The Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials I: Mellin Transform Techniques |
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| Authors: | Giorgio Mantica Sandro Vaienti |
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| Institution: | (1) Center for Non-linear and Complex Systems, Università dell’Insubria, Via Valleggio 11, I-22100 Como, Italy;(2) CNISM and INFN sez., Como, Italy;(3) Centre de Physique Théorique, CNRS, U.M.R. 6207, Luminy, Marseille, France;(4) Universites d’Aix-Marseille I, II Université du Sud, Toulon-Var, France;(5) Fédération de Recherche des Unités de Mathématiques de Marseille, Marseille, France |
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| Abstract: | The Fourier transform of orthogonal polynomials with respect to their own orthogonality measure defines the family of Fourier–Bessel
functions. We study the asymptotic behaviour of these functions and of their products, for large real values of the argument.
By employing a Mellin analysis we construct a general framework to exhibit the relation of the asymptotic decay laws to certain
dimensions of the orthogonality measure, that are defined via the divergence abscissas of suitable integrals. The unifying
r?le of Mellin transform techniques in deriving classical and new results is underlined.
Submitted: November 5, 2004. Accepted: January 6, 2006. |
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