A $q$-sampling theorem related to the $q$-Hankel transform
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Abstract:
A $q$-version of the sampling theorem is derived using the $q$-Hankel transform introduced by Koornwinder and Swarttouw. The sampling points are the zeros of the third Jackson $q$-Bessel function.References
- 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.
- 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.
- 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
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- J. R. Higgins, An interpolation series associated with the Bessel-Hankel transform, J. London Math. Soc. (2) 5 (1972), 707–714. MR 320616, DOI 10.1112/jlms/s2-5.4.707
- 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
- H. T. Koelink and R. F. Swarttouw, On the zeros of the Hahn-Exton $q$-Bessel function and associated $q$-Lommel polynomials, J. Math. Anal. Appl. 186 (1994), no. 3, 690–710. MR 1293849, DOI 10.1006/jmaa.1994.1327
- 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
- Andrei A. Kvitsinsky, Spectral zeta functions for $q$-Bessel equations, J. Phys. A 28 (1995), no. 6, 1753–1764. MR 1338058, DOI 10.1088/0305-4470/28/6/026
- László Máté, Hilbert space methods in science and engineering, Adam Hilger, Ltd., Bristol, 1989. MR 1065137
Additional Information
- L. D. Abreu
- Affiliation: Department of Mathematics, Universidade de Coimbra, Portugal
- Email: daniel@mat.uc.pt.
- 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
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2117222