On the inversion of the convolution and Laplace transform

Abstract:

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform $\hat f$ 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 $\hat f$ 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 $f\in L^{1}$, 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 ${\mathcal K}:f\mapsto k\star f$; i.e., for given $g$ and $k$ we construct a sequence of continuous functions $f_{n}$ such that $k\star f_{n}\to g$.