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 L2 class is proved, that is, if f ∈ L2(TN) and  f = 0 on an open set Ω ⊂ TN, then it is shown that the spherical partial sums of this function converge to zero almost-everywhere on Ω. It has been previously known that the generalized localization is not valid in Lp(TN) when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \leqslant p < 2$$\end{document}. 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.