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Jordan's theorem for fourier and hankel transforms

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Title: Jordan's theorem for fourier and hankel transforms
Authors: Bingham, N.H Browse this author
Inoue, A. Browse this author
Issue Date: 1-Aug-1995
Journal Title: Hokkaido University Preprint Series in Mathematics
Volume: 305
Start Page: 1
End Page: 30
Abstract: Passing from regular variation of a function f to regular variation of a Mellin convo­lution k* f with kernel k is an Abelian problem; its converse, under suitable Tauberian con­ditions, a Tauberian one. In either case, one has a comparison statement ( k * f)( x) / f ( x) → c (x → ∞), in which c is the Mellin transform k(p) of the kernel k at the index p of regular variation. Passing from a comparison statement to a regular-variation statement is a Merce­rian problem. The prototype results here are the Drasin-Shea theorem (fork non-negative) and Jordan's theorem (for kernels which may change sign). In each case, the Mellin trans­forms k( s) is assumed absolutely convergent in the relevant strip in the complex s-plane. We extend Jordan's theorem to cases where k( s) is only conditionally convergent, at the price of restricting from general to special (Fourier cosine and sine, and Hankel) kernels - and considerable extra complication in an already long proof. We need Korenblum's extension to the Wiener Tauberian theory, and results of Weber, Sonine and Gegenbauer on Bessel functions. 1 2
Type: bulletin (article)
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

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