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Asymptotic inversion of Laplace transforms: a class of counterexamples

Author: John S. Lew
Journal: Proc. Amer. Math. Soc. 39 (1973), 329-336
MSC: Primary 44A10
MathSciNet review: 0324325
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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

$\displaystyle Lf(s) = o({s^n}){\text{ near }}0 + \quad {\text{for}}\;n = 0,1,2, \cdots .$

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.

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Keywords: Laplace transform, asymptotic inversion, inverse Abelian theorem, Tauberian theorem, Mellin series
Article copyright: © Copyright 1973 American Mathematical Society