Proceedings of the American Mathematical Society

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

Biorthogonal exponential sequences with weight function $\exp(ax^2+ibx)$ on the real line and an orthogonal sequence of trigonometric functions

Author(s): Mohammad Masjed-Jamei
Journal: Proc. Amer. Math. Soc. 136 (2008), 409-417.
MSC (2000): Primary 05E35, 42C05, 33C47
Posted: November 1, 2007
MathSciNet review: 2358478
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Some orthogonal functions can be mapped onto other orthogonal functions by the Fourier transform. In this paper, by using the Fourier transform of Stieltjes-Wigert polynomials, we derive a sequence of exponential functions that are biorthogonal with respect to a complex weight function like $\exp(q_1(ix+p_1)^2+q_2(ix+p_2)^2)$ on $(-\infty,\infty)$ . Then we restrict these introduced biorthogonal functions to a special case to obtain a sequence of trigonometric functions orthogonal with respect to the real weight function $\exp(-qx^2)$ on $(-\infty,\infty)$ .

References:

1.: T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, N.Y. 1978. MR 0481884 (58:1979)
2.: J. S. Christiansen, The moment problem associated with the Stieltjes-Wigert polynomials, J. Math. Anal. Appl., 277 (2003), no. 1, 218-245. MR 1954473 (2004b:44007)
3.: J. S. Christiansen and E. Koelink, Self-adjoint difference operators and classical solutions to the Stieltjes-Wigert moment problem, J. Approx. Theory, 140 (2006), no. 1, 1-26. MR 2226673
4.: J. S. Christiansen and M. E. H Ismail, A moment problem and a family of integral evaluations, Trans. Amer. Math. Soc,. 358 (2006), no. 9, 4071-4097. MR 2219011 (2007a:33015)
5.: A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. 2, McGraw-Hill, 1954.
6.: W. Groenevelt, The Wilson function transform, Int. Math. Res. Not., (2003), no. 52, 2779-2817. MR 2058035 (2006a:33009)
7.: M. E. H Ismail and D. R. Masson, -Hermite polynomials, biorthogonal rational functions, and -beta integrals, Trans. Amer. Math. Soc., 346 (1994), no. 1, 63-116. MR 1264148 (96a:33022)
8.: R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its -analogue, Report no. 98-17, Technical Universiteit Delft, Faculty of Technical Mathematics and Informatics, Delft, (1998), Web site: http://aw.twi.tudelft.nl/ koekoek/askey/
9.: H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124 (1996), 997-898. MR 1307541 (96f:33018)
10.: W. Koepf and M. Masjed-Jamei, Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials, Proc. Amer. Math. Soc., 135 (2007), no. 11, 3599-3606.
11.: T. H. Koornwinder, Special orthogonal polynomial systems mapped onto each other by the Fourier-Jacobi transform, Polynômes Orthogonaux et Applications (C. Brezinski, A. Draux, A. P. Magnus, P. Maroni and A. Ronveaux, Eds.), Lecture Notes Math., 1171, Springer, (1985), 174-183. MR 838982 (87g:33007)
12.: T. H. Koornwinder, Meixner-Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys., 30 (1989), 767-769. MR 987105 (90e:33037)
13.: M. Masjed-Jamei, Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation, J. Integral Transforms and Special Functions, 13 (2002), no. 2, 169-190. MR 1915513 (2003i:33011)
14.: M. Masjed-Jamei, Classical orthogonal polynomials with weight function $((ax+b)^2+(cx+d)^2)^{-p}\exp(q\operatorname{arctan}(ax+b)/(cx+d))$ ; $x\in(-\infty,\infty)$ and a generalization of and distributions, J. Integral Transforms and Special Functions, 15 (2004), no. 2, 137-153. MR 2053407 (2005b:33011)
15.: T.J. Stieltjes, Recherches sur les fractions continues, Annales de la faculte des sciences de Toulous, 8 (1894), J1-122; 9 (1895), A1-47; Qeuvres, vol.2, 398-566.
16.: S. Wigert, Sur les polynomes orthogonaux et l'approximation des functions continues, Arkiv for matematik, astronomi och fysik, 17 (1923), no. 18, 15 pp.

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