On the Weak Localization Principle of the Eigenfunction Expansions of the Laplace-Beltrami Operator by Riesz Method

TL;DR

This paper investigates the weak localization principle of eigenfunction expansions for the Laplace-Beltrami operator on the sphere, comparing Riesz and Cesaro summation methods and highlighting the influence of function behavior at opposite points.

Contribution

It establishes conditions for weak localization of Fourier-Laplace series on the sphere, emphasizing the role of behavior at diametrically opposite points, and compares Riesz and Cesaro methods.

Findings

Weak localization depends on behavior at opposite points.

Riesz and Cesaro methods are compared for eigenfunction expansions.

Conditions for weak localization are established for functions on the sphere.

Abstract

In this paper we deal with the problems of the weak localization of the eigenfunction expansions related to Laplace-Beltrami operator on unit sphere. The conditions for weak localization of Fourier-Laplace series are investigated by comparing the Riesz and Cesaro methods of summation for eigenfunction expansions of the Laplace-Beltrami operator. It is shown that the weak localization principle for the integrable functions f(x) at the point x depends not only on behavior of the function around x but on the behavior of the function around diametrically opposite point \overline{x}.