Generalized Localization for Spherical Partial Sums of Multiple Fourier Series

Abstract

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the $$L_2$$-class is proved, that is, if $$f\in L_2({\mathbb {T}}^N)$$ and $$f=0$$ on an open set $$\Omega \subset {\mathbb {T}}^N$$, then it is shown that the spherical partial sums of this function converge to zero almost-everywhere on $$\Omega $$. It has been previously known that the generalized localization is not valid in $$L_p({\mathbb {T}}^N)$$ when $$1\le p<2$$. Thus the problem of generalized localization for the spherical partial sums is completely solved in $$L_p({\mathbb {T}}^N)$$, $$p\ge 1$$: if $$p\ge 2$$ then we have the generalized localization and if $$p<2$$, then the generalized localization fails.