Boris Bäumer, Günter Lumer, and Frank Neubrander

Convolution Kernels and Generalized Functions

1. Introduction

The operation of convolution k ⋆ f : t ↦ ∫_{−∞}^{∞} k(t − s)f(s) ds is basic in classical analysis. It generalizes derivatives δ^{(n)} ⋆ f, antiderivatives \frac{t^{n}}{n!}χ_{[0,∞)} ⋆ f and arises when two Laplace or Fourier integrals are multiplied together. The linearity and time invariance of the convolution operator (i.e., convolution commutes with translation) puts it in the center of signal processing applications and the theory of Markov processes. Moreover, convolution defines a zero divisor free multiplication of functions with appropriate support (see below). Thus, the totality of complex valued functions with support in some interval [a, ∞) extends to a field of generalized functions. This field provides a mathematically clear and technically simple basis for an “operational calculus” applicable to linear ordinary differential equations, difference equations, and integral equations (see for example J. Mikusiński [Mi1], K. Yosida [Yo], or B. Bäumer [Ba]).

Concerning the convolution transform, two results are of particular importance. One is E. C. Titchmarsh’s “Injectivity Theorem” ([Ti]), the other is C. Foiaş’ “Dense Range Theorem” ([Fo]). The aim of this paper is to extend these results to (a) Banach space valued functions f, (b) defined on intervals or point inflection invariant bounded open sets Ω ⊂ IR^{n}, and (c) scalar or semigroup valued kernels K. For Banach space valued functions f on real intervals we give outlines of two proofs of Titchmarsh’s and Foiaş’ results: an elegant proof by contradiction following the original proofs in the numerical case and a new, constructive proof yielding a sequence f_{n} of smooth functions “converging” to the generalized solution f of the convolution equation k⋆f = g (for given k, g). We extend Foiaş’ result to functions defined on regions Ω ⊂ IR^{n} that are bounded and symmetric to a point; i.e., we show that the injectivity of the convolution transform as a mapping from C(Ω; X) → C(Ω; X) implies that the closure of the image contains C_{0}(Ω; X). As example of an operator valued kernel we consider strongly continuous semigroups K and show that the convolution operator T f := ∫_{0}^{t}K(t − s)f(s) ds is injective and has dense range in C_{0}([0, a]; X).

In the following, Banach spaces are denoted by X, their dual by X^{⋆}. We denote by C_{0}([0, a]; X) the space of continuous functions f : [0, a] → X, f(0) = 0, endowed with the supremums norm; if X = C, then we write C_{0}[0, a] instead. We loosely identify any function defined on some interval with its zero extension to the whole real line. The convolution between two functions f ∈ C([a, b]; X) and k ∈ L^{1}[c, d] is therefore defined to be the convolution of their zero extensions^{(1)}.

2. Extensions to Banach space valued functions on intervals

This section contains outlines of two proofs of Banach space valued versions of statements obtained for the scalar case by E. C. Titchmarsh in 1926 [Ti] and C. Foiaş in 1961 [Fo] (see also J. Mikusiński [Mi2]).

Theorem 1 (Titchmarsh-Foiaş). Let k ∈ L_{loc}^{1}[0, ∞) and a > 0. Consider the convolution T : f ↦ k ⋆ f as an operator from C([0, a]; X) into C_{0}([0, a]; X). Then the following are equivalent.

(i) T has dense range,

(ii) T is injective,

(iii) 0 ∈ supp(k). Moreover, if 0 ∈ supp(k), then ∥f∥_{T} := ∥T f∥ defines a norm on C([0, a]; X) and T extends to an isometric isomorphism between the completion of C([0, a]; X) under that norm and C_{0}([0, a]; X).

First proof (using the scalar versions of Titchmarsh’s and Foiaş’ results).

(i) => (ii): Let T̃ : h ↦ k ⋆ h for h ∈ C[0, a]. Let g ∈ C_{0}[0, a], x ∈ X with ∥x∥ = 1 and let x^{⋆} ∈ X^{⋆} be such that ⟨x, x^{⋆}⟩ = 1. Since the range of T is dense in C_{0}([0, a]; X), there exist f_{n} ∈ C([0, a]; X) such that k⋆f_{n} → g·x and thus k⋆⟨f_{n}, x^{⋆}⟩ = ⟨k⋆f_{n}, x^{⋆}⟩ → ⟨g · x, x^{⋆}⟩ = g. Hence, the range of T̃ is dense in C_{0}[0, a].

(1)In particular, if k ∈ L_{loc}^{1}[0, ∞) and f ∈ C([0, a]; X), then (k ⋆ f)(t) = ∫_{0}^{t}k(t − s)f(s) ds for t ∈ [0, a].

Now let f ∈ C([0, a]; X). Suppose T f = 0. Choose g_{n} ∈ C[0, a] such that T g_{n} → Id. Then 0 = k ⋆ f = k ⋆ f ⋆ g_{n} = lim k ⋆ g_{n} ⋆ f = Id ⋆ f. Hence ∫_{0}^{t}(t − s)f(s) ds = 0 for all t ≥ 0 and thus f = 0.

(ii) => (iii): Suppose 0 is not in the support of k. Then k = 0 a.e. on [0, ϵ] for some ϵ > 0. Then (T f)(t) = ∫_{0}^{t−ϵ} k(t − s)f(s) ds for all t ∈ [0, a] and therefore T f = 0 for all f ∈ C([0, a]; X) with f = 0 on [0, a − ϵ], contradicting (ii).

(iii) => (ii): Suppose k ⋆ f = 0 for some f ∈ C([0, a]; X). Then 0 = ⟨k ⋆ f, x^{⋆}⟩ = k ⋆ ⟨f, x^{⋆}⟩ for all x^{⋆} ∈ X^{⋆}. By the full^{(2)} theorem of Titchmarsh, this implies that ⟨f, x^{⋆}⟩ = 0 for all x^{⋆} ∈ X^{⋆} and thus f = 0.

(ii) => (i): Let g ∈ C_{0}([0, a]; X). Let (h_{n})_{n∈IN} ⊂ C_{0}[0, a] be an approximate identity; i.e., h_{n} ≥ 0, ∫_{0}^{a} h_{n}(t) dt = 1 and h_{n} = 0 on [1/n, a]. By Foiaş’ theorem, which is proved by contradiction using the Hahn-Banach theorem on the dual of C[0, a] (see, for example, [Fo], [Ki-Sk], or [Mi2]) there exist f_{n} ∈ C[0, a] such that ∥k⋆f_{n}−h_{n}∥ → 0. This implies that

k ⋆ (f_{n} ⋆ g) = (k ⋆ f_{n}) ⋆ g → g.

It remains to show that if 0 ∈ supp(k), then the convolution operator T : f ↦ k ⋆ f defines an isometric isomorphism between the completion of the vector space C([0, a]; X) with norm ∥f∥_{T} := ∥T f∥ and C_{0}([0, a]; X) (with the usual supremums norm). This follows from the following proposition, whose elementary proof is omitted (see [Ba], Theorem 2.1 or [Na]).

Proposition 1. Let Z, Y be Banach spaces and let T ∈ L(Z, Y) be injective with dense range in Y. Let \overline{Z}^{T} be the completion of the normed vector space Z^{T} := (Z, ∥·∥_{T}), where ∥z∥_{T} := ∥T z∥. Then T extends to an isometric isomorphism between \overline{Z}^{T} and Y. ♢

(2)E. C. Titchmarsh proved that f, g ∈ C[0, ∞) and f ⋆ g = 0 on [0, a] implies that f = 0 on [0, a_{1}] and g = 0 on [0, a_{2}] for some a_{1}, a_{2} ≥ 0 with a_{1} + a_{2} ≥ a. The proof (see, for example, [Ti] or [Mi1])

is slightly more complicated than the well known ones for the half-line.

Second proof (by approximations): The implications (i) => (ii) and (ii) => (iii) are proved as above. The proof of (iii) => (i) is based on the following new inversion formula for Laplace transforms

Let (β_{n}) be a Müntz sequence; i.e., 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)

For all n ∈ IN, choose N_{n} ∈ IN such that ∑_{j=1}^{N_{n}} \frac{1}{β_{nj}} ≥ a and define

α_{n,i} := \frac{1}{e} ∏_{\begin{matrix}j=1 \\ j≠i \end{matrix}}^{N_{n}} \frac{β_{nj}}{β_{nj} − β_{ni}} e^{− \frac{β_{ni}}{β_{nj}}} .

Then |α_{n,i}| ≤ e^{2β_{ni}/nδ} and for all f ∈ L^{1}([0, a]; X) and q(λ) = ∫_{0}^{a} e^{−λt}f(t) dt one has

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

where the limit is uniform for t ∈ [0, a]. For a proof of the “Phragmén-Mikusiński” inversion formula (1) see [Ba].

Now let g ∈ C_{0}([0, a]; X) and 0 < ϵ < a. Then there exists ϵ′ > 0 and g_{ϵ} ∈ C_{0}^{1}([0, a]; X) with ∥g_{ϵ} − g∥ ≤ ϵ and supp(g_{ϵ}) ⊂ [ϵ′, a]. Thus, it suffices to show that there exists a sequence f_{n} such that k ⋆ f_{n} → g_{ϵ}. Define ĝ′_{ϵ}(λ) := ∫_{0}^{a} e^{−λt}g′_{ϵ}(t) dt and k̂(λ) := ∫_{0}^{a} e^{−λt}k(t) dt. Choose an integer Müntz sequence \binom{}{β̃_{n}}_{n∈IN} ⊂ IN. Drop all elements for which |k̂(β̃_{n})| ≤ e^{−β̃_{n}ϵ′/2} and call the obtained sequence (β_{n})_{n∈IN}. The dropped elements cannot form a Müntz sequence (γ_{n})_{n∈IN}, because otherwise the Phragmén-Mikusiński inversion (1) would yield

|∫_{0}^{t}k(s) ds| = | lim_{n→∞} ∑_{i=1}^{N_{n}} α_{n,i}e^{γ_{ni}t}k̂(γ_{ni})| ≤ lim_{n→∞} ∑_{i=1}^{N_{n}} |α_{n,i}|e^{γ_{ni}t}|k̂(γ_{ni})|
≤ lim_{n→∞} ∑_{i=1}^{N_{n}} e^{2γ_{ni}/n}e^{γ_{ni}t}e^{−γ_{ni}2ϵ/3}) ≤ lim_{n→∞} ∑_{i=1}^{N_{n}} e^{2ni(1/n+t−ϵ′/2)} = 0

for all t ∈ [0, ϵ′/2) contradicting 0 ∈ supp(k). Thus ∑_{i=1}^{∞} \frac{1}{γ_{i}} < ∞, and therefore (β_{n})_{n∈IN} is still a Müntz sequence satisfying |\frac{1}{k̂(β_{n})}| < e^{β_{n}ϵ′/2}.

Now consider the sum of exponential functions

f_{n}(t) := ∑_{i=1}^{N_{n}} α_{n,i} \frac{ĝ′_{ϵ}(β_{ni})}{k̂(β_{ni})}e^{β_{ni}t},  (2)

where α_{n,i} and N_{n} are as above. Then

(k ⋆ f_{n})(t) = ∑_{i=1}^{N_{n}} α_{n,i} \frac{ĝ′_{ϵ}(β_{ni})}{k̂(β_{ni})}∫_{0}^{t}k(s)e^{β_{ni}(t−s)} ds
= ∑_{i=1}^{N_{n}} α_{n,i} \frac{ĝ′_{ϵ}(β_{ni})}{k̂(β_{ni})}e^{β_{ni}t}∫_{0}^{t}e^{−β_{ni}s}k(s) ds
= ∑_{i=1}^{N_{n}} α_{n,i} \frac{ĝ′_{ϵ}(β_{ni})}{k̂(β_{ni})}e^{β_{ni}t} ∫_{0}^{a} e^{−β_{ni}s}k(s) ds

− ∑_{i=1}^{N_{n}} α_{n,i} \frac{ĝ′_{ϵ}(β_{ni})}{k̂(β_{ni})}e^{β_{ni}t} ∫_{t}^{a} e^{−β_{ni}s}k(s) ds

= ∑_{i=1}^{N_{n}} α_{n,i}e^{β_{ni}t}ĝ′_{ϵ}(β_{ni}) − ∑_{i=1}^{N_{n}} α_{n,i} \frac{ĝ′_{ϵ}(β_{ni})}{k̂(β_{ni})}e^{β_{ni}t} ∫_{t}^{a} e^{−β_{ni}s}k(s) ds.

By the Phragmén-Mikusiński inversion (1), the first term converges uniformly for t ∈ [0, a] to g_{ϵ}. We conclude the proof by showing that the second term converges to zero. This follows from the fact that there exists a constant C such that, as n → ∞,

∥ ∑_{i=1}^{N_{n}} α_{n,i} \frac{ĝ′_{ϵ}(β_{ni})}{k̂(β_{ni})}e^{β_{ni}t} ∫_{t}^{a} e^{−β_{ni}s}k(s) ds∥ ≤ ∑_{i=1}^{∞} e^{2β_{ni}/n} \frac{Ce^{−ϵ′β_{ni}}}{e^{−β_{ni}ϵ′/2}} e^{β_{ni}t} ∫_{t}^{a} e^{−β_{ni}s}∥k(s)∥ ds
≤ ∑_{i=1}^{∞} Ce^{2β_{ni}/n−β_{ni}ϵ′/2+β_{ni}t−β_{ni}t}
≤ C ∑_{i=1}^{∞} e^{β_{ni}(2/n−ϵ′/2)} ≤ C ∑_{i=1}^{∞} e^{ni(2/n−ϵ′/2)} → 0.
♢

3. Remarks

Usually generalized functions are defined as functionals on a space of test functions. Starting from this definition, one sheds 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 θ give a regularizing function k such that k⋆θ is a continuous function g; or, at least formally, that θ = \frac{1}{k̂(D)}g, where k̂(λ) := ∫_{0}^{∞} e^{−λt}k(t) dt and D denotes the first derivative operator.

The Titchmarsh-Foiaş result leads to an alternative approach to generalized functions suggested earlier by S. Bochner ([Bo]) and J. Mikusiński and R. Sikorski [Mi-Si]. There, one takes the (local) structure theorems as (local) definitions and defines generalized functions θ : [0, a] → X as elements of the (Banach space) completion C^{[k]}([0, a]; X) of the vector space^{(3)} C([0, a]; X) with norm ∥f∥_{T} := ∥k ⋆ f∥; i.e., generalized functions θ are regarded as equivalence classes [f_{n}] of continuous^{(4)} functions f_{n} such that k ⋆ f_{n} → g = k ⋆ θ ∈ C_{0}([0, a]; X) for some k ∈ L_{loc}^{1}[0, ∞). Note that (2) allows to construct for each θ ∈ C^{[k]}([0, a]; X) a concrete approximating sequence [f_{n}] = θ, where f_{n}(t) = ∑_{i=1}^{N_{n}} γ_{i,n}e^{β_{in}t} for some Müntz sequence (β_{n})_{n∈IN} ⊂ IN and coefficients γ_{i,n} ∈ X.

Example 1. Let k(t) = 1 for t ≥ 0. By the Titchmarsh-Foiaş theorem, the antiderivative operator (T f)(t) := ∫_{0}^{t}f(s) ds extends to an isometric isomorphism between the completion R([0, 1]; X) of the space C([0, 1]; X) with ‘mean’-norm ∥f∥_{M} := ∥T f∥ = sup_{t∈[0,1]} ∥∫_{0}^{t}f(s) ds∥ and the space C_{0}([0, 1]; X) with the supremums norm. Thus, the Banach space R([0, 1]; X) consists of (generalized) derivatives of functions in C_{0}([0, 1]; X). Let θ ∈ R([0, 1]; X). Then there exists f_{n} ∈ C([0, 1]; X) and g, g_{n} ∈ C_{0}([0, 1]; X) such that T θ = g = lim g_{n} = lim T f_{n}. Thus, one may define for 0 ≤ a < b ≤ 1

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

If C([0, 1]; X) ∋ f_{n} → θ ∈ R([0, 1]; X), then f_{n} might not converge pointwise, but f_{n} converges towards f in the mean; i.e., for all 0 ≤ a < b ≤ 1,

(3)Since the convolution is shift invariant; i.e., k⋆f = S_{−a}(k⋆S_{a}(f)) for any shift S, the Titchmarsh-Foiaş theorem also holds if the convolution operator is considered as an operator on C([a, b]; X) for a < b ∈ IR. Furthermore, using the Frechét topology, the same holds for the spaces C([a, ∞); X), a ∈ IR.

(4)One could also take C^{∞}-functions, polynomials or any dense set of continuous functions as core elements.

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

In fact, a sequence f_{n} ∈ C([0, 1]; X) is Cauchy in R([0, 1]; X) if and only if f_{n} is Cauchy in the mean. The Banach space R([0, 1]; X) strictly contains the spaces L^{p}([0, 1]; X) for 1 ≤ p ≤ ∞. In general, R([0, 1]; X) also contains non-measurable functions. To see this, take X = L^{∞}[0, 1] and f(t) := χ_{[0,t]} (the characteristic function of the interval [0, t]. Then f : [0, 1] → X is nowhere continuous. Since ∥f(s) − f(t)∥ = 1 for t ≠ s, f is not almost separably valued; by Pettis’ theorem, f is not measurable. However, f ∈ R([0, 1]; X). Moreover, f is Riemann integrable and the antiderivative (T f)(t) = F (t) := ∫_{0}^{t}f(s) ds = (t−·)χ_{[0,t]} is a Lipschitz continuous function in C_{0}([0, 1]; X) which is nowhere differentiable (in the classical sense). However, F is differentiable in the extended sense and F′ = f. In fact, any function H ∈ C_{0}([0, 1]; X) is differentiable in the extended sense; i.e., let H_{n} ∈ C_{0}^{1}([0, 1]; X) → H in C_{0}([0, 1]; X). Then H′_{n} is Cauchy in the Banach space R([0, 1]; X) and θ := [H′_{n}] is called the generalized derivative of H. This yields the following version of the fundamental theorem of calculus.

Corollary 1 (Fundamental Theorem of Calculus). Let X be a Banach space. Any H ∈ C_{0}([0, 1]; X) is differentiable in the extended sense, H′ ∈ R([0, 1]; X) is integrable and ∫_{a}^{b} H′(s) ds = H(b) − H(a) for all 0 ≤ a < b ≤ 1. Moreover, each θ ∈ R([0, 1]; X) is integrable, H(t) := ∫_{0}^{t}θ(s) ds is differentiable (in the extended sense) and H′ = θ. ♢

J. Mikusiński developed his ‘Operational Calculus’ on the basis of a field extension of the zero-divisor free convolution ring of continuous functions on [0, ∞). We obtain the same field by considering generalized functions f ∈ C^{[k]}[a, ∞) for some a ∈ IR and k ∈ C[0, ∞) with 0 ∈ supp(k). The convolution between two generalized functions f_{1} ∈ C^{[k_{1}]}[a_{1}, ∞) and f_{2} ∈ C^{[k_{2}]}[a_{2}, ∞) is defined via

f_{1} ⋆ f_{2} := T_{k_{1}⋆k_{2}}^{−1}(k_{1} ⋆ f_{1} ⋆ k_{2} ⋆ f_{2}) ∈ C^{[k_{1}⋆k_{2}]}[a_{1} + a_{2}, ∞)

and two generalized functions f_{1}, f_{2} are identified if k_{2} ⋆ k_{1} ⋆ f_{1} = k_{1} ⋆ k_{2} ⋆ f_{2} (see [Ba] for more details).

In the vector-valued case one has to face the problem that the convolution between two vector-valued functions is in general not defined. However, vector-valued functions form a torsion-free convolution module over the ring of scalar-valued functions, and therefore can be extended to a vector space over the field of generalized scalar-valued functions. This vector space can be identified with the space of generalized functions

V_{X} := {f ∈ C^{[k]}([a, ∞); X) : a ∈ IR and k ∈ C[0, ∞) with 0 ∈ supp(k)}.

If (K(t))_{t≥0} ⊂ L(X, Y) is a strongly continuous operator family, then T_{K}f := K ⋆f defines a bounded operator from C([0, a]; X) into C_{0}([0, a]; Y). In general, it does not seem to be known to date how to characterize those K for which T_{K} is injective and under which additional conditions on K the operator has the dense range property (for some preliminary results employing estimates of the growth of ∥K̂(λ)∥ as λ → ∞, see [Ba]). However, for an important class of strongly continuous operator families with 0 ∈ supp(K) one obtains the following result.

Proposition 2. Let K be a strongly continuous semigroup on a Banach space X and a > 0. Then the convolution operator T_{K} : C([0, a]; X) → C_{0}([0, a]; X) defined by T_{K}f := ∫_{0}^{t}K(t − s)f(s) ds is one-to-one and has dense range.^{(5)} Proof. Let A denote the generator of K and D(A) its domain. To see that T_{K}f is one-to-one, observe that T̂_{K}f = ∫_{0}^{∞} e^{−λt}∫_{0}^{t}K(t − s)f(s) ds dt = 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 proof of the density of the range of T_{K}f in C_{0}([0, a]; X) we show first that C_{0}^{1}([0, a]; [D(A)]) is dense in C_{0}([0, a]; X), where [D(A)] denotes the Banach space D(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 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, a]; X). Then

(5)Since for all g ∈ C([0, ∞); X), and for all a > 0, ϵ > 0 there exist f_{a} ∈ C([0, ∞); X) such that sup_{t∈[0,a]} ∥∫_{0}^{t}K(t − s)f_{a}(s) ds − g(t)∥ < ϵ, the image is also dense in the Frechét space C([0, ∞); X).

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

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

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

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

Results of this type have important applications. For example, 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 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).

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

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

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

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

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

uniformly in t ∈ [0, a], 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, a]; 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}}.

4. Extensions to Banach space valued functions defined on Ω ⊂ IR^{n}

It seems to be unknown whether the Titchmarsh-Foiaş theorem holds for domains in more than one dimension. J. Lions [Li] showed in 1951 that for scalar valued continuous functions f, g with compact^{(6)} support in IR^{n} the convex hulls of their supports add up to the convex hull of the support of the convolution product f ⋆g(t) = ∫_{IR}_{n} f(t − s)g(s) ds; i.e.,

co[supp(f ⋆ g)] = co[supp(f)] + co[supp(g)].

This implies that 0 ∈ supp(k) implies injectivity on cones or certain ‘triangles’ (see J. Mikusiński and C. Ryll-Nardzewski [Mi-Ry], J. Mikusiński [Mi1], or T. K. Boehme [Boe]). However, for cones or triangles it seems not to be known whether injectivity of the convolution operator implies the crucial dense range property. We present a theorem that gives this implication on domains which are symmetric to a point and on some subsets thereof. We do not know whether 0 ∈ supp(k) implies on these domains the injectivity of the convolution product. For many k however, the injectivity is often known by other means; for example, if k comes from a fundamental solution with respect to a differential, ultradifferential, or hyperdifferential, linear local operator L, Lk = δ, then k ⋆ μ = 0 on a region Ω gives L(k ⋆ μ) = Lk ⋆ μ = δ ⋆ μ = μ = 0 on Ω for any distribution μ that is supported by Ω.

(6)The theorem fails, if we do not require compact support. For example, take h ∈ C^{∞}[0, ∞) with supp(h) ⊂ (0, 1). Then e^{t}χ_{[0,∞)} ⋆ (h − h′) = −h.

Theorem 2. Let V ⊂ IR^{n} be a bounded open set. Let b ∈ IR^{n}. Let Ω = b + W, where W = V ∪ (−V) or W = V ∩ (−V). Let k be a distribution with compact support in IR_{+}^{n}, and T_{k} : C_{0}^{∞}(Ω) → C(\overline{Ω}) be defined via T_{k}ϕ := ∫_{IR}_{n} k(t − s)ϕ(s) ds|_{Ω}. Suppose k ⋆ μ = 0 implies that μ = 0 for all measures μ that are supported by Ω; i.e., μ(IR^{n}\Ω) = 0 and k ⋆ μ = 0 implies that μ = 0. Then the closure of the range of T_{k} contains C_{0}(Ω).

Proof. Suppose the closure of the range of T_{k} does not contain C_{0}(Ω). Then there exists a measure μ ≠ 0 supported by Ω such that 0 = ⟨k ⋆ ϕ, μ⟩ for all ϕ ∈ C_{0}^{∞}(Ω). However,

0 = ⟨k ⋆ ϕ, μ⟩ = ((k^{−} ⋆ ϕ^{−}) ⋆ μ) (0) = ((k^{−} ⋆ μ) ⋆ ϕ^{−}) (0) = ⟨k^{−} ⋆ μ, ϕ⟩,

where k^{−}(t) := k(−t). Thus k^{−} ⋆ μ = 0 on Ω. By a change of variables, k^{−} ⋆ μ = k ⋆ ν on 2b − Ω = Ω, where ν(m) := μ(2b − m) for any measurable set m. Thus ν = 0 on Ω and hence μ = 0 on 2b − Ω = Ω. ♢

Of course functions in the image of T_{k} are in general not zero on the boundary. Restricting the region Ω to Ω̃ ⊂ Ω, where (Ω\Ω̃ + supp(k)) ∩ Ω̃ = ∅ leads to a better understanding of the size of \overline{Im(T_{k})}, at least on Ω̃.

Corollary 2. Let Ω̃ ⊂ Ω with (Ω\Ω̃ + supp(k)) ∩ Ω̃ = ∅. Then

\overline{Im(T_{k})} ⊃ Z := \{f ∈ C(\overline{Ω̃}) : f(t) = 0 for t ∈ ∂(Ω)\}.

Proof. Clearly, any function ϕ̃ ∈ Z can be extended to a function ϕ ∈ C_{0}(Ω). It remains to be shown that k ⋆ ϕ̃|Ω̃ = k ⋆ ϕ|Ω̃. This holds since

supp(k ⋆ (ϕ − ϕ̃)) ∩ Ω̃ ⊂ (supp(k) + Ω\Ω̃) ∩ Ω̃ = ∅.
♢

This result can now be lifted to the vector-valued case by a partition of unity argument. Any f ∈ C_{0}(Ω; X) can be approximated by a finite sum f(t) ≈ ∑_{i=1}^{N} f_{i}(t)x_{i}, where f_{i} ∈ C_{0}(Ω) and x_{i} ∈ X. Since each f_{i} can be approximated by k ⋆ g_{i} ≈ f_{i}, k ⋆ ∑_{i=1}^{N} g_{i}(t)x_{i} → f. Thus we have proved the following theorem.

Theorem 3. Let V ⊂ IR^{n} be a bounded open set. Let b ∈ IR^{n}. Let Ω = b + W, where W = V ∪(−V) or W = V ∩(−V). Let k be a distribution with compact support in IR_{+}^{n}, and T_{k} : C_{0}^{∞}(Ω; X) → C(\overline{Ω}; X) be defined via T_{k}ϕ := ∫_{IR}_{n} k(t − s)ϕ(s) ds|_{Ω}. Suppose k ⋆ μ = 0 implies that μ = 0 for all measures μ supported by Ω. Then the closure of the range of T_{k} contains C_{0}(Ω; X). Furthermore, for Ω̃ ⊂ Ω with (Ω\Ω̃ + supp(k)) ∩ Ω̃ = ∅ we obtain that

\overline{Im(T_{k})} ⊃ Z := \{f ∈ C(\overline{Ω̃}; X) : f(t) = 0 for t ∈ ∂(Ω)\}.
♢

5. References

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[Bo] S. Bochner: Lectures on Fourier Integrals. Annals of Mathematical Studies 42, Princeton University Press, Princeton, NJ, 1959.

[Boe] T. K. Boehme: A proof of the theorem of supports. Studia Math. 81 (1985), 323-328.

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[Ki-Sk] W. Kierat and K. Skórnik: A remark on the Foiaş theorem on convolution. Bulletin Polish Academy of Sciences 34 (1986), 15–17.

[Li] J.-L. Lions: Supports de produits de composition I, C. R. Acad. Sci. Paris 232 (1951), 1530–1532.

[Mi1] J. Mikusiński: Operational Calculus, Vol. 1-2. Pergamon Press, 2nd edition, 1987.

[Mi2] J. Mikusiński: On the Foiaş theorem on convolution. Bulletin Polish Academy of Sciences 33 (1985), 285–288.

[Mi-Ry] J. Mikusiński and C. Ryll-Nardzewski: Un théorème sur le produit de composition des fonctions de plusieurs variables. Studia Math. 13 (1953), 62–68.

[Mi-Si] J. Mikusiński and R. Sikorski: The Elementary Theory of Distributions. Rozprawy Matemat. 12, Warszawa, 1957.

[Na] R. Nagel: Sobolev spaces and semigroups. Semesterbericht Funktionalanalysis, Sommersemester 1983, 1–19.

[Ti] E. C. Titchmarsh: The zeros of certain integral functions. Proceedings London Mathematical Society 25 (1926), 283–302.

[Yo] K. Yosida: Operational Calculus: A Theory of Hyperfunctions. Applied Mathematical Sciences 55, Springer-Verlag, New York – Berlin, 1984.

B. Bäumer, F. Neubrander Department of Mathematics Louisiana State University Baton Rouge, LA 70803 USA G. Lumer Department of Mathematics University of Mons-Hainaut 7000 Mons Belgium