This is the abstract of the paper
  "On the Inversion of the Convolution and Laplace Transform"
by Boris Baeumer.

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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 M\"untz 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 Frechet 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.