The almost everywhere convergence of the fourier-laplace series

Abstract

In this paper we study the almost everywhere convergence of the spectral expansions related to the self-adjoint extension of the Laplace-Beltrami operator on the unit sphere. The sufficient conditions for summability of the Fourier-Laplace series is obtained. We have established the positive results on the almost everywhere convergence of Fourier-Laplace series by Riesz means of critical order N−1 2 . The more general properties and representation in terms of eigenfunction expansion of the Laplace-Beltrami operator is used. We have constructed different method for investigating the convergence problems of Fourier-Laplace series, which based on the theory of spectral decompositions property of self-adjoint Laplace-Beltrami operator on unit sphere.