Abstract
We present a new inversion formula for the classical, finite, and asymptotic Laplace transform f̂ of continuous or generalized functions ƒ. 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 ƒ is continuous, it is in L¹ if ƒ ∈ L¹, and converges in an appropriate norm or Fréchet topology for generalized functions ƒ. As a corollary we obtain a new constructive inversion procedure for the convolution transform 𝘒 : ƒ ↦ 𝘬 ⋆ ƒ; i.e., for given 𝑔 and 𝘬 we construct a sequence of continuous functions ƒₙ such that 𝘬 ⋆ ƒₙ → 𝑔.