Asymptotic inversion of laplace transforms: A class of counterexamples

Let f be a complex-valued locally integrable function on [0, + ∞), and let Lf be its Laplace transform, whenever and wherever it exists. We some known methods, exact and approximate, for recovering f from Lf. Since numerical algorithms need auxiliary information about f near + ∞, we note that the behavior of f near +∞ 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 Re(s)>0 and satisfies (equation omited). 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. © 1973 American Mathematical Society.