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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Tauberian theorems for the Laplace-Stieltjes transform


Author: C. J. K. Batty
Journal: Trans. Amer. Math. Soc. 322 (1990), 783-804
MSC: Primary 44A10; Secondary 30B50, 40E05, 47A60
MathSciNet review: 1013326
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Abstract: Let $ \alpha :[0,\infty ) \to {\mathbf{C}}$ be a function of locally bounded variation, with $ \alpha (0) = 0$, whose Laplace-Stieltjes transform $ g(z) = \int_0^\infty {{e^{ - zt}}d\alpha (t)} $ is absolutely convergent for $ \operatorname{Re} z > 0$. Let $ E$ be the singular set of $ g$ in $ i{\mathbf{R}}$, and suppose that $ 0 \notin E$. Various estimates for $ \lim {\sup _{t \to \infty }}\vert\alpha (t) - g(0)\vert$ are obtained. In particular, $ \alpha (t) \to g(0)$ as $ t \to \infty $ if

\begin{displaymath}\begin{gathered}({\text{i)}}\quad E\,{\text{is null,}} \hfill... ...vert\alpha (s) - \alpha (t)\vert = 0. \hfill \\ \end{gathered} \end{displaymath}

This result contains Tauberian theorems for Laplace transforms, power series, and Dirichlet series.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1013326-6
PII: S 0002-9947(1990)1013326-6
Keywords: Tauberian theorem, Laplace-Stieltjes transform, Laplace transform, power series, Dirichlet series, $ {C_0}$-semigroup
Article copyright: © Copyright 1990 American Mathematical Society