A Paley-Wiener theorem for the Askey-Wilson function transform

Abreu, Luís; Bouzeffour, Fethi

doi:10.1090/S0002-9939-10-10327-X

A Paley-Wiener theorem for the Askey-Wilson function transform
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by Luís Daniel Abreu and Fethi Bouzeffour PDF

Proc. Amer. Math. Soc. 138 (2010), 2853-2862 Request permission

Abstract:

We define an analogue of the Paley-Wiener space in the context of the Askey-Wilson function transform, compute explicitly its reproducing kernel and prove that the growth of functions in this space of entire functions is of order two and type $\ln q^{-1}$, providing a Paley-Wiener Theorem for the Askey-Wilson transform. Up to a change of scale, this growth is related to the refined concepts of exponential order and growth proposed by J. P. Ramis. The Paley-Wiener theorem is proved by combining a sampling theorem with a result on interpolation of entire functions due to M. E. H. Ismail and D. Stanton.

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