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

Ismail, Mourad; Zayed, Ahmed

doi:10.1090/S0002-9939-03-07208-3

A $q$-analogue of the Whittaker-Shannon-Kotel’nikov sampling theorem
HTML articles powered by AMS MathViewer

by Mourad E. Ismail and Ahmed I. Zayed PDF

Proc. Amer. Math. Soc. 131 (2003), 3711-3719 Request permission

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.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
R. Askey and Mourad E. H. Ismail, A generalization of ultraspherical polynomials, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 55–78. MR 820210
Joaquin Bustoz and Sergei K. Suslov, Basic analog of Fourier series on a $q$-quadratic grid, Methods Appl. Anal. 5 (1998), no. 1, 1–38. MR 1631331, DOI 10.4310/MAA.1998.v5.n1.a1
George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
Mourad E. H. Ismail, The zeros of basic Bessel functions, the functions $J_{\nu +ax}(x)$, and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), no. 1, 1–19. MR 649849, DOI 10.1016/0022-247X(82)90248-7
M. E. H. Ismail, Orthogonality and completeness of $q$-Fourier type systems, Z. Anal. Anwendungen 20 (2001), no. 3, 761–775. MR 1863946, DOI 10.4171/ZAA/1044
Mourad E. H. Ismail and Ruiming Zhang, Diagonalization of certain integral operators, Adv. Math. 109 (1994), no. 1, 1–33. MR 1302754, DOI 10.1006/aima.1994.1077
Mourad E. H. Ismail and Dennis W. Stanton (eds.), $q$-series from a contemporary perspective, Contemporary Mathematics, vol. 254, American Mathematical Society, Providence, RI, 2000. MR 1768918, DOI 10.1090/conm/254
Tom H. Koornwinder and René F. Swarttouw, On $q$-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333 (1992), no. 1, 445–461. MR 1069750, DOI 10.1090/S0002-9947-1992-1069750-0
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019
S. K. Suslov, “Addition” theorems for some $q$-exponential and $q$-trigonometric functions, Methods Appl. Anal. 4 (1997), no. 1, 11–32. MR 1457202, DOI 10.4310/MAA.1997.v4.n1.a2
Ahmed I. Zayed, Advances in Shannon’s sampling theory, CRC Press, Boca Raton, FL, 1993. MR 1270907