The Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials I: Mellin Transform Techniques

Sandro Vaienti (1, 2) , Giorgio Mantica (3)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332
3 Uninsubria - Universitá degli Studi dell’Insubria = University of Insubria [Varese]

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 rôle of Mellin transform techniques in deriving classical and new results is underlined.

Domaines: Systèmes dynamiques [math.DS]

Sandro Vaienti, Giorgio Mantica. The Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials I: Mellin Transform Techniques. Annales Henri Poincaré, 2007, 8 (2), pp.265-300. ⟨10.1007/s00023-006-0308-2⟩. ⟨hal-00476268⟩