Proceedings of the American Mathematical Society

Author(s): Ahmed I. Zayed
Journal: Proc. Amer. Math. Soc. 135 (2007), 2193-2203.
MSC (2000): Primary 33C47, 44A05; Secondary 42C05, 33C45
Posted: March 2, 2007
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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

The associated Legendre functions, and more generally, the spheroidal wave functions are also an orthogonal basis of

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

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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Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33C47, 44A05, 42C05, 33C45

Ahmed I. Zayed
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illiniois 60614
Email: azayed@condor.depaul.edu

DOI: 10.1090/S0002-9939-07-08739-4
PII: S 0002-9939(07)08739-4
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
Posted: March 2, 2007
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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