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Jordan's theorem for fourier and hankel transforms
| 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 |
| Publisher: | Department of Mathematics, Hokkaido University |
| 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) |
| URI: | http://hdl.handle.net/2115/69056 |
| 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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Submitter: 数学紀要登録作業用
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