⋆ T u_{n} → 1 ⋆ v in C([0, a); [D(A)]). Hence

T (1 ⋆ Au_{n}) = A(1 ⋆ T u_{n}) → A(1 ⋆ v) = v − T (1x).

Therefore T^{−1}v =: u ∈ \overline{C([0, a); X)}^{T} is a generalized limit solution of ACP. ♢ In terms of k-regularized and C-regularized solutions, this translates into the following Corollary. For an introduction to the theory of C-regularized solutions, see, for example, R. deLaubenfels [deL], for k-regularized solutions, see, for example, I. Cioranescu and G. Lumer [Ci-Lu].

Corollary 1.4. Let A be a closed linear operator on a Banach space X, let k ∈ L_{loc}^{1}[0, a) with 0 ∈ supp(k), and let C ∈ L(X) be one-to-one. Suppose for all x ∈ D(A) we have that Cx ∈ D(A) and CAx = ACx. Suppose furthermore that C as an operator on [D(A)] has a dense image in [D(A)]. Let T_{k} : f ↦ k ⋆ f be the convolution operator, and define T_{C} : f ↦ Cf. Then

(a) u ∈ \overline{C[0, a); X)}^{T_{k}} is a generalized limit solution if and only if v := k ⋆ u is a k-regularized solution of (ACP).

(b) u ∈ \overline{C[0, a); X)}^{T_{C}} is a generalized limit solution if and only if v := Cu is a C-regularized solution of (ACP).

Next we will show that existence and uniqueness of generalized limit solutions of (ACP) for all x ∈ X implies continuous dependence of the generalized solutions of the initial data; i.e., that a Cauchy problem describes a well-posed physical phenomena if it admits unique solutions for all x ∈ X.

Proposition 1.5 Let A be a closed operator on a Banach space X, a > 0 and k ∈ L_{loc}^{1}[0, a) with 0 ∈ supp(k). If (ACP) has a unique generalized limit solution u = u(·, x) ∈ \overline{C([0, a); X)}^{T_{k}} for all x ∈ X, then (ACP) is well-posed; i.e., for all 0 < T < a there exists M_{T} > 0 such that ∥u(·, x)∥_{T_{k}} ≤ M_{T} ∥x∥ on the interval [0, T].^{(3)}

(3) Conjecture. If for all x ∈ X there exists a k ∈ L_{loc}^{1}[0, a) with 0 ∈ supp(k) such that (ACP) has a unique generalized solution u_{x} ∈ \overline{C([0, a); X)}^{T_{k}}, then there exists k_{0} ∈ L_{loc}^{1}[0, a) such that u_{x} ∈ \overline{C([0, a); X)}^{T_{k_{0}}} is a generalized solution and hence (ACP) is well-posed.

Proof. Let S(·)x := k ⋆ u(·, x). Thus, by Theorem 1.3,

S(t)x = A∫_{0}^{t}S(s)x ds + (k ⋆ 1)(t)x.

By the existence and uniqueness property, we know that S : X → C([0, a); X) is a linear operator. Furthermore, S is a closed operator, since x_{n} → x and Sx_{n} → v implies that 1 ⋆ Sx_{n} → 1 ⋆ v and, by the closedness of A, that

A(1 ⋆ v) = v − k ⋆ 1x.

By the uniqueness property, Sx = v, and hence S is a closed linear operator. By the closed graph theorem, S is bounded and hence for all 0 ≤ T < a there exists M_{T} > 0 such that

∥u(·, x)∥_{T_{k}} = ∥Sx∥ = sup_{t∈[0,T]} ∥S(t)x∥ ≤ M_{T} ∥x∥.

♢

IV.2 Existence and Uniqueness of Solutions

In this section we show the equivalence between the abstract Cauchy problem and its characteristic equation

(λI − A)ũ(λ) ≈_{T} x,  (CE)

where ũ ∈ L_{A}(u). We show that u is a generalized solution of (ACP) if and only if there exists ũ solving (CE). Thus existence and uniqueness of solutions of (ACP) is equivalent to existence and uniqueness of solutions to (CE). Furthermore, recall from Section III.2 that we can deduce the regularity of a generalized function; i.e., we can determine the regularizing functions K such that u ∈ \overline{C([0, a); X)}^{K} by the growth behaviour of its asymptotic Laplace transform in the region of analyticity.

For the sake of clarity we introduce the following notation for the asymptotic Laplace transform:

{f}_{T} := L_{T} (f).

In order to investigate the abstract Cauchy problem with our operational calculus, we have to establish how the asymptotic Laplace transform interacts with closed or relatively closed linear operators A. Recall that we define the graph space [D(A)] for a (X_{A} ↪ X)-closed operator A via

[D(A)] := (D(A) : ∥x∥_{[D(A)]} := ∥x∥_{X_{A}} + ∥Ax∥_{X}) .

By the definition of a (X_{A} ↪ X)-closed operator, this is a Banach space (for closed operators, X_{A} = X). Clearly, we need some conditions on f such that {Af} = A{f}. It is well known (see, for example, Hille-Phillips [Hi-Ph], Theorem 3.7.12) that if A is a closed operator, f ∈ L^{1}[(0, t]; X) with f(s) ∈ D(A) almost everywhere, and if Af ∈ L^{1}([0, t]; X), that then ∫_{0}^{t}f(s) ds ∈ D(A) and

∫_{0}^{t}Af(s) ds = A∫_{0}^{t}f(s) ds.

However, this does not imply that {Af} = A{f}, since the remainder term on the right hand side might not be of exponential decay anymore. However, f ∈ L^{1}[(0, t]; X) and Af ∈ L^{1}([0, t]; X) implies that f ∈ L_{loc}^{1}([0, T); [D(A)]). Hence, we can consider the asymptotic Laplace transform

{f}_{[D(A)]} := {q ∈ A(P; [D(A)]) : q ≈_{t} ∫_{0}^{t}e^{−λs}f(s) ds for all 0 < t < T }

for f ∈ L_{loc}^{1}([0, T); [D(A)]) and, as we will see in Proposition 2.1., we can obtain

{Af} = {Af}_{[D(A)]} + {0} = A{f}_{[D(A)]} + {0}.

On the other hand, let f ∈ L_{loc}^{1}([0, T); X). If f̃(λ) ∈ D(A) for f̃ ∈ {f} and Af̃ ∈ {g} for some g ∈ L_{loc}^{1}([0, T); X), we can deduce by the Phragmén inversion formula and the closedness of A that A1 ⋆ f = 1 ⋆ g and hence Af = g.

Reformulating these statements for the general setting of relatively closed operators yields the following proposition. Proposition 2.1. Let A be a (X_{A} ↪ X)-closed operator. Suppose f ∈ L_{loc}^{1}([0, T); X_{A}) with f(t) ∈ D(A) almost everywhere. Let g ∈ L_{loc}^{1}([0, T); X). Then

(a) {f}_{X_{A}} + {0} = {f}.

(b) Af = g implies that f ∈ L_{loc}^{1}([0, T); [D(A)]), {f} = {f}_{[D(A)]} + {0}, and

{g} = {Af} = A{f}_{[D(A)]} + {0}.

(c) for f̃ ∈ {f}_{X_{A}} and f̃(λ) ∈ D(A), the condition that Af̃ ∈ {g} implies that Af = g.

Proof. (a) Let f̃ ∈ {f}_{X_{A}}. Then for all t ∈ [0, T) there exists a remainder term r_{t} with r_{t} ≈_{t} 0 in X_{A} and f̃(λ) = ∫_{0}^{t}e^{−λt}f(t) dt + r_{t}(λ). Since X_{A} ↪ X we know that r_{t} ≈_{t} 0 in X and thus f̃ ∈ {f}. Thus for all q ∈ {f}, f̃ ≈_{t} q in X. Hence {f}_{X_{A}} + {0} = {f}.

(b) The proof follows Theorem 3.7.11 of [Hi-Ph]. Since f is locally integrable in X_{A}, for each ϵ > 0 and t ∈ [0, T), there exists a countable subdivision S_{n} of [0, t] of disjoint sets S_{n} of positive measure, such that the measure of ⋃ S_{n} is t, and ∥f(s)−f_{ϵ}(s)∥_{X_{A}} ≤ ϵ for all s ∈ ⋃ S_{n} and a countably valued step function f_{ϵ} that is constant on S_{n}. Since Af is locally integrable in X, we obtain another subdivision for Af. Take a common refinement S̃_{n}. Then ∥f(s)−f_{ϵ}(s)∥_{X_{A}} +∥Af(s)−g(s)∥ ≤ 2ϵ for all s ∈ ⋃ S̃_{n} for countably valued step functions f_{ϵ}, g_{ϵ} that are constant on S̃_{n}. Without loss of generality we can assume that for all n, f_{ϵ}(s) = f(s) for some s ∈ S̃_{n}. Let g̃(s) := Af_{ϵ}(s). Then ∥g̃(s) − g(s)∥ ≤ 2ϵ and ∥f(s) − f_{ϵ}(s)∥_{[D(A)]} ≤ 3ϵ for all s ∈ ⋃ S̃_{n}. Thus, the restriction of f onto [0, t] is the uniform limit almost everywhere of countably valued functions, and is therefore strongly measurable in [D(A)]. Since

∫_{0}^{t}∥f(s)∥_{[D(A)]} ds = ∫_{0}^{t}∥f(s)∥_{X_{A}} ds + ∫_{0}^{t}∥Af(s)∥ ds < ∞,

we obtain that f ∈ L^{1}([0, t]; [D(A)]) and therefore f ∈ L_{loc}^{1}([0, T); [D(A)]).

Suppose Af = g. Let f̃ ∈ {f}_{[D(A)]}. Then for all t ∈ [0, T) there exists a function r_{t} such that r_{t} ≈_{t} 0 in [D(A)], and

Af̃(λ) = A∫_{0}^{t}e^{−λt}f(s) ds + Ar_{t}(λ) = ∫_{0}^{t}e^{−λt}Af(s) ds + Ar_{t}(λ).

Note that r_{t} ≈_{t} 0 and Ar_{t} ≈_{t} 0 in X. Therefore f̃ ∈ {f}, Af̃ ∈ {Af}, and thus {f}_{[D(A)]} + {0} = {f} and A{f}_{[D(A)]} + {0} = {Af}.

(c) Suppose that f̃ ∈ {f}_{X_{A}}, f̃(λ) ∈ D(A), and Af̃ ∈ {g}. Then

Af̃(λ) = A(∫_{0}^{t}e^{−λs}f(s) ds + r_{t}(λ)) = ∫_{0}^{t}e^{−λs}g(s) ds + q_{t}(λ),

for some r_{t} ≈_{t} q_{t} ≈_{t} 0. Since by the generalized Phragmén inversion (Theorem II.3.1)

∫_{0}^{s} f(r) dr = X_{A} − lim_{k→∞} ∑_{k=1}^{N_{k}} a_{k,n}e^{sβ_{kn}} f̃(β_{kn})

and

∫_{0}^{s} g(r) dr = lim_{k→∞} ∑_{k=1}^{N_{k}} a_{k,n}e^{sβ_{kn}} Ãf(β_{kn}),

the relative closedness of A implies that A ∫_{0}^{s} f(r) dr = ∫_{0}^{s} g(r) dr. Again, by the relative closedness of A, Af(s) = g(s) for all s ∈ [0, t] and all t ∈ [0, T). Thus Af = g. ♢

With these operational rules at hand, we can now easily prove the following theorem.

Theorem 2.2. Let A be a (X_{A} ↪ X)-closed, linear operator on a Banach space X. Let T > 0, let k ∈ C[0, T) with 0 ∈ supp(k), and let v ∈ L_{loc}^{1}([0, T); X) with 1 ⋆ v ∈ L_{loc}^{1}([0, T); X_{A}). Then the following are equivalent.

(i) v is a solution of (ACP_{k}); i.e., ∫_{0}^{t}v(s) ds ∈ D(A) and v(t) = A∫_{0}^{t}v(s) ds + ∫_{0}^{t}k(s)x ds.

(ii) There exists ṽ ∈ \{v\} with λ ↦ \frac{ṽ(λ)}{λ} ∈ \{1 ⋆ v\}_{X_{A}} and k̃ ∈ \{k\} such that \frac{ṽ}{k̃} solves (CE); i.e., ṽ(λ) ∈ D(A) and (λI − A)\frac{ṽ(λ)}{k̃(λ)} ≈_{T} x.

Proof. Suppose (i) holds. Then v(t) = A∫_{0}^{t}v(s) ds+k⋆x(t), and by Proposition 2.1, 1 ⋆ v ∈ L_{loc}^{1}([0, T); [D(A)]). Thus

{v} = {A1 ⋆ v} + {k ⋆ x} = A{1 ⋆ v}_{[D(A)]} + {k ⋆ x} + {0}.

Let q ∈ \{1 ⋆ v\}_{[D(A)]} ⊂ \{1 ⋆ v\}_{X} = \frac{\{v\}}{λ}. Hence q(λ) = \frac{ṽ(λ)}{λ} for some ṽ ∈ \{v\} and

A\frac{ṽ(λ)}{λ} + k̃(λ)\frac{x}{λ} ∈ \{v\} = ṽ(λ) + \{0\}.

Thus (λ − A)\frac{ṽ(λ)}{k̃(λ)} ≈_{T} x and thus (ii) holds.

Suppose (ii) holds. Then

A\frac{ṽ(λ)}{λ} ∈ ṽ(λ) + k̃(λ)\frac{x}{λ} + \{0\} ⊂ \{v − k ⋆ x\}.

By Proposition 2.1 (c), (1 ⋆ v)(t) ∈ D(A) for all 0 ≤ t < T and

A∫_{0}^{t}v(s) ds = v(t) − ∫_{0}^{t}k(s)x ds.
♢

The above theorem yields the following extension of the celebrated Lyubich uniqueness theorem (see [Ly] or [Pa],Thm. IV.1.2). Corollary 2.3 (Lyubich). Let A be a closed linear operator on a Banach space X. Let a > 0 and k ∈ L_{loc}^{1}[0, a) with 0 ∈ supp(k). Suppose there exists a Müntz sequence (β_{n})_{n∈IN} ⊂ IR^{+} such that β_{n} is in the resolvent set of A for all n ∈ IN and

lim sup_{n→∞} \frac{1}{β_{n}} ln ∥R(β_{n}, A)∥ ≤ 0.

Then every k-generalized solution to (ACP) is unique.

Proof. Suppose (β_{n} − A)\frac{ṽ(β_{n})}{k̃(β_{n})} = q(β_{n}) ≈_{a} 0 for some k̃ ∈ \{k\} and some ṽ ∈ A(P; X). Then

∥\frac{ṽ(β_{n})}{k̃(β_{n})}∥ ≤ ∥R(β_{n}, A)∥∥q(β_{n})∥ ≈_{a} 0.

Since 0 ∈ supp(k), there exists a Müntz subsequence β_{n_{j}}, such that

lim_{j→∞} \frac{1}{β_{n_{j}}} ln ∥k̃(β_{n_{j}})∥ ≤ 0.

Hence ∥ṽ(β_{n_{j}})∥ ≈_{a} 0, implying that any regularized solution v is zero. ♢ The condition that the resolvent exists at points of a Müntz sequence and is of subexponential growth along those points, is also a necessary condition if we require existence of solutions for all initial data. Theorem 2.4. Let A be a closed linear operator on a Banach space X. Let k ∈ L_{loc}^{1}[0, a) with 0 ∈ supp(k). Suppose (ACP) has a unique generalized limit solution u = u(·, x) ∈ \overline{C([0, a); X)}^{T_{k}} for all x ∈ X. Then every Müntz sequence (β_{n})_{n∈IN} has a subsequence (β_{n_{j}})_{j∈IN} such that β_{n_{j}} is in the resolvent set of A and

lim sup_{j→∞} \frac{1}{β_{n_{j}}} ln ∥R(β_{n_{j}}, A)∥ ≤ 0.

Proof. By Proposition 1.5 there exists a strongly continuous operator family (S(t))_{t∈[0,a)} such that

S(t)x = A∫_{0}^{t}S(s)x ds + (k ⋆ 1)(t)x.

Furthermore, let 0 < T < a. Let S̃(λ)x := ∫_{0}^{T} e^{−λt}S(t)x dt.

Since (1 ⋆ S)(t) ∈ L(X, [D(A)]) for all 0 ≤ t < a, we obtain that S̃(λ) ∈ L(X, [D(A)]) and (S̃(λ))_{λ∈C} is a strongly analytic operator family. Let k̃ ∈ {k}_{T}. Then, by Theorem 2.2,

(λ − A)S̃(λ)x = k̃(λ)x + r_{x}(λ),

where r_{x} ≈_{T} 0 for all x ∈ X. Furthermore, r_{(·)}(λ) ∈ L(X). Let 0 < ϵ < T/2. Since r_{x} ≈_{T} 0, there exists M_{x} such that

e^{λ(T −ϵ)}∥r_{x}(λ)∥ ≤ M_{x}

for all λ > 0. By the uniform boundedness principle there exists a constant M such that r(λ) : x ↦ r_{x}(λ) satisfies

∥r(λ)∥ ≤ Me^{−λ(T −ϵ)}.

Let (β_{n})_{n∈IN} be a Müntz sequence. By Theorem II.3.6 there exists a subsequence (β_{n_{j}})_{j∈IN} such that lim_{j→∞} \frac{1}{β_{n_{j}}} ln |k̃(β_{n_{j}})| = 0 and thus there exists a constant J such that

|k̃(β_{n_{j}})| ≥ e^{−β_{n_{j}} ϵ} > 2∥r(β_{n_{j}})∥

for all j > J. Then ∥\frac{r(β_{n_{j}})}{k̃(β_{n_{j}})}∥ < 1/2. Thus Id + \frac{r(β_{n_{j}})}{k̃(β_{n_{j}})} is an invertible operator that maps onto X and

∥ (Id + \frac{r(β_{n_{j}})}{k̃(β_{n_{j}})})^{−1} ∥ = ∥ ∑_{i=0}^{∞}(−1)^{i} (\frac{r(β_{n_{j}})}{k̃(β_{n_{j}})})^{i} ∥ ≤ 2.

Since k̃(β_{n_{j}}) ≠ 0 is scalar-valued, the operator

(β_{n_{j}} − A)S̃(β_{n_{j}}) = k̃(β_{n_{j}})(Id + \frac{r(β_{n_{j}})}{k̃(β_{n_{j}})})

is a continuously invertible operator. Let

Q(β_{n_{j}}) := (k̃(β_{n_{j}})(Id + \frac{r(β_{n_{j}})}{k̃(β_{n_{j}})}))^{−1}

and define

R(β_{n_{j}}) := S̃(β_{n_{j}})Q(β_{n_{j}}).

Then clearly, ∥Q(β_{n_{j}})∥ ≤ _{\frac{2}{k̂(β_{n_{j}})}} and (β_{n_{j}} − A)R(β_{n_{j}}) = Id.

We show next that (β_{n_{j}} − A) is one-to-one. Suppose β_{n_{j}}x = Ax for some x ∈ D(A). Then

(e^{β_{n_{j}} t} − 1)x = A\frac{1}{β_{n_{j}}} (e^{β_{n_{j}} t} − 1)x = A∫_{0}^{t}e^{β_{n_{j}} s}x ds.

Since the solutions to ACP are unique, we deduce that S(t)x = (k ⋆ e^{β_{n_{j}} (·)})(t)x. Thus S̃(λ)x is a scalar multiple of x for all λ ∈ C. Hence,

0 = (β_{n_{j}} − A)S̃(β_{n_{j}})x = k̃(β_{n_{j}})(x + \frac{r(β_{n_{j}})x}{k̃(β_{n_{j}})}).

Therefore, ∥ − x∥ = ∥\frac{r(β_{n_{j}})x}{k̃(β_{n_{j}})}∥ ≤ ∥x∥/2, and thus x = 0.

Hence (β_{n_{j}} − A) is one-to-one and thus R(β_{n_{j}}) = R(β_{n_{j}}, A) is the resolvent of A at β_{n_{j}}.

Since ∥Q(β_{n_{j}})∥ ≤ 2/|k̃(β_{n_{j}})|, we know that

lim sup_{j→∞} \frac{1}{β_{n_{j}}} ln ∥Q(β_{n_{j}})∥ = 0

and hence the same holds for R(β_{n_{j}}) = S̃(β_{n_{j}})Q(β_{n_{j}}). ♢

References

[Ad-We] W. A. Adkins and S. H. Weintraub: Algebra. An Approach via Module Theory. Graduate Texts in Mathematics 136, Springer-Verlag, New York, 1992.

[Ba-Ne1] B. Bäumer and F. Neubrander: Laplace transform methods for evolution equations. Conference del Seminario di Matematica dell’ Universitá di Bari 259 (1994), 27–60.

[Ba-Ne2] B. Bäumer and F. Neubrander: Existence and uniqueness of solutions of ordinary linear differential equations in Banach spaces. Seminar Notes in Functional Analysis and PDE’s, Department of Mathematics, Louisiana State University, 1994.

[Bar] J. Barros-Neto: An Intoduction to the Theory of Distributions. Pure and Applied Mathematics 14, Marcel Dekker, Inc. New York, 1973.

[Bel] E. J. Beltrami: Some alternative approaches to distributions. SIAM Rev. 5 (1963), 351–357.

[Ber] L. Berg: Einführung in die Operatorenrechnung. VEB Deutscher Verlag der Wissenschaften, Berlin, 1965.

[Ben] Ph. Bénilan: Solutions faibles d’équations d’évolution dans un espace de Banach réflexif. Séminaire sur les semi-group, Publ. Math. Orsay, 1970–71.

[Be-Cr-Pa] Ph. Bénilan, M. Crandall, and A. Pazy: ”Bonnes solutions” d’un probléme d’évolution semi-linéaire. C. R. Acad. Sci. Paris 306 (1988), 527–530.

[Bo] S. Bochner: Lectures on Fourier Integrals. Annals of Mathematics Studies 42, Princton University Press, Princeton, NJ, 1959.

[Ca] S. R. Caradus: Semiclosed operators. Pacific J. Math. 44 (1973), 75–79.

[Ci-Lu] I. Cioranescu and G. Lumer: Problèmes d’évolution régularisés par un noyau général K(t). Formule de Duhamel, prolongements, théorèmes de génération. C. R. Acad. Sci. Paris Sr. I Math. 319 (1994), 1273–1278.

[Co] J. L. B. Cooper: Heaviside and the operational calculus. Math. Gaz. 36 (1952), 5–19.

[DaP-Gr] G. Da Prato and P. Grisvard: On extrapolation spaces. Rend. Accad. Naz. Lincei 72 (1982), 330–332.

[De] M. Deakin: Laplace transforms for superexponential functions. J. Austral. Math. Soc. Ser. B 36 (1994),17–25.

[De2] M. Deakin: The Ascendancy of the Laplace transform and how it came about. Arch. Hist. Exact Sci. 44 (1992), 265–286.

[deL] R. deLaubenfels: Existence Families, Functional Calculi and Evolution Equations. Lecture Notes in Mathematics 1570, Springer-Verlag, Berlin, 1994.

[Di] W. A. Ditkin: On the theory of operational calculus. Dokl. Akad. Nauk SSSR 123 (1958), 395–396.

[Di-Pr2] W. A. Ditkin and A. P. Prudnikov: Integral Transforms and Operational Calculus. Pergamon Press, Oxford-Edinburgh-New York, 1965.

[Do1-3] G. Doetsch: Handbuch der Laplace Transformation. Vol. I-III, Birkhäuser Verlag, Basel-Stuttgart, 1950-1956.

[Ev] L. C. Evans: Partial Differential Equations. Berkeley Mathematics Lecture Notes, Center for Pure and Applied Mathematics at the Department of Mathematics, University of California, Berkeley, 1994.

[Fo] C. Foias: Approximation des opérateurs de J. Mikusinski par des fonctions continues. Studia Mathematica 21 (1961), 73–74.

[Ge-Sh] I. M. Gelfand and G. E. Shilov:Generalized Functions. vols. 1-3, Academic Press Inc., New York, 1964.

[Go] J. A. Goldstein: Semigroups of Linear Operators and Applications. Oxford University Press, 1985.

[Ha] J. Hadamard: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, 1923.

[He1-3] O. Heaviside: Electromagnetic Theory. Vol. I-III, London, 1893-1899.

[Hi-Ph] E. Hille and R.S. Phillips: Functional Analysis and Semigroups. Amer. Math. Soc. Colloquium Publ. 31, Providence, Rhode Island, 1957.

[Ke] W. Kecs: The Convolution Product, D. Reidel Publishing Company, Boston, 1982.

[Ki-Sk] W. Kierat and K. Skórnik: A remark on the Foias theorem on convolution. Bulletin of the Polish Academy of Sciences 34 (1986), 15–17.

[Ko] H. Komatsu: Operational calculus and semi-groups of operators. Functional Analysis and Related Topics, 1991 (Kyoto), Lecture Notes in Math. 1540, Springer Verlag, Berlin, 1993, 213–234.

[Li] M. J. Lighthill: Fourier Analysis and Generalized Functions. Cambridge University Press, New York, 1958.

[Lu-Ne] G. Lumer and F. Neubrander: Asymptotic Laplace transforms. Preprint

[Ly] Y. I. Lyubich: The classical and local Laplace transformation in an abstract Cauchy problem. Usepi Math. Nauk 21 (1966), 3–51.

[Mi] J. Mikusinski: Sur la méthode de généralisation de Laurant Schwartz et sur la convergence faible. Fundamenta math. 35 (1948), 235–239.

[Mi2] J. Mikusinski: Operational Calculus. v. 1-2, Pergamon Press, 2nd edition, 1987.

[Mi3] J. Mikusinski: On the Foias theorem on convolution. Bulletin of the Polish Academy of Sciences 33 (1985), 285–288.

[Mi-Si] J. Mikusinski and R. Sikorski: The Elementary Theory of Distributions. Rozprawy Matemat. 12, Warszawa,1957.

[Na1] R. Nagel: Sobolev spaces and semigroups. Semesterbericht Funktionalanalysis, Sommersemester 1983, 1–19.

[Na2] R. Nagel: Extrapolation spaces for semigroups. Preprint.

[Pa] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44, Springer-Verlag, New York-Berlin, 1983.

[Pr] J. Prüss: Evolutionary Integral Equations and Applications. Monographs in Mathematics 87, Birkhäuser Verlag, Basel, 1993

[Re] R. Remmert: Theory of Complex Functions. Graduate Texts in Mathematics 122, Springer-Verlag, New York, 1991.

[Ru] W. Rudin: Functional Analysis. International Series in Pure and Applied Mathematics, 2d ed., McGraw-Hill, Inc., New York, 1991.

[Sa] N. Sauer: Linear evolution equations in two Banach spaces. Proc. Royal Soc. Edinburgh, ser. A 91 (1982), 287–303.

[Scha] H. H. Schaefer: Topological Vector Spaces. The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1966.

[Scha2] H. H. Schaefer: Banach lattices and Positive Operators. Springer-Verlag, Berlin Heidelberg New York, 1974.

[Schw] L. Schwartz: Théorie des Distributions. Vol. I and II, Actualités Scientifiques et Industrielles, Hermann & Cie, Paris, 1957, 1959.

[Sk] K. Skornik: On the Foias theorem on convolution of continuous functions. In Complex Analysis and Applications ’85, Bulgar. Acad. Sci., Sofia, 1986, 604–608.

[Te] G. Temple: The theory of generalized functions. Proc. Roy. Soc. (London), ser. A 228 (1955), 175–190.

[Ti] E. C. Titchmarsh: The zeros of certain integral functions. Proceedings of the London Mathematical Society 25 (1926), 283–302.

[Tr] F. Treves: Topological Vector Spaces, Distributions, and Kernels. Academic Press, New York, 1964.

[Ve] A. Verrusio: Abstracte Hölderräume und Maximale Regularität. Dissertation, Tübingen, 1997.

[Vi] J. C. Vignaux: Sugli integrali di Laplace asintotici, Atti Accad. naz. Lincei, Rend. Cl. Sci. fis. mat. (6) 29 (1939), 345–400.

[Vi2] J. C. Vignaux: Asymptotic series and integrals. Univ. Nac. La Plata. Publ. Fac. Ci. Fisicomat.(2) 3(14) (1944), 401–412.

[Vi-Co] J. C. Vignaux and M. Cotlar: Asymptotic Laplace-Stieltjes integrals. Univ. Nac. La Plata. Publ. Fac. Ci. Fisicomat.(2) 3(14) (1944), 345–400.

[Wa] T. Walther: Abstrakte Sobolevräume und ihre Anwendungen auf die Störungstheorie für Generatoren von C_{0}-Halbgruppen. Dissertation, Tübingen, 1986.

[Wi] D. V. Widder: The Laplace Transform. Princeton Mathematical Series v. 6, Princeton University Press, Princeton, N. J., 1941.

[Yo] K. Yosida: Operational Calculus: A Theory of Hyperfunctions. Applied Mathematical Sciences 55, Springer-Verlag, New York-Berlin, 1984.

[Ze] A. H. Zemanian: Distribution Theory and Transform Analysis, an Introduction to Generalized Functions, with Applications. Dover Publications, 1965.

Appendix: Relatively Closed Operators

The appendix is part of a preprint of F. Neubrander and the author ([Ba-Ne2]), and modified only slightly.

Some initial and/or boundary value problems for partial differential equations in function spaces on I × Ω (I interval, Ω ⊂ IR^{N}) lead to an equation

u′(ξ) = Au(ξ), u(0) = x, ξ ∈ [0, ξ_{0}],

in which the operator A is not closed. Since much of the significant information on the problem is contained in the space X and in the domain D(A), it is not always desirable to change the norm (by changing X) or the domain (by switching to the closure of A, if possible). As already remarked by Agmon and Nirenberg, in these cases it is ‘convenient’ to consider the graph of A not in X × X but in an auxiliary space X_{A} × X, where D(A) ⊂ X_{A} ↪ X. We will show in this section that for a large class of linear operators A ⊂ X × X there exists an auxiliary Banach space X_{A} with D(A) ⊂ X_{A} ↪ X such that the graph of A is closed in X_{A} × X. Such operators will be called relatively closed with respect to X_{A} or also (X_{A} ↪ X)-closed. An operator A is called relatively closed if it is relatively closed with respect to some X_{A}.

The main result is Theorem A.4 below. There it is shown that the class of relatively closed operators with respect to a fixed Banach space Y is invariant under compositions, additions, and limits. In particular, compositions, additions, and limits of closed operators are relatively closed.

It is important to notice that a relatively closed, linear operator commutes with the Bochner and Stieltjes integral for sufficiently regular functions. We recall the following classical result (see 3.3 and 3.7 in [Hi-Ph]): Let A be (X_{A} ↪ X)- closed, let u : [a, b] → D(A) be Bochner integrable in X_{A}, and let Au : [a, b] → X be Bochner integrable in X. Then ∫_{a}^{b} u(t) dt ∈ D(A) and ∫_{a}^{b} Au(t) dt = A ∫_{a}^{b} u(t)dt.

Examples. (1) A natural class of examples of relatively closed operators are sums and compositions of operators A, B which are closed in X×X. In general, the sum S := A+B with domain D(S) = D(A)∩D(B) and the composition C := BA with domain D(C) = {x ∈ D(A) : Ax ∈ D(B)} will not be closed or closable in X ×X. However, both S and C are closed in [D(A)]×X, where [D(A)] denotes the Banach space D(A) endowed with the graph norm. Since D(A) ⊂ [D(A)] ↪ X, sums and compositions of closed operators are relatively closed.

(2) We remark that the sum S and composition C can be relatively closed even if the operators A, B are not closed themselves. As example take a pair of jointly closed operators A, B on a Banach space X; i.e., D(A) ∩ D(B) ∋ x_{n} → x, Ax_{n} → y_{1}, and Bx_{n} → y_{2} implies that x ∈ D(A) ∩ D(B), Ax = y_{1}, and Bx = y_{2} (see also [Sa]). Then X_{S} := [D(A)∩D(B)] with norm ∥x∥_{X_{S}} := ∥x∥+∥Ax∥+∥Bx∥ is a Banach space and D(S) ⊂ X_{S} ↪ X. It is easy to see that the sum S is closed in X_{S} × X.

(3) There are operators which are not closable in X × X, but are relatively closed. As example, let A be the first derivative on X = C[0, 1] with maximal domain and let B be the bounded operator Bf := f(0)g, where 0 ≠ g ∈ X. As seen above, the composition Cf := BAf = f′(0)g with domain D(C) = D(A) = C^{1}[0, 1] is relatively closed. However, since there is a sequence f_{n} ∈ D(C) with f_{n} → 0 and f′_{n}(0) = 1, it follows from Cf_{n} = f′_{n}(0)g = g ≠ 0 that C is not closable; i.e., the closure C[0, 1] × Cg of the graph of C in C[0, 1] × C[0, 1] is not the graph of a single-valued operator. Because the multivalued closure of C does not contain any information about the original operator and because closedness is absolutely necessary for most operations, it is necessary to consider the graph as a subset of X_{A} × X.

(4) In the example above one might take the domain D(C_{max}) := {f ∈ X : f′(0) ex.} instead of C^{1}[0, 1] and define bounded operators on X by

A_{t}f := \frac{f(t) − f(0)}{t}g.

Then, for each f ∈ D(C_{max}) one has that A_{t}f → C_{max}f as t → 0. Because C_{max} is the pointwise limit of bounded operators, it follows from Theorem 2.4 below that C_{max} is relatively closed with respect to

X_{C_{max}} := \{f ∈ C[0, 1] : ∥f∥_{C_{max}} := ∥f∥ + sup_{t∈[0,1]} ∥\frac{f(t) − f(0)}{t}∥ < ∞\}.

(5) Operators with “small” domains might not be relatively closed. Consider the identity operator I on X = C[0, 1] with the polynomials P as its domain. Clearly, I is closable. Assume it would be relatively closed. Then there exists a Banach space X_{I} such that the graph G = {(p, p) : p ∈ P} of I is closed in X_{I} ×X.

Thus G is a complete metric space. However, G is also the union of countably many finite-dimensional subspaces and is thus of first category. By Baire’s theorem, complete metric spaces are of second category, which is a contradiction. Thus, the operator I with domain P is not relatively closed. ♢

In the following proposition we collect some continuity properties of relatively closed operators and their restrictions to some Banach space Y ↪ X. The straightforward proofs are ommitted.

Proposition A.1. Let A be (X_{A} ↪ X)-closed and Y ↪ X. The following hold.

(a) Equipped with the graph norm, the domain

[D(A)] := (D(A), ∥x∥_{A} := ∥x∥_{X_{A}} + ∥Ax∥_{X})

is a Banach space and A ∈ L([D(A)], X).

(b) Let A_{Y} be the restriction of A to Y (i.e. D(A_{Y}) = D(A) ∩ Y). Then

[D(A_{Y})] := (D(A_{Y}), ∥x∥_{A_{Y}} := ∥x∥_{X_{A}} + ∥x∥_{Y} + ∥Ax∥_{X})

is a Banach space and A_{Y} ∈ L([D(A_{Y})], X).

(c) Let A|_{Y} be the restriction of A in Y; i.e., D(A|_{Y}) := {x ∈ D(A) : Ax ∈ Y }.

Then

[D(A|_{Y})] := (D(A|_{Y}), ∥x∥_{A|_{Y}} := ∥x∥_{X_{A}} + ∥Ax∥_{Y})

is a Banach space, A|_{Y} is closed in X_{A} × Y and hence A|_{Y} ∈ L([D(A|_{Y})], Y).

The following statement about relatively closed operators is an immediate consequence of the proposition above. Further properties of this type can be found in [Ca].

Corollary A.2. An operator A ⊂ X × X is relatively closed if and only if there exists a Banach space Y such that D(A) ⊂ Y ↪ X and A ∈ L(Y, X). Moreover, if A is closed in X_{A} × X, where D(A) ⊂ X_{A} ↪ X, then the restriction A|_{X_{A}} is closed in X_{A} × X_{A}. ♢

In applications, the following weak formulation of relative closedness is helpful.

Proposition A.3. Let D(A) ⊂ X_{A} ↪ X. The following are equivalent.

(i) A is closed in X_{A} × X. If x_{n} ∈ D(A), x_{n} → x in X_{A}, and Ax_{n}→y in X, then x ∈ D(A) and Ax = y.

(ii) If x_{n} ∈ D(A), x_{n} → x in X_{A} and Ax_{n}→y weakly in X, then x ∈ D(A) and Ax = y.

Proof. Clearly, (ii) implies (i). Assume that (i) holds and that x_{i} → x in X_{A} and y_{i} := Ax_{i} → y weakly in X. Let V_{n} := \{x_{i} : ∥x_{i} − x∥_{X_{A}} ≤ \frac{1}{n}\} and consider y_{i} = Ax_{i} for x_{i} ∈ V_{n}. By Mazur’s Theorem there exists a finite convex combination ỹ_{n} := ∑(a_{i}y_{i}) = A ∑(a_{i}x_{i}) such that ∥ỹ_{n} − y∥ ≤ \frac{1}{n}. Define x̃_{n} := ∑(a_{i}x_{i}). Then ∥x̃_{n} − x∥_{X_{A}} ≤ \frac{1}{n} and ∥Ax̃_{n} − y∥ ≤ \frac{1}{n}. Thus there exists a sequence (x̃_{n}) ⊂ D(A) with x̃_{n} → x in X_{A} and Ax̃_{n} → y. By (i), x ∈ D(A) and Ax = y. ♢ Most linear operators that appear in applications can be decomposed into sums, products and/or limits of relatively closed operators. As we will see in the following, such operators are always relatively closed.

Notice that if {A_{α}}_{α∈I} is a family of (X_{A_{α}} ↪ X)-closed operators such that Y ↪ X_{A_{α}} for all α ∈ I, then the restrictions of A_{α} to Y are (Y ↪ X)-closed.

Theorem A.4. Finite sums and compositions of relatively closed operators are relatively closed. Also, limits, infinite sums and integrals of operators which are all relatively closed with respect to some Y ↪ X are relatively closed.

Proof. 1) \underline{Finite Sums.} Let A := ∑_{n=0}^{N} A_{n} with D(A) := ⋂_{0≤n≤N} D(A_{n}). If the operators A_{n} are (X_{A_{n}} ↪ X)-closed, then A is (X_{A} ↪ X)-closed, where

X_{A} := X_{A_{N}} ∩ ⋂_{0≤n≤N−1}[D(A_{n})] , ∥x∥_{X_{A}} := sup_{0≤n≤N} ∥x∥_{X_{An}} + sup_{0≤n≤N−1} ∥A_{n}x∥.

2) \underline{Finite Products.} Let C_{n} := A_{n} ⋯ A_{0} for n ∈ IN_{0} with D(C_{0}) := D(A_{0}) and D(C_{n}) := {x ∈ D(C_{n−1}) : C_{n−1}x ∈ D(A_{n})} for n > 0. If the operators A_{n} are (X_{n} ↪ X)-closed, then C_{n} is (X_{C_{n}} ↪ X)-closed, where

X_{C_{n}} := [D(C_{n−1}|_{X_{An}})] , ∥x∥_{C_{n}} := ∥x∥_{X_{A_{0}}} + sup_{1≤i≤n} ∥C_{i−1}x∥_{X_{A_{i}}} .

3) \underline{Limits.} Let Ax := lim_{t→0} A_{t}x with D(A) := \{x ∈ ⋂_{t∈I} D(A_{t}) : lim_{t→0} A_{t}x exists }. Assume that the operators A_{t} are (Y ↪ X)-closed for all t ∈ (0, 1]. Let

X_{A} := {x ∈ ⋂_{t∈I}[D(A_{t})] : ∥x∥_{X_{A}} := ∥x∥_{Y} + sup_{t∈I} ∥A_{t}x∥ < ∞}.

Then X_{A} is a Banach space and D(A) ⊂ X_{A} ↪ X. Suppose D(A) ∋ x_{n} → x in X_{A}. We show first that x ∈ D(A). Let ϵ > 0. Choose n_{0} such that ∥A_{t}x−A_{t}x_{n}∥ < ϵ/3 for all n ≥ n_{0} and all t ∈ (0, 1]. Then

∥A_{t}x − A_{s}x∥ ≤ ∥A_{t}x − A_{t}x_{n_{0}}∥ + ∥A_{t}x_{n_{0}} − A_{s}x_{n_{0}}∥ + ∥A_{s}x_{n_{0}} − A_{s}x∥ < ϵ
for all t, s < t_{0} for some t_{0} ∈ (0, 1]. Hence x ∈ D(A) and

∥Ax − Ax_{n}∥ = lim_{t→0} ∥A_{t}x − A_{t}x_{n}∥ ≤ ϵ/3

for all n > n_{0}. Thus D(A) is a closed subspace of X_{A} and A is a bounded linear operator from D(A) into X.

4) \underline{Infinite Sums.} Let Ax := ∑_{n=0}^{∞} A_{n}x with D(A) := \{x ∈ ⋂_{n∈IN} D(A_{n}) : ∑_{n=0}^{∞} A_{n}x exists}. Assume that the operators A_{n} are (Y ↪ X)-closed for all n ∈ IN. Applying the limit case to partial sums we obtain that A (X_{A} ↪ X)- closed, where

X_{A} := {x ∈ ⋂_{n∈IN} D(A_{n}) : ∥x∥_{X_{A}} := ∥x∥_{Y} + sup_{n∈IN} ∥ ∑_{i=1}^{n} A_{i}x∥ < ∞}.

5) \underline{Integrals.} Let Ax := ∫ A_{t}x dt with D(A) := \{x ∈ D(A_{t}) for almost all t and A_{(·)}x ∈ L^{1}(I, X)}. Assume that the operators A_{t} are (Y ↪ X)-closed for almost all t ∈ I. Let

X_{A} := {x ∈ D(A) : ∥x∥_{X_{A}} := ∥x∥_{Y} + ∫ ∥A_{t}x∥ dt}.

To see that X_{A} is a Banach space, let (x_{i})_{i∈IN} be a Cauchy sequence in X_{A}. Then (A_{(·)}x_{i})_{i∈IN} is a Cauchy sequence in L^{1}(I, X). Hence there exists a function y_{(·)} ∈ L^{1}(I, X) such that A_{(·)}x_{i} → y_{(·)} in L^{1}(I, X). It follows that there exists a subsequence (x_{i_{k}}) for which A_{t}x_{i_{k}} → y_{t} pointwise for almost all t. Since (x_{i}) is also a Cauchy sequence in Y, there exists x such that x_{i} → x in Y. The relative closedness of A_{t} implies that x ∈ D(A_{t}) and A_{t}x = y_{t} for almost all t. Since y_{(·)} ∈ L^{1}(I, X) we obtain that A_{(·)}x ∈ L^{1}(I, X) and therefore x ∈ X_{A}. It follows from

∥x_{i} − x∥_{X_{A}} = ∥x_{i} − x∥_{Y} + ∫ ∥A_{t}x_{i} − y_{t}∥ dt

that X_{A} is a Banach space. Since

∥Ax − Ax_{i}∥ = ∥ ∫ A_{t}x − A_{t}x_{i} dt∥ ≤ ∫ ∥A_{t}(x − x_{i})∥ dt ≤ ∥x − x_{i}∥_{X_{A}}

we obtain that A is a bounded operator from X_{A} into X. ♢

Considering infinite compositions of relatively closed operators, one has to make sure that there exists a Banach space Y such that the finite composition operators are all (Y ↪ X)-closed. In the special case of the infinite composition of closed operators, Y can be chosen as follows.

Corollary A.5. Compositions of closed operators are relatively closed.

Proof. Let A_{n} be closed operators in X × X and let C_{n} := A_{n} ⋯ A_{0} for n ∈ IN. Then the restriction C_{n Y} of C_{n} to Y is (Y ↪ X)-closed, where

Y := {x ∈ ⋂_{n∈IN} D(C_{n}) : ∥x∥_{Y} := ∥x∥ + sup_{n∈IN} ∥C_{n}x∥ < ∞}.

Now it follows that C := lim_{n→∞} C_{n} is (Y ↪ X)-closed. ♢

Vita

The author was born on July 1, 1968 in Schramberg, Germany. After obtaining his Abitur in 1987, he spent two years of civil service in a youth camp. In the fall of 1989, he began his mathematical career at the University of Tübingen and passed the Vordiplom in 1991 with a major in mathematics and a minor in chemistry. He joined the exchange program between the University of Tübingen and Louisiana State University in the fall of 1992 for a one year visit and, after the year stay was over, chose to remain at LSU. He received a master of sciences degree in mathematics from LSU in the spring of 1994 and received his doctor of philosophy degree in mathematics on August 1, 1997.