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


Authors: Luís Daniel Abreu and Fethi Bouzeffour
Journal: Proc. Amer. Math. Soc. 138 (2010), 2853-2862
MSC (2010): Primary 33D45, 30D15; Secondary 44A20
DOI: https://doi.org/10.1090/S0002-9939-10-10327-X
Published electronically: April 15, 2010
MathSciNet review: 2644898
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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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Additional Information

Luís Daniel Abreu
Affiliation: Department of Mathematics, Centre for Mathematics, School of Science and Technology (FCTUC), University of Coimbra, 3001-454 Coimbra, Portugal
Email: daniel@mat.uc.pt

Fethi Bouzeffour
Affiliation: Faculté des Sciences, Institut Préparatoire aux Études D’Ingénieur de Bizerte, 7021 Jarzouna, Bizerte, Tunisie
Email: Fethi.Bouzeffour@ipeib.rnu.tn

DOI: https://doi.org/10.1090/S0002-9939-10-10327-X
Keywords: Askey-Wilson function, Paley-Wiener theorem, reproducing kernels, sampling theorem.
Received by editor(s): June 18, 2008
Received by editor(s) in revised form: December 10, 2009
Published electronically: April 15, 2010
Additional Notes: The research of the first author was partially supported by CMUC/FCT and FCT postdoctoral grant SFRH/BPD/26078/2005, POCI 2010 and FSE
Communicated by: Peter A. Clarkson
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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