On Axer's theorem

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:388

Article

Waterman, Daniel

关键词: Generalized bounded variation;    Tauberian theorems;   

DOI  :  10.1016/j.jmaa.2011.08.082

来源: Elsevier

【 摘 要 】

In Axer's theorem, conditions on a function of bounded variation and an infinite series of real numbers imply an order condition on the multiplicative convolution of the series and the function. We prove an extension of Axer's theorem. We show that although the converse of Axer's theorem is false without further restriction on the function, with such a restriction one can prove the converse of the strengthened theorem. We also consider the changes in Axer's theorem resulting from assuming the function to be in Lambda BV or phi BV. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free