About the almost everywhere convergence of the spectral expansions of functions from $L_1^\a(S^N)$

TL;DR

This paper investigates the almost everywhere convergence of spectral expansions of functions in $L_1^eta(S^N)$ on the sphere, establishing conditions for summability and introducing new methods based on spectral properties of the Laplace-Beltrami operator.

Contribution

It provides new sufficient conditions for the summability of spectral expansions and introduces a novel summability method based on the spectral properties of the Laplace-Beltrami operator.

Findings

Positive summability results for Riesz means orders greater than the critical index.

Construction of new summability methods based on spectral expansion properties.

Identification of limitations and alternative approaches for orders below the critical index.

Abstract

In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace-Beltrami operator on the unit sphere. The sufficient conditions for summability is obtained. The more general properties and representation by the eigenfunctions of the Laplace-Beltrami operator of the Liouville space $L_1^\a$ is used. For the orders of Riesz means, which greater than critical index $\frac{N-1}{2}$ we proved the positive results on summability of Fourier-Laplace series. Note that when order $\alpha$ of Riesz means is less than critical index then for establish of the almost everywhere convergence requests to use other methods form proving negative results. We have constructed different method of summability of Laplace series, which based on spectral expansions property of self-adjoint Laplace-Beltrami operator on the unit sphere.