Extension of the Drasin-Shea-Jordan theorem
type: Item
instance of: scholarly article
title: Extension of the Drasin-Shea-Jordan theorem
zbMATH Open document ID: 0964.44002
author: N. H. Bingham
author: Akihiko Inoue
published in: Journal of the Mathematical Society of Japan
publication date: 2001-07-22T00:00:00Z
review text: Consider a positive function \(f\) and a kernel \(K\) such that the Mellin convolution \(k*f(x):=\int^\infty_0K(x/t)f(t)dt/t\) exists absolutely for \(x>0\). We may consider the statements i) \(f\in RV_\rho\), i.e., is regularly varying of order \(\rho\); ii) \(K*f\in RV_\rho\); iii) \((K*f)(x)/f(x)\to c\neq 0\), as \(x\to\infty\). Under mild conditions i) implies ii) (Abelian conclusion), whereas the Tauberian counterpart ii) implies i) needs additional assumptions on \(f\), so-called Tauberian conditions. Finally, the Drasin-Shea theorem says, that for positive kernels \(K\) iii) implies i) and hence ii) (Mercerian conclusion). By a result of Jordan \(K\) may also have sign changes but has to obey other technical conditions [see e.g., the book of \textit{N. H. Bingham}, \textit{C. M. Goldie} and \textit{J. L. Teugels}, Regular variation (1987; Zbl 0617.26001)]. The aim of the present paper is to give a Mercerian result for kernels which may have infinitely many changes of signs, but satisfy otherwise only some mild conditions and functions \(f\) being of bounded increase or decrease. Particular attention is given to Hankel transforms with index \(\nu>-1/2\). The proofs are simpler than before, in particular the Polyá peak theorem is not used, but absolute convergence of the integrals is assumed instead. This paper continues earlier work of the authors in this context [see \textit{N. H. Bingham} and \textit{A. Inoue}, Q. J. Math., Oxf. II. Ser. 48, No. 191, 279-307 (1997; Zbl 0889.42005) and, Ratio Mercerian theorems with applications to Fourier and Hankel transforms, Proc. Lond. Math. Soc., III. Ser. 79, No. 3, 626-648 (1999; Zbl 1030.44005)].
reviewed by: Ulrich Stadtmüller
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 40E05
zbMATH DE Number: 1529887
zbMATH Keywords: Mercerian theorem
zbMATH Keywords: Mellin convolutions
zbMATH Keywords: regular variation
zbMATH Keywords: kernels with sign changes
zbMATH Keywords: Hankel transform
zbMATH Keywords: Drasin-Shea theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05230545
OpenAlex ID: W1978687166
DOI: 10.2969/JMSJ/05230545
description: scientific article; zbMATH DE number 1529887