Equiconvergence theorems for Chébli-Trimèche hypergroups

Abstract

We consider a Sturm-Liouville operator of the kind d 2 dt2 + A (t) A(t) d dt on (0;+1) and the related eigenfunction expansion. We prove that, under suitable assumptions on A (t), the partial sums of the Fourier integral associated to such expansion behave like the partial sums of the classical Fourier-Bessel transform. This implies an almost everywhere convergence result for L (A (t) dt) functions. Our methods rely on asymptotic expansions for the eigenfunctions and the Harish-Chandra function that we prove under very weak hypotheses. MSC2000: Primary 43A62. Secondary 43A32, 34L10 Di¤erential operators of the kind L = d 2 dt2 + A0 (t) A (t) d dt (0.1) and the associated spectral decompositions arise naturally in harmonic analysis. For example when A (t) = t , the operator L is the radial part of the Laplacian in R and in this case the associated spectral decomposition is the so called Fourier-Bessel expansion that corresponds to the harmonic analysis of radial functions in R. This transform is de ned for any > 1 2 by F f ( ) = Z +1 0 f (t) 2 ( + 1) J ( t) ( t) t 2 dt but of course it can be interpreted as a Fourier transform of a radial function only when = (n 2)=2. The inversion formula associated to this transform is given by