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(T^N) and  f = 0 on an open set Ω ⊂ T^N, 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 L_p(T^N) when $1 \leqslant p < 2$. Thus the problem of generalized localization for the spherical partial sums is completely solved in L_p(T^N), p ≥ 1: if p ≥ 2 then we have the generalized localization and if p < 2, then the generalized localization fails.