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Abel and Tauberian theorems for integrals

Authors: A. F. Grishin and I. V. Poedintseva
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 3.
Journal: St. Petersburg Math. J. 26 (2015), 357-409
MSC (2010): Primary 40E05; Secondary 30D20
Published electronically: March 20, 2015
MathSciNet review: 3289177
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

A. F. Grishin
Affiliation: Department of Mathematics and Mechanics, V. N. Kazarin Kharkov National University, pl. Svobody 4, Kharkov 61022, Ukraine

I. V. Poedintseva
Affiliation: Department of Mathematics and Mechanics, V. N. Kazarin Kharkov National University, pl. Svobody 4, Kharkov 61022, Ukraine

Keywords: Valiron's proximate order, Radon measure, Azarin's cluster set for a measure, Azarin's regular measure, Wiener Tauberian theorem
Received by editor(s): September 5, 2013
Published electronically: March 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society