On the weak localization principle of the eigenfunction expansions of the laplace-beltrami operator by Riesz Method

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 x.