Asymptotic inversion of Laplace transforms: a class of counterexamples

Abstract:

Let $f$ be a complex-valued locally integrable function on $[0, + \infty )$, and let $Lf$ be its Laplace transform, whenever and wherever it exists. We review some known methods, exact and approximate, for recovering $f$ from $Lf$. Since numerical algorithms need auxiliary information about $f$ near $+ \infty$, we note that the behavior of $f$ near $+ \infty$ depends on the behavior of $Lf$ near 0 +, hence that our ability to retrieve $f$ is limited by the class of momentless functions, namely, all functions $f$ such that $Lf(s)$ converges absolutely for $\operatorname {Re} (s) > 0$ and satisfies \[ Lf(s) = o({s^n}){\text { near }}0 + \quad {\text {for}}\;n = 0,1,2, \cdots .\] We discuss the space $Z$ of momentless functions and complex distributions, then construct a family of elements in this space which defy various plausible conjectures.