Biorthogonal exponential sequences with weight function 𝑒𝑥𝑝(𝑎𝑥²+𝑖𝑏𝑥) on the real line and an orthogonal sequence of trigonometric functions

Masjed-Jamei, Mohammad

doi:10.1090/S0002-9939-07-09139-3

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6826(online) ISSN 0002-9939(print)

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

Author: 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
Full-text 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 $(-\infty,\infty)$ .

1. T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York, 1978. Mathematics and its Applications, Vol. 13. MR 0481884 (58 #1979)
2. Jacob Stordal Christiansen, The moment problem associated with the Stieltjes-Wigert polynomials, J. Math. Anal. Appl. 277 (2003), no. 1, 218–245. MR 1954473 (2004b:44007), http://dx.doi.org/10.1016/S0022-247X(02)00534-6
3. Jacob S. Christiansen and Erik 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 (2007g:47042), http://dx.doi.org/10.1016/j.jat.2005.11.010
4. Jacob S. Christiansen and Mourad 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), http://dx.doi.org/10.1090/S0002-9947-05-03785-2
5. A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. 2, McGraw-Hill, 1954.
6. Wolter Groenevelt, The Wilson function transform, Int. Math. Res. Not. 52 (2003), 2779–2817. MR 2058035 (2006a:33009), http://dx.doi.org/10.1155/S107379280313190X
7. M. E. H. Ismail and D. R. Masson, 𝑞-Hermite polynomials, biorthogonal rational functions, and 𝑞-beta integrals, Trans. Amer. Math. Soc. 346 (1994), no. 1, 63–116. MR 1264148 (96a:33022), http://dx.doi.org/10.1090/S0002-9947-1994-1264148-6
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), no. 3, 887–898. MR 1307541 (96f:33018), http://dx.doi.org/10.1090/S0002-9939-96-03190-5
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, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. 174–183. MR 838982 (87g:33007), http://dx.doi.org/10.1007/BFb0076542
12. Tom H. Koornwinder, Meixner-Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys. 30 (1989), no. 4, 767–769. MR 987105 (90e:33037), http://dx.doi.org/10.1063/1.528394
13. Mohammad Masjedjamei, Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation, Integral Transforms Spec. Funct. 13 (2002), no. 2, 169–191. MR 1915513 (2003i:33011), http://dx.doi.org/10.1080/10652460212898
14. Mohammad Masjed Jamei, Classical orthogonal polynomials with weight function ((𝑎𝑥+𝑏)²+(𝑐𝑥+𝑑)²)^{-𝑝}𝑒𝑥𝑝(𝑞𝐴𝑟𝑐𝑡𝑔((𝑎𝑥+𝑏)/(𝑐𝑥+𝑑))), 𝑥∈(-∞,∞) and a generalization of 𝑇 and 𝐹 distributions, Integral Transforms Spec. Funct. 15 (2004), no. 2, 137–153. MR 2053407 (2005b:33011), http://dx.doi.org/10.1080/10652460410001663456
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.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|