publication . Article . Preprint . 2014

The finite Fourier transform of classical polynomials

Atul Dixit; Lin Jiu; Victor H. Moll; Christophe Vignat;
Open Access English
  • Published: 22 Feb 2014
  • Publisher: HAL CCSD
  • Country: France
Abstract
<jats:title>Abstract</jats:title><jats:p>The finite Fourier transform of a family of orthogonal polynomials is the usual transform of these polynomials extended by<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S1446788714000500_inline1" /><jats:tex-math>$0$</jats:tex-math></jats:alternatives></jats:inline-formula>outside their natural domain of orthogonality. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families.</jats:p>
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Subjects
free text keywords: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics - Functional Analysis, Mathematics - Classical Analysis and ODEs, 42A16, 33C45, General Mathematics, Orthogonal polynomials, Pure mathematics, Chebyshev filter, Legendre polynomials, Mathematics, Expression (mathematics), Orthogonality, Polynomial, Finite fourier transform

[1] A. Erd´elyi. Tables of Integral Transforms, volume I. McGraw-Hill, New York, 1st edition, 1954.

[2] A. S. Fokas, A. Iserles, and S. A. Smitherman. The unified method in polygonal domains via the explicit Fourier transform of Legendre polynomials. In A. S. Fokas and B. Pelloni, editors, Unified Transforms, page ?? SIAM, 2014.

[3] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. Edited by A. Jeffrey and D. Zwillinger. Academic Press, New York, 7th edition, 2007.

[4] R. Graham, D. Knuth, and O. Patashnik. Concrete Mathematics. Addison Wesley, Boston, 2nd edition, 1994.

[5] N. Greene. Formulas for the Fourier series of orthogonal polynomials in terms of special functions. International Journal of Mathematical Models and Methods in Applied Sciences, 2:317-320, 2008.

[6] G. H. Hardy and W. W. Rogosinski. Fourier Series. Cambridge University Press, 2nd edition, 1950.

[7] I. Nemes, M. Petkovsek, H. Wilf, and D. Zeilberger. How to do MONTHLY problems with your computer. Amer. Math. Monthly, 104:505-519, 1997.

[8] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors. NIST Handbook of Mathematical Functions. Cambridge University Press, 2010.

[9] M. Petkovˇsek, H. Wilf, and D. Zeilberger. A=B. A. K. Peters, Ltd., 1st edition, 1996.

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