The asymptotic behaviour of the Fourier transform of orthogonal polynomials I: Mellin transform techniques

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