Proceedings of the American Mathematical Society

Author(s): Óscar Ciaurri; Juan L. Varona
Journal: Proc. Amer. Math. Soc. 135 (2007), 2939-2947.
MSC (2000): Primary 94A20; Secondary 42A38
Posted: May 8, 2007
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Abstract: A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an orthonormal system which is complete in $L^2((-1,1),\vert x\vert^{2\alpha+1}\,dx)$ . This orthonormal system is a generalization of the classical exponential system defining Fourier series.

1.: L. D. Abreu, A -sampling theorem related to the -Hankel transform, Proc. Amer. Math. Soc. 133 (2005), 1197-1203. MR 2117222 (2006f:33014)
2.: N. B. Andersen and M. de Jeu, Elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line, Int. Math. Res. Not. 30 (2005), 1817-1831. MR 2172939 (2006h:42016)
3.: J. Betancor, Ó. Ciaurri, and J. L. Varona, The multiplier of the interval for the Dunkl transform on the real line, J. Funct. Anal., to appear.
4.: Ó. Ciaurri, J. J. Guadalupe, M. Pérez, and J. L. Varona, Mean and almost everywhere convergence of Fourier-Neumann series, J. Math. Anal. Appl. 236 (1999), 125-147. MR 1702679 (2001a:42025)
5.: C. F. Dunkl, Differential-difference operators associated with reflections groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.MR 0951883 (90k:33027)
6.: C. F. Dunkl, Integral kernels with reflections group invariance, Canad. J. Math. 43 (1991), 1213-1227.MR 1145585 (93g:33012)
7.: M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), 147-162.MR 1223227 (94m:22011)
8.: A. G. García, Orthogonal sampling formulas: a unified approach, SIAM Rev. 42 (2000), 499-512.MR 1786936 (2001i:94034)
9.: J. R. Higgins, An interpolation series associated with the Bessel-Hankel transform, J. Lond. Math. Soc. 5 (1972), 707-714. MR 0320616 (47:9152)
10.: M. E. Ismail and A. I. Zayed, A -analogue of the Whittaker-Shannon-Kotel'nikov sampling theorem, Proc. Amer. Math. Soc. 131 (2003), 3711-3719.MR 1998178 (2004e:33013)
11.: L. Máté, ``Hilbert space methods in science and engineering'', Adam Hilger, Bristol, 1989.MR 1065137 (92e:46002)
12.: C. T. Roosenraad, ``Inequalities with orthogonal polynomials'', Ph.D. thesis, University of Wisconsin-Madison, 1969.
13.: M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, Oper. Theory Adv. Appl. 73 (1994), 369-396.MR 1320555 (96b:33005)
14.: C. E. Shannon, Communication in the presence of noise, Proc. IRE 137 (1949), 10-21.MR 0028549 (10:464e)
15.: G. N. Watson, ``A treatise on the theory of Bessel functions'', Cambridge University Press, Cambridge, 1958.MR 1349110 (96i:33010)
16.: E. T. Whittaker, On the functions which are represented by the expansion of the interpolation theory, Proc. Roy. Soc. Edinburgh Sect. A 35 (1915), 181-194.
17.: J. M. Whittaker, ``Interpolatory Function Theory'', Cambridge University Press, Cambridge, 1935.MR 0185330 (32:2798)
18.: A. I. Zayed, ``Advances in Shannon's sampling theory'', CRC Press, Boca Raton, FL, 1993.MR 1270907 (95f:94008)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 94A20, 42A38

Óscar Ciaurri
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J. L. Vives, Calle Luis de Ulloa s/n, 26004 Logroño, Spain
Email: oscar.ciaurri@dmc.unirioja.es

Juan L. Varona
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J. L. Vives, Calle Luis de Ulloa s/n, 26004 Logroño, Spain
Email: jvarona@dmc.unirioja.es

DOI: 10.1090/S0002-9939-07-08831-4
PII: S 0002-9939(07)08831-4
Keywords: WSK sampling theorem, reproducing kernel, Dunkl transform, orthonormal system, Bessel functions
Received by editor(s): February 9, 2006
Received by editor(s) in revised form: June 6, 2006
Posted: May 8, 2007
Additional Notes: Research supported by grant MTM2006-13000-C03-03 of the DGI
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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