Malaysian Journal of Mathematical Sciences, May 2015, Vol. 9, No. 2
On the Weak Localization Principle of the Eigenfunction Expansions of the Laplace-Beltrami Operator by Riesz Method
Anvarjon Ahmedov and Ahmad Fadly Nurullah Rasedee
Corresponding Email: ahmadfadlynurullah@yahoo.com
Received date: -
Accepted date: -
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 $\bar{x}$.
Keywords: Distributions, Fourier-Laplace series, localization, Riesz method, Sphere, Laplace-Beltrami operator
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