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J. Korean Math. Soc. 2000; 37(4): 545-563

Printed July 1, 2000

Copyright © The Korean Mathematical Society.

Plancherel and Paley-Wiener theorems for an index integral transform

Vu Kim Tuan, Ali Ismail, and Megumi Saigo

Kuwait University, Kuwait University, Fukuoka University

Abstract

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An integral transform with the Bessel function $J_\nu (z)$ in the kernel is considered. The transform is related to a singular Sturm-Liouville problem on a half line. This relation yields a Plancherel's theorem for the transform. A Paley-Wiener-type theorem for the transform is also derived.

Keywords: Plancherel theorem, Paley-Wiener theorem, singular Sturm-Liouville problem, index transform, Bessel function

MSC numbers: Primary 44A15, 34B24; Secondary 33C10

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