On the Inversion of the Convolution and Laplace Transform

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 L1 if f ∈ L1, and converges in an appropriate norm or Fréchet topology for generalized functions f . As a corollary we obtain a new constructive inversion procedure for the convolution transform K : f 7→ k ? f ; i.e., for given g and k we construct a sequence of continuous functions fn such that k ? fn → g. Introduction C. Foiaş [Fo] showed in 1961 that the image of the convolution transform f 7→ k ? f := ∫ t 0 k(t− s)f(s) ds is dense in L[0, T ] for k, f ∈ L[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[0, T ] with 0 ∈ supp(k) and g ∈ C0([0, T ];X) or g ∈ L([0, T ];X), where X is a Banach space, find a sequence fn ∈ C([0, T ];X) such that k?fn → g uniformly, or in the L-norm respectively. The sequence (fn) 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 the like, but consists of the limit of (finite) linear combinations of exponential functions ∑N j=1 aje j, where the coefficients aj are determined by the (classical, finite, or asymptotic) Laplace transform f̂ of f , evaluated at Müntz points (βn). This sequence of exponential functions converges uniformly if f is continuous, it converges in L if f ∈ L, and it converges in an appropriate norm or Fréchet topology for generalized functions f . Received by the editors January 25, 1999 and, in revised form, August 5, 2002. 2000 Mathematics Subject Classification. Primary 44A35, 44A10, 44A40.