Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A $q$-analogue of the Whittaker-Shannon-Kotel'nikov sampling theorem


Authors: Mourad E. Ismail and Ahmed I. Zayed
Journal: Proc. Amer. Math. Soc. 131 (2003), 3711-3719
MSC (2000): Primary 33B10, 33D15; Secondary 42C15, 94A11
Published electronically: July 17, 2003
MathSciNet review: 1998178
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Abstract: The Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem plays an important role not only in harmonic analysis and approximation theory, but also in communication engineering since it enables engineers to reconstruct analog signals from their samples at a discrete set of data points. The main aim of this paper is to derive a $q$-analogue of the Whittaker-Shannon-Kotel'nikov sampling theorem. The proof uses recent results in the theory of $q$-orthogonal polynomials and basic hypergeometric functions, in particular, new results on the addition theorems for $q$-exponential functions.


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Additional Information

Mourad E. Ismail
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: ismail@math.usf.edu

Ahmed I. Zayed
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois, 60614
Email: azayed@math.depaul.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07208-3
Keywords: Shannon sampling theorem, band-limited and sinc functions, $q$-trigonometric series, basic hypergeometric functions
Received by editor(s): February 19, 2002
Published electronically: July 17, 2003
Additional Notes: Research partially supported by NSF grant DMS 99-70865 and the Liu Bie Ju Centre of Mathematical Sciences
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society