Generalized localization for spherical partial sums of multiple Fourier series
Ravshan AshurovNational University of Uzbekistan named after Mirzo Ulugbek; Institute of Mathematics named after V.I. Romanovskiy of the Academy of Sciences of Uzbekistan
ABI
Abstract
In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2-class is proved, that is, if f L2 (ТN) and f = 0 on an open set ТN then it is shown that the spherical partial sums of this function converge to zero almost - everywhere on . It has been previously known that the generalized localization is not valid in Lp (TN) when 1 p 2. Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp (TN), p 1: if p 2 then we have the generalized localization and if p 2, then the generalized localization fails.