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THE FINITE FOURIER TRANSFORM OF CLASSICAL POLYNOMIALS

Part of: Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  04 December 2014

ATUL DIXIT ,
LIN JIU ,
VICTOR H. MOLL  and
CHRISTOPHE VIGNAT
ATUL DIXIT
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA email adixit@tulane.edu
LIN JIU
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA email ljiu@tulane.edu
VICTOR H. MOLL*
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA email vhm@tulane.edu
CHRISTOPHE VIGNAT
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA email cvignat@math.tulane.edu LSS-Supelec, Universit’e Orsay Paris Sud 11, France
*
✉vhm@tulane.edu
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Abstract

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The finite Fourier transform of a family of orthogonal polynomials is the usual transform of these polynomials extended by $0$ outside their natural domain of orthogonality. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families.

Keywords

Fourier transformorthogonal polynomialsJacobi polynomials

MSC classification

Primary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Secondary: 33C47: Other special orthogonal polynomials and functions 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$ 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 2 , April 2015 , pp. 145 - 160
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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