Abel and Tauberian theorems for integrals

Grishin, A.; Poedintseva, I.

doi:10.1090/S1061-0022-2015-01343-5

Abel and Tauberian theorems for integrals
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by A. F. Grishin and I. V. Poedintseva
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 26 (2015), 357-409

DOI: https://doi.org/10.1090/S1061-0022-2015-01343-5

Published electronically: March 20, 2015

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Abstract:

A new method is suggested for obtaining Abel and Tauberian Theorems for integrals of the form $\int _0^\infty K\big (\frac {t}{r}\big ) d\mu (t)$. It is based on properties of limit sets for measures. Accordingly, a version of Azarin’s cluster set theory for Radon measures on the half-line $(0,\infty )$ is created. Theorems of new sort are proved, in which the asymptotic behavior of the above integrals is described in terms of cluster sets for $\mu$. With the use of these results and a stronger version (also proved in the paper) of Karleman’s well-known analytic continuation lemma, the second Tauberian theorem by Wiener is refined considerably.

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