A -sampling theorem related to the -Hankel transform

Author:
L. D. Abreu

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1197-1203

MSC (2000):
Primary 33D15, 33D05; Secondary 94A20

DOI:
https://doi.org/10.1090/S0002-9939-04-07589-6

Published electronically:
October 14, 2004

MathSciNet review:
2117222

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Abstract | References | Similar Articles | Additional Information

Abstract: A -version of the sampling theorem is derived using the -Hankel transform introduced by Koornwinder and Swarttouw. The sampling points are the zeros of the third Jackson -Bessel function.

**1.**L. D. Abreu, J. Bustoz,*Complete sets of**q-Bessel functions*, to appear in ``Theory and Applications of Special Functions. A volume dedicated to Mizan Rahman'', (eds. M. E. H. Ismail and E. Koelink), Developments in Mathematics, Kluwer Acad. Publ.**2.**L. D. Abreu, J. Bustoz, J. L. Cardoso,*The roots of the third Jackson q-Bessel function*, Internat. J. Math. Math. Sci.**67**, (2003), 4241-4248.**3.**G. Gasper and M. Rahman, ``*Basic Hypergeometric Series*,'' Cambridge University Press, Cambridge, UK, 1990. MR**91d:33034****4.**G. H. Hardy,*Notes on special systems of orthogonal functions IV*, Proc. Cambridge Phil. Soc., 37 (1941), 331-348. MR**3:108b****5.**J. R. Higgins,*An interpolation series associated with the Bessel-Hankel transform*, J. Lond. Math. Soc.**5**, (1972) 707-714. MR**47:9152****6.**M. E. H. Ismail,*The Zeros of Basic Bessel functions, the functions*,*and associated orthogonal polynomials*, J. Math. Anal. Appl.**86**(1982), 1-19. MR**83c:33010****7.**H. T. Koelink, R. F. Swarttouw,*On the zeros of the Hahn-Exton q-Bessel function**and associated q-Lommel polynomials,*J. Math. Anal. Appl.**186**, (1994), 690-710. MR**95j:33050****8.**T. H. Koornwinder, R. F. Swarttouw,*On q-analogues of the Fourier and Hankel transforms*. Trans. Amer. Math. Soc. 333 (1992), no. 1, 445-461. MR**92k:33013****9.**A. A. Kvitsinsky,*Spectral zeta functions for q-Bessel equations*, J. Phys. A: Math. Gen.**28**, 1753-1764 (1995), no. 6. MR**96f:58181****10.**L. Máté, ``*Hilbert space methods in science and engineering*''. Adam Hilger, Ltd., Bristol, 1989.MR**92e:46002**

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Additional Information

**L. D. Abreu**

Affiliation:
Department of Mathematics, Universidade de Coimbra, Portugal

Email:
daniel@mat.uc.pt.

DOI:
https://doi.org/10.1090/S0002-9939-04-07589-6

Keywords:
Sampling theorem,
reproducing kernel,
$q$-Bessel functions,
$q$-Hankel transform

Received by editor(s):
November 21, 2003

Received by editor(s) in revised form:
December 12, 2003

Published electronically:
October 14, 2004

Additional Notes:
Partial financial assistance by Fundação para a Ciência e Tecnologia and Centro de Matemática da Universidade de Coimbra

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2004
American Mathematical Society