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ISSN 1088-6826(e) ISSN 0002-9939(p)

A Paley-Wiener theorem for the Askey-Wilson function transform

Author(s): Luís Daniel Abreu; Fethi Bouzeffour
Journal: Proc. Amer. Math. Soc. 138 (2010), 2853-2862.
MSC (2010): Primary 33D45, 30D15; Secondary 44A20
Posted: April 15, 2010
MathSciNet review: 2644898
Retrieve article in: PDF

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Abstract: We define an analogue of the Paley-Wiener space in the context of the Askey-Wilson function transform, compute explicitly its reproducing kernel and prove that the growth of functions in this space of entire functions is of order two and type $\ln q^{-1}$ , providing a Paley-Wiener Theorem for the Askey-Wilson transform. Up to a change of scale, this growth is related to the refined concepts of exponential order and growth proposed by J. P. Ramis. The Paley-Wiener theorem is proved by combining a sampling theorem with a result on interpolation of entire functions due to M. E. H. Ismail and D. Stanton.

References:

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.: M. H. Annaby, -type sampling theorems, Result. Math. 44 (3-4) (2003), 214-225. MR 2028677 (2004k:33035)
3.: R. Askey, J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319. MR 783216 (87a:05023)
4.: C. Berg, H. L. Pedersen, with an appendix by Walter Hayman, Logarithmic order and type of indeterminate moment problems, Proceedings of the International Conference : ``Difference equations, special functions and orthogonal polynomials'', Munich, July 25-30, 2005. World Scientific Publishing Company, Singapore, 2007. MR 2450118 (2009i:44007)
5.: F. Bouzeffour, A -sampling theorem and product formula for continuous -Jacobi functions, Proc. Amer. Math. Soc 135 (2007), 2131-2139. MR 2299491 (2008k:33055)
6.: F. Bouzeffour, A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Askey-Wilson functions, J. Nonlinear Math. Phys. 14 (2007), no. 3, 367-380. MR 2350096 (2008h:94031)
7.: Ó. Ciaurri and J. L. Varona, A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform , Proc. Amer. Math. Soc. 135 (2007), 2939-2947. MR 2317972 (2008c:94013)
8.: E. Koelink, J.V. Stokman, The Askey-Wilson function transform, Int. Math. Res. Not. IMRN, no. 22, 1203-1227 (2001). MR 1862616 (2003a:33010)
9.: E. Koelink, J.V. Stokman, The Askey-Wilson function transform scheme, ``Special functions 2000: Current perspective and future directions'' (Tempe, AZ), 221-241, NATO Sci. Ser. II Math. Phys. Chem., 30, Kluwer Acad. Publ., Dordrecht, 2001. MR 2006290 (2005b:33019)
10.: T. H. Koornwinder, A new proof of a Paley-Wiener type theorem for the Jacobi transform, Ark. Mat. 13 (1975), 145-159. MR 0374832 (51:11028)
11.: A. G. Garcia, Orthogonal sampling formulas: A unified approach, SIAM Rev. 42 (2000), no. 3, 499-512. MR 1786936 (2001i:94034)
12.: G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge University Press, Cambridge, UK, 1990. MR 1052153 (91d:33034)
13.: G. H. Hardy, Notes on special systems of orthogonal functions. IV, Proc. Cambridge Phil. Soc. 37 (1941), 331-348. MR 0005145 (3:108b)
14.: J. R. Higgins, An interpolation series associated with the Bessel-Hankel transform , J. Lond. Math. Soc. 5 (1972), 707-714. MR 0320616 (47:9152)
15.: J. R. Higgins, A sampling principle associated with Saitoh's fundamental theory of linear transformations. Analytic extension formulas and their applications (Fukuoka, 1999/Kyoto, 2000), 73-86, Int. Soc. Anal. Appl. Comput., 9, Kluwer Acad. Publ., Dordrecht, 2001. MR 1830378 (2002c:46053)
16.: M. E. H. Ismail, M. Rahman, The associated Askey-Wilson polynomials, Trans. Amer. Math. Soc. 328 (1991), 201-239. MR 1013333 (92c:33019)
17.: M. E. H. Ismail, 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)
18.: M. E. H. Ismail, D. Stanton, -Taylor theorems, polynomial expansions, and interpolation of entire functions. J. Approx. Theory 123 (2003), no. 1, 125-146. MR 1985020 (2004g:30040)
19.: M. E. H. Ismail, D. Stanton, Applications of -Taylor theorems, J. Comput. Appl. Math. 153 (2003), no. 1-2, 259-272. MR 1985698 (2004f:33035)
20.: J. P. Ramis, About the growth of entire functions solutions of linear algebraic -difference equations, Ann. Fac. Sci. Toulouse Math. (6) 1 (1992), no. 1, 53-94. MR 1191729 (94g:39003)
21.: S. Saitoh, Integral transforms, reproducing kernels and their applications (Harlow: Longman), Pitman Research Notes in Mathematics Series, 369, Longman, Harlow, 1997. MR 1478165 (98k:46041)
22.: J. V. Stokman, An expansion formula for the Askey-Wilson function. J. Approx. Theory 114 (2002), no. 2, 308-342. MR 1883410 (2003k:33025)
23.: S. K. Suslov, Some orthogonal very-well-poised $_{8}\varphi _{7}$ -functions, J. Phys. A 30 (1997), 5877-5885. MR 1478393 (98m:33047)
24.: S. K. Suslov, Some orthogonal very-well-poised $_{8}\varphi _{7}$ -functions that generalize the Askey-Wilson polynomials, Ramanujan J. 5 (2001), 183-218. MR 1857183 (2002m:33018)
25.: R. M. Young, An introduction to nonharmonic Fourier series, revised first edition, Academic Press, New York, 2001. MR 1836633 (2002b:42001)
26.: A. I. Zayed, Advances in Shannon's sampling theory, CRC Press, Boca Raton, FL, 1993. MR 1270907 (95f:94008)

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