Open Access
July, 2000 Extension of the Drasin-Shea-Jordan theorem
Nicholas H. BINGHAM, Akihiko INOUE
J. Math. Soc. Japan 52(3): 545-559 (July, 2000). DOI: 10.2969/jmsj/05230545


Passing from regular variation of a function f to regular variation of its integral transform k*f of Mellin-convolution form with kernel k is an Abelian problem; its converse, under suitable Tauberian conditions, is a Tauberian one. In either case, one has a comparison statement that the ratio of f and k*f tends to a constant at infinity. Passing from a comparison statement to a regular-variation statement is a Mercerian problem. The prototype results here are the Drasin-Shea theorem (for non-negative k) and Jordan's theorem (for k which may change sign). We free Jordan's theorem from its non-essential technical conditions which reduce its applicability. Our proof is simpler than the counter-parts of the previous results and does not even use the Pólya Peak Theorem which has been so essential before. The usefulness of the extension is highlighted by an application to Hankel transforms.


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Nicholas H. BINGHAM. Akihiko INOUE. "Extension of the Drasin-Shea-Jordan theorem." J. Math. Soc. Japan 52 (3) 545 - 559, July, 2000.


Published: July, 2000
First available in Project Euclid: 10 June 2008

zbMATH: 0964.44002
MathSciNet: MR1760604
Digital Object Identifier: 10.2969/jmsj/05230545

Primary: 40E05
Secondary: 44A15

Keywords: Hankel transform , Mercerian theorem , regular variation

Rights: Copyright © 2000 Mathematical Society of Japan

Vol.52 • No. 3 • July, 2000
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