On the Inversion of the Convolution and Laplace Transform

by Boris Bäumer

Abstract

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform f̂ of continuous or generalized functions f. The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of f̂ evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if f is continuous, it is in L^{1} if f ∈ L^{1}, and converges in an appropriate norm or Frechét topology for generalized functions f. As a corollary we obtain a new constructive inversion procedure for the convolution transform K : f ↦ k ⋆ f; i.e., for given g and k we construct a sequence of continuous functions f_{n} such that k ⋆ f_{n} → g. Keywords: Operational calculus, generalized functions, integral transforms. 1991 Mathematics Subject Classification: 44A35, 44A10, 44A40

Introduction.

C. Foiaş [Fo] showed in 1961 that the image of the convolution transform

f ↦ k ⋆ f := ∫_{0}^{t}k(t − s)f(s) ds

is dense in L^{1}[0, T] for k, f ∈ L^{1}[0, T] and 0 ∈ supp(k). This result was later lifted by K. Skórnik [Sk] to the continuous case. However, the proof is done by contradiction and is not constructive. We will answer the following question: given k ∈ L^{1}[0, T] with 0 ∈ supp(k) and g ∈ C_{0}([0, T]; X) or g ∈ L^{1}([0, T]; X), where X is a Banach space, find a sequence f_{n} ∈ C([0, T]; X) such that k ⋆ f_{n} → g uniformly, or in L^{1}-norm respectively. The sequence (f_{n}) is the convolution inverse in the sense of the operational calculus of J. Mikusiński (see [Mi] or [Ba]) and converges to a generalized function f in an appropriate norm induced by the function k.

We solve this problem by introducing a new inversion formula which can be used for the Laplace transform, the finite Laplace transform and the asymptotic Laplace transform. It is noteworthy that the inversion formula does not involve infinite integrals, infinite sums, derivatives of all orders or alike, but consists of the limit of (finite) linear combinations of exponential functions ∑_{j=1}^{N} a_{j}e^{β_{j}t}, where the coefficients a_{j} are determined by the (classical, finite, or asymptotic) Laplace transform f̂ of f, evaluated at Müntz points (β_{n}). This sequence of exponential functions converges uniformly if f is continuous, it converges in L^{1} if f ∈ L^{1}, and it converges in an appropriate norm or Frechét topology for generalized functions f.

We refer to the new inversion formula as the Phragmén-Mikusiński inversion since it generalizes the classical Phragmén-Doetsch inversion of Laplace transform theory (see [Do] or [B-N]) and since the proof was inspired by a proof of J. Mikusiński of Titchmarsh’s theorem ([Mi], Chapter VII).

1. The Phragmén-Mikusiński Inversion.

The first theorem deals with the inversion of the finite Laplace transform. As a corollary we obtain that the inversion formula is indiscriminate towards perturbations of exponential decay which in turn allows the extension to the Laplace transform and to asymptotic Laplace transforms.^{(1)}

We say that a sequence (β_{n}) ⊂ IR^{+} is a Müntz sequence,^{(2)} if for all n ∈ IN,

β_{n+1} − β_{n} ≥ 1, and ∑_{n=1}^{∞} \frac{1}{β_{n}} = ∞.

In the following let X be an arbitrary complex Banach space.

Theorem 1 (Phragmén-Mikusiński inversion on a finite interval). Let f ∈ C_{0}([0, T]; X) and q(λ) := ∫_{0}^{T} e^{−λt}f(t) dt. Let (β_{n})_{n∈IN} be a Müntz sequence and let N_{n} ∈ IN be such that ∑_{j=1}^{N_{n}} \frac{1}{β_{nj}} ≥ T. Define

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

(1) It is important to realize that the inversion procedures of Laplace transforms are invariant under certain perturbations. This defines co-classes of functions in the Laplace domain, representing the same function in the time domain.

(2) The condition β_{n+1} − β_{n} ≥ 1 can be relaxed to β_{n+1} − β_{n} ≥ δ > 0; however, then there exists a subsequence that satisfies the condition.

Then |α_{n,i}| ≤ β_{ni}e^{\frac{1+ln 2}{n} β_{ni}} and

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

where the limit is uniform on [0, S] for all 0 < S < T.

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

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

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

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

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

converges “in the mean” to the δ-function; i.e., the antiderivatives Φ_{n} := 1 ⋆ ϕ_{n} converge towards the Heaviside function H : t ↦= χ_{(0,∞)}(t) (pointwise for all t ≠ 0 and uniformly for |t| > ϵ).

(A) We show that Φ_{n}(t) = 1 − ∑_{i=1}^{N_{n}} \frac{α_{n,i}}{β_{ni}} e^{−β_{ni}t} for t ≥ −c_{n}. Consider ψ_{n} := 1 ⋆ β_{n}e^{−β_{n}(·)} ⋆ … ⋆ β_{nN_{n}}e^{−β_{nNn} (·)}. Then

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

where

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

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

Φ_{n}(t) = ψ_{n}(t + c_{n}) = 1 + ∑_{i=1}^{N_{n}} γ_{n,i}e^{−β_{ni}(t+c_{n})}

for all t ≥ −c_{n}. Since

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

we obtain that Φ_{n}(t) = 1 − ∑_{i=1}^{N_{n}} \frac{α_{n,i}}{β_{ni}} e^{−β_{ni}t} for all t ≥ −c_{n}.

(B) We show that |α_{n,i}| ≤ β_{ni}e^{\frac{1+ln 2}{n} β_{ni}}. We have that

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

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

S_{2} ≤ ∑_{j=1}^{i−1} ln \frac{β_{ni} − n(i − j)}{n(i − j)} = ∑_{j=1}^{i−1} ln \frac{β_{ni} − nj}{nj} .

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

S_{2} ≤ ∫_{0}^{i−1} ln \frac{β_{ni} − nt}{nt}dt = \frac{β_{ni}}{n} ∫_{0}^{n(i−1)/β_{ni}} 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

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

In a similar fashion we find an estimate for S_{1} + S_{3}. Since the function t ↦ −\frac{β_{ni}}{t} + ln \frac{t}{t−β_{ni}} is positive and decreasing on (β_{ni}, ∞), and since β_{ni} + n(j − i) ≤ β_{nj}, we obtain that

S_{1} + S_{3} = − ∑_{j=1}^{N_{n}} \frac{β_{ni}}{β_{nj}} + ∑_{j=i+1}^{N_{n}} ln \frac{β_{nj}}{β_{nj} − β_{ni}} < ∑_{j=i+1}^{∞} −\frac{β_{ni}}{β_{nj}} + ln \frac{β_{nj}}{β_{nj} − β_{ni}}
≤ ∑_{j=i+1}^{∞} (−\frac{β_{ni}}{β_{ni} + n(j − i)} + ln \frac{β_{ni} + n(j − i)}{n(j − i)})
= ∑_{j=1}^{∞} (−\frac{β_{ni}}{β_{ni} + nj} + ln \frac{β_{ni} + nj}{nj} ) < ∫_{0}^{∞} (−\frac{β_{ni}}{β_{ni} + nt} + ln \frac{β_{ni} + nt}{nt}) dt
=\frac{β_{ni}}{n} ∫_{0}^{∞} (−\frac{1}{1 + t} + ln \frac{1 + t}{t}) dt = \frac{β_{ni}}{n} (t ln \frac{1 + t}{t})|_{t=0}^{t=∞} = \frac{β_{ni}}{n} .

Hence, ln |\frac{α_{n,i}}{β_{ni}} | < \frac{β_{ni}(1+ln 2)}{n}.

(C) We show that Φ_{n}(t) → 1 for all t > 0. Let t > 0, and let n be such that \frac{2}{n} < \frac{t}{2}. Then

\begin{matrix}|Φ_{n}(t) − 1| ≤ & ∑_{i=1}^{N_{n}} |\frac{α_{n,i}}{β_{ni}} |e^{−β_{ni}t} ≤ ∑_{i=1}^{∞} e^{\frac{2β_{ni}}{n}} e^{−β_{ni}t} \\ ≤ & ∑_{i=1}^{∞} e^{−β_{ni}t/2} ≤ ∑_{i=1}^{∞} e^{−nit/2} = \frac{e^{−nt/2}}{1 − e^{−nt/2}} . \end{matrix}

Thus, Φ_{n}(t) → 1 as n → ∞, uniformly for t > ϵ > 0.

It follows from the definition of ϕ_{n} that ϕ_{n}, as a convolution of positive functions, is positive. Hence Φ_{n} = 1 ⋆ ϕ_{n} is positive and monotonically increasing. Therefore,

∫_{0}^{∞} e^{−t}Φ_{n}(t) dt → 1.

(D) We show that ∫_{−∞}^{∞} e^{−t}Φ_{n}(t) dt → 1, which implies – again by the positivity and monotonicity of Φ_{n} – that Φ_{n}(t) → 0 for all t < 0^{(3)} and thus uniformly for all t < −ϵ < 0. Since

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

we know that

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

By the definition of Φ_{n},

\begin{matrix}∫_{−∞}^{∞} e^{−t}Φ_{n}(t) dt = & ∫_{−c_{n}}^{∞} e^{−t}Φ_{n}(t) dt = e^{c_{n}} ∫_{0}^{∞} e^{−t}Φ_{n}(t − c_{n}) dt \\ = & e^{∑{Nn \atop i=1} 1/β_{ni}} ∏_{i=1}^{N_{n}} \frac{β_{ni}}{1 + β_{ni}} = ∏_{i=1}^{N_{n}} \frac{β_{ni}}{1 + β_{ni}} e^{1/β_{ni}}. \end{matrix}

Thus,

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

(3) Suppose there exists t < 0 such that Φ_{n}(t) does not converge to 0. Since ∫_{t}^{0} e^{−s}Φ_{n}(s) ds ≥ Φ_{n}(t)(e^{−t} − 1), it would follow that ∫_{−∞}^{0} e^{−t}Φ_{n}(s) ds does not converge to 0 either.

(E) Finally, we have the tools necessary to show convergence of the Phragmén-Mikusiński inversion. We know that

\begin{matrix}∥f(t) & − ∑_{i=1}^{N_{n}} α_{n,i}r(β_{ni})e^{β_{ni}t}∥ = ∥f(t) − ∫_{0}^{T} ∑_{i=1}^{N_{n}} α_{n,i}e^{β_{ni}t}e^{−β_{ni}s}f(s) ds∥ \\ = & ∥f(t) − ∫_{0}^{T} ϕ_{n}(s − t)f(s) ds∥. \end{matrix}  (1)

Let ϵ > 0. Choose δ > 0 such that ∥f(t) − f(s)∥ < ϵ for |t − s| < 2δ, and choose n_{0} such that Φ_{n}(−δ) + 1 − Φ_{n}(δ) < ϵ for all n > n_{0}. Then, for t ∈ [δ, T − δ],

∥f(t) − ∫_{0}^{T} ϕ_{n}(s − t)f(s) ds∥ ≤ ∫_{0}^{t−δ} ϕ_{n}(s − t)∥f(s)∥ ds + ∫_{t+δ}^{T} ϕ_{n}(s − t)∥f(s)∥ ds
+ ∥f(t) − ∫_{t−δ}^{t+δ} ϕ_{n}(s − t)f(t) ds∥ + ∫_{t−δ}^{t+δ} ϕ_{n}(s − t)∥f(t) − f(s)∥ ds
≤ ∥f∥ ∫_{0}^{t−δ} ϕ_{n}(s − t) ds + ∥f∥ ∫_{t+δ}^{T} ϕ_{n}(s − t) ds
+ ∥f∥|1 − ∫_{t−δ}^{t+δ} ϕ_{n}(s − t) ds| + ϵ ∫_{t−δ}^{t+δ} ϕ_{n}(s − t) ds
≤ ∥f∥(Φ_{n}(−δ) − Φ_{n}(−t)) + ∥f∥(Φ_{n}(T − t) − Φ_{n}(δ))
+ ∥f∥|1 − Φ_{n}(δ) + Φ_{n}(−δ)| + ϵ(Φ_{n}(t + δ) − Φ_{n}(t − δ)) ≤ ϵ (3∥f∥ + 1).

For t ∈ [0, δ] we have the following estimate (using f(0) = 0).

\begin{matrix}∥f(t) & − ∫_{0}^{T} ϕ_{n}(s − t)f(s) ds∥ \\ ≤ & ∥f(t)∥ + ∫_{0}^{t+δ} ϕ_{n}(s − t)∥f(s)∥ ds + ∫_{t+δ}^{T} ϕ_{n}(s − t)∥f(s)∥ ds \\ ≤ & ϵ + ϵ(Φ_{n}(δ) − Φ_{n}(−t)) + ∥f∥(Φ_{n}(T − t) − Φ_{n}(δ)) ≤ ϵ(2 + ∥f∥) \end{matrix}

Thus ∑_{i=1}^{N_{n}} α_{n,i}r(β_{ni})e^{β_{ni}t} converges uniformly on [0, S] to f(t). ♢

Changing only the final argument of the proof of the previous theorem we obtain convergence in L^{1}-norm if f ∈ L^{1}. In fact, as we will see in Section 2, if f is in a space of generalized functions, then, modified slightly, the above sum of exponential functions converges in a suitable norm (which is optimal).

Corollary 2. Let (β_{n})_{n∈IN}, α_{n,i}, and N_{n} be as in Theorem 1, and let f ∈ L^{1}([0, T]; X). Then

f = lim_{n→∞} ∑_{i=1}^{N_{n}} α_{n,i}q(β_{ni})e^{β_{ni}(·)},

where the limit is in L^{1}([0, T]; X)-norm.

Proof. By (1) we have to find an estimate for ∥f(·) − ∫_{0}^{T} ϕ_{n}(s − ·)f(s) ds∥_{L^{1}}. Let ϵ > 0. Choose δ > 0 such that ∫_{δ}^{T −δ} ∥f(s + t) − f(s)∥ ds < ϵ for |t| < δ. Choose n_{0} such that Φ_{n}(−δ) + 1 − Φ_{n}(δ) < ϵ for all n > n_{0}. Extending f by zero we obtain

∥f(·) − ∫_{0}^{T} ϕ_{n}(s − ·)f(s) ds∥_{L^{1}} ≤ ∫_{0}^{T} ∥f(t)∥(1 − ∫_{t−δ}^{t+δ} ϕ_{n}(s − t) ds) dt
+ ∫_{0}^{T} ∫_{t−δ}^{t+δ} ϕ_{n}(s − t)∥f(t) − f(s)∥ ds dt
+ ∫_{0}^{T} ∫_{0}^{t−δ} ϕ_{n}(s − t)∥f(s)∥ ds dt + ∫_{0}^{T} ∫_{t+δ}^{T} ϕ_{n}(s − t)∥f(s)∥ ds dt
≤∥f∥_{L^{1}}(1 − Φ_{n}(δ) + Φ_{n}(−δ)) + ∫_{−δ}^{δ} ∫_{0}^{T} ϕ_{n}(s)∥f(t) − f(s + t)∥ dt ds
+ ∫_{0}^{T −δ} ∫_{s+δ}^{T} ϕ_{n}(s − t)∥f(s)∥ dt ds + ∫_{δ}^{T} ∫_{0}^{s−δ} ϕ_{n}(s − t)∥f(s)∥ dt ds
≤ϵ(∥f∥ + 1) + ∫_{0}^{T −δ} ∥f(s)∥(Φ_{n}(−δ) − Φ_{n}(s − T)) ds + ∫_{δ}^{T} ∥f(s)∥(Φ_{n}(s) − Φ_{n}(δ)) ds
≤ϵ(3∥f∥ + 1).
♢

In order to extend the inversion formula to Laplace or asymptotic Laplace transforms of functions defined on [0, ∞), we need the notion of exponential decay. We say a function is of exponential decay T > 0 if

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

This notion is important since, as we see next, the Phragmén-Mikusiński inversion does not register perturbations of exponential decay T; i.e., if for t ∈ [0, T),

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

for some function q, then

f(t) = lim_{n→∞} ∑_{i=1}^{N_{n}} α_{n,i}q̃(β_{ni})e^{β_{ni}t}

for all perturbed functions q̃ = q + r, where r is some perturbation of exponential decay T.

Corollary 3. Let r : (ω, ∞) → X be a function, and let (β_{n})_{n∈IN} ⊂ IR^{+} be a Müntz sequence. If r is of exponential decay T > 0, then

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

for all 0 ≤ t < T, where α_{n,i} and N_{n} 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 n_{0} such that ∥r(β_{n})∥ ≤ \frac{1}{β_{n}} e^{− \frac{2T +t}{3} β_{n}} for all n ≥ n_{0} and such that 2/n_{0} < (T − t)/3. By the previous theorem we know that |α_{n,i}| < β_{ni}e^{\frac{2}{n} β_{ni}} < β_{ni}e^{\frac{T −t}{3} β_{ni}}. Thus

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

as n → ∞, uniformly for all t ∈ [0, S] for all 0 < S < T. ♢

Asymptotic Laplace transforms appear as an extension of the usual Laplace transform to map functions which are not exponentially bounded and were first introduced by J.C. Vignaux in 1939 (see [Vi] or [Lu-Ne]). The asymptotic Laplace transform of a function f ∈ L_{loc}^{1}([0, ∞); X) is defined to be an equivalence class of analytic functions defined in a post sectorial region Ω of the complex plane with the following property:

L_{A}(f) = {r ∈ A(Ω, X) : r ≈_{T} λ ↦ ∫_{0}^{T} e^{−λt}f(s) ds for all T > 0}

where r ≈_{T} q if r − q is of exponential decay T. This set is always nonempty and the properties of the Laplace transform extend fully (see [Lu-Ne] or [Ba]). With the help of Corollaries 2 and 3 we obtain the following result.

Corollary 4. Let f ∈ C_{0}([0, ∞); X) or f ∈ L_{loc}^{1}([0, ∞); X) and f̂(λ) ∈ L_{A}(f). Let (β_{n})_{n∈IN} be a Müntz sequence. Let N_{n} be such that ∑_{i=1}^{N_{n}} \frac{1}{β_{ni}} → ∞, as n → ∞, and let α_{n,i} be as in Theorem 1. Then, for all t ≥ 0,

f = lim_{n→∞} ∑_{i=1}^{N_{n}} α_{n,i}f̂(β_{ni})e^{β_{ni}(·)},

where the limit is uniform on compact sets or taken in L_{loc}^{1}, respectively.

Proof. Let 0 < S < T, q(λ) := ∫_{0}^{T} e^{−λt}f(t) dt, and r(λ) := f̂(λ) − q(λ). 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 1, for f ∈ C_{0}([0, ∞); X), the first term converges uniformly on [0, S] to f(t). By Corollary 2, the first term converges in L^{1} for f ∈ L_{loc}^{1}([0, ∞); X). By Corollary 3, the second term converges uniformly to 0 on [0, S]. ♢

An immediate consequence is the inversion formula for the Laplace transform.

Corollary 5. Let f ∈ C_{0}([0, ∞); X) or f ∈ L_{loc}^{1}([0, ∞); X) be a Laplace transformable function and f̂(λ) = ∫_{0}^{∞} e^{−λt}f(t) dt for sufficiently large Re(λ). Let (β_{n})_{n∈IN} be a Müntz sequence. Let N_{n} be such that ∑_{i=1}^{N_{n}} \frac{1}{β_{ni}} → ∞, as n → ∞, and let α_{n,i} be as in Theorem 1. Then, for all t ≥ 0,

f = lim_{n→∞} ∑_{i=1}^{N_{n}} α_{n,i}f̂(β_{ni})e^{β_{ni}(·)},

where the limit is uniform on compact sets or taken in L_{loc}^{1}, respectively. ♢

Another consequence is the following statement characterizing the maximal interval [0, T] on which a function can vanish in terms of the growth of its asymptotic Laplace transform evaluated at Müntz points. It will furthermore allow us to avoid singularities that might appear in the Laplace transform of generalized functions (see below).

Theorem 6. Let 0 ≤ T and let f ∈ L_{loc}^{1}([0, ∞); X) with f̂ ∈ L_{A}(f). Then the following are equivalent.^{(4)}

(i) Every Müntz sequence (β_{n})_{n∈IN} satisfies

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

(ii) For every Müntz sequence (β_{n})_{n∈IN} there exists a Müntz subsequence (β_{n_{k}})_{k∈IN} satis-

fying

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

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

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.

Proof. We show first that (i) => (ii) => (iii) => (iv) => (i) and then (iv) <=> (v). Suppose (i) holds, and let (β_{n})_{n∈IN} be a Müntz 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 (γ_{n}^{ϵ}) satisfies

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

Since (i) is assumed to hold, the dropped sequence (γ_{n}^{ϵ}) can not satisfy the Müntz condition. Thus ∑ \frac{1}{γ_{n}^{ϵ}} < ∞, and hence (β_{n}^{ϵ})_{n∈IN} is still a Müntz sequence. 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 end up with a subsequence having the properties stated in (ii).

(4) G. Doetsch ([Do], 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_{n→∞} \frac{1}{β_{n}} ln ∥f̂(β_{n})∥ = −T is replaced by lim sup_{n→∞} \frac{1}{β_{n}} ln ∥f̂(β_{n})∥ ≤ −T.

Clearly (ii) implies (iii). Suppose (iii) holds. In the case that T > 0, we combine Corollary 3 with Corollary 2 and obtain that ∫_{0}^{t} f(s) ds = 0 for all t ∈ [0, T). Now let T ≥ 0 and suppose that f = 0 almost everywhere on [0, T + ϵ]. However, by the definition of asymptotic Laplace transforms, there exists a remainder function r ≈_{T +ϵ} 0 such that f̂(λ) = ∫_{0}^{T +ϵ} e^{−λt}f(t) dt + r(λ) = r(λ), contradicting (iii). Thus (iv) holds.

Suppose (iv) holds. Then lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ ≤ −T. Suppose

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

Then Corollary 2 and Corollary 3 imply that ∫_{0}^{t}f(s) ds = 0 for t ∈ [0, T + ϵ), contradicting T ∈ supp(f). Thus (i) holds. Furthermore, suppose lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ < −T. This would contradict (i) and therefore (v) has to hold.

Suppose (v) holds. Thus f vanishes on [0, T]. Suppose f vanishes on [0, T + ϵ]. Then lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥ ≤ −T − ϵ, contradicting (v). Thus (iv) holds. ♢

As a Corollary we obtain a short and elegant proof of Titchmarsh’s theorem, following an idea of J. Mikusiński for the scalar valued case [Mi].

Corollary 7 (E. C. Titchmarsh’s Theorem). Let 0 < T, and let k ∈ L_{loc}^{1}[0, ∞) and f ∈ L_{loc}^{1}([0, ∞); X). Then k ⋆ f = 0 on [0, T] implies that there exist x_{1}, x_{2} ≥ 0 with x_{1} + x_{2} ≥ T s.t. k = 0 a.e. on [0, x_{1}] and f = 0 a.e. on [0, x_{2}].

Proof. Let k̂ ∈ L_{A}(k) and f̂ ∈ L_{A}(f). Let x_{1} := lim sup_{λ→∞} \frac{1}{λ} ln |k̂(λ)| and x_{2} := lim sup_{λ→∞} \frac{1}{λ} ln ∥f̂(λ)∥. By Theorem 5, k and f are 0 a.e. on [0, x_{1}] and on [0, x_{2}] respectively. Furthermore, by taking subsequences, there exists a Müntz sequence (β_{n})_{n∈IN} such that x_{1} = − lim_{n→∞} \frac{1}{β_{n}} ln |k̂(β_{n})| and x_{2} = − lim_{n→∞} \frac{1}{β_{n}} ln ∥f̂(β_{n})∥. Since

lim_{n→∞} \frac{1}{β_{n}} ln ∥k̂(β_{n})f̂(β_{n})∥ = lim_{n→∞} \frac{1}{β_{n}} ln |k̂(β_{n})| + lim_{n→∞} \frac{1}{β_{n}} ln ∥f̂(β_{n})∥ = −x_{1} − x_{2}

we obtain that x_{1} + x_{2} ≥ T. ♢ 2. The Inversion of the Convolution Transform

J. Mikusiński introduced generalized functions as elements in the quotient field of continuous functions with convolution as multiplication. Into this purely algebraic approach he inserted a topology. A sequence of generalized functions \frac{f_{n}}{g_{n}} is said to converge to \frac{f}{g} if there exists a k ∈ C[0, ∞) such that k ⋆ \frac{f_{n}}{g_{n}} ∈ C[0, ∞) for all n and k ⋆ \frac{f_{n}}{g_{n}} → k ⋆ \frac{f}{g}. See [Mi] for details.

We use this topology to introduce Banach spaces (Frechét spaces, Fréchet lattices, etc.) of generalized functions. We define generalized functions g : [0, ∞) → X as elements of the completion C^{[k]}([0, ∞); X) of the vector space C([0, ∞); X) equipped with seminorms

∥f∥_{n} := ∥k ⋆ f∥_{n} = sup_{t∈[0,n]} ∥∫_{0}^{t}k(t − s)f(s) ds∥,

where k ∈ L_{loc}^{1}[0, ∞] with 0 ∈ supp(k) (See [Ba] or [B-L-N]). Similarly, we define the spaces L^{1,[k]}([0, ∞); X) as completions, measuring a function by the L^{1}-norm of its convolution image. Hereby, the following theorem is crucial (see also [Fo], [Ti], or [B-L-N]).

Theorem 8 (Titchmarsh–Foiaş). Let k ∈ L^{1}[0, T] and consider the convolution K : f ↦ k ⋆ f as an operator from C([0, T]; X) into C_{0}([0, T]; X) or as an element of L(L^{1}([0, T]; X)). Then the following are equivalent.

(i) 0 ∈ supp(k) (ii) K is injective. (iii) The range of K is dense. Moreover, in that case, K extends to an isometric isomorphism between C^{[k]}([0, T]; X) and C_{0}([0, T]; X) as well as between L^{1,[k]}([0, T]; X) and L^{1}([0, T]; X).

Notice that in Corollary 7 we proved the equivalence of (i) and (ii). In Theorem 9 we will not only prove the equivalence of (i) and (iii), but give an approximating sequence and thus provide an inversion for the convolution transform.

With the above definition, a generalized function g can be regarded as an equivalence class [g_{n}]_{k} of continuous (or L^{1}) functions g_{n} such that k ⋆ g_{n} → f =: k ⋆ g. The union of these spaces will, for the scalar valued case and up to translation, give us back the