On the inversion of the convolution and Laplace transform

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform (f) over cap 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 (f) over cap evaluated on a Muntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if f is continuous, it is in (1) if f is an element of L-1, and converges in an appropriate norm or Frechet topology for generalized functions f. As a corollary we obtain a new constructive inversion procedure for the convolution transform K : f bar right arrow k star f; i. e., for given g and k we construct a sequence of continuous functions f(n) such that k star f(n) --> g.