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

 

Absolute Tauberian constants for Cesàro means of a function


Author: Soraya Sherif
Journal: Trans. Amer. Math. Soc. 224 (1976), 231-242
MSC: Primary 40D10
MathSciNet review: 0420059
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Abstract: This paper is concerned with introducing two estimates of the forms $ F \leqslant C{A_k}(\alpha ),F \leqslant D{B_k}(\alpha ),(\alpha > 0)$, where $ F = \smallint_0^\infty {\vert d\{ f(\alpha x) - {\sigma _k}(x)\} \vert,{\sigma _k}(x)} $ denote the Cesàro transform of order k of the function $ f(x) = \smallint_0^x {g(t)\;dt,g(t)} $ is a function of bounded variation in every finite interval of $ t \geqslant 0,{A_k}(\alpha ),{B_k}(\alpha )$ are absolute Tauberian constants, $ C = \smallint_0^\infty {\vert d\{ tg(t)\} \vert < \infty ,D = \smallint_0^\infty {\vert d\{ \phi (t)\} \vert < \infty } } $ and $ \phi (t) = {t^{ - 1}}\smallint_0^t {ug(u)du} $. The constants $ {A_k}(\alpha ),{B_k}(\alpha )$ will be determined.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0420059-6
PII: S 0002-9947(1976)0420059-6
Article copyright: © Copyright 1976 American Mathematical Society