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A generalization of the prolate spheroidal wave functions

Author: Ahmed I. Zayed
Journal: Proc. Amer. Math. Soc. 135 (2007), 2193-2203
MSC (2000): Primary 33C47, 44A05; Secondary 42C05, 33C45
Published electronically: March 2, 2007
MathSciNet review: 2299497
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Abstract: Many systems of orthogonal polynomials and functions are bases of a variety of function spaces, such as the Hermite and Laguerre functions which are orthogonal bases of $ L^2(-\infty, \infty)$ and $ L^2(0,\infty),$ and the Jacobi polynomials which are an orthogonal basis of a weighted $ L^2(-1,1).$ The associated Legendre functions, and more generally, the spheroidal wave functions are also an orthogonal basis of $ L^2(-1,1).$

The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property. They are an orthogonal basis of both $ L^2(-1,1)$ and a subspace of $ L^2(-\infty, \infty),$ known as the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. This raises the question of whether there are other systems possessing this property.

The aim of the article is to answer this question in the affirmative by providing an algorithm to generate such systems and then demonstrating the algorithm by a new example.

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Additional Information

Ahmed I. Zayed
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illiniois 60614

Keywords: Prolate and oblate spheroidal wave functions, orthogonal polynomials and functions, reproducing-kernel Hilbert spaces, bandlimited functions.
Received by editor(s): October 20, 2005
Received by editor(s) in revised form: March 27, 2006
Published electronically: March 2, 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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