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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
Published electronically: November 1, 2007
MathSciNet review: 2358478
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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)$.

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Additional Information

Mohammad Masjed-Jamei
Affiliation: Department of Applied Mathematics, K. N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran

Keywords: Stieltjes--Wigert polynomials, Fourier transform, Parseval identity, normal and log-normal distributions.
Received by editor(s): September 14, 2006
Published electronically: November 1, 2007
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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