Two-dimensional Waterman classes and -convergence of Fourier series

Abstract

New results on the -convergence of the double Fourier series of functions from Waterman classes are obtained. It turns out that none of the Waterman classes wider than ensures even the uniform boundedness of the -sums of the double Fourier series of functions in this class. On the other hand, the concept of -convergence is introduced (the sums are taken over regions that are forbidden to stretch along coordinate axes) and it is proved that for functions belonging to the class , where , the corresponding -partial sums are uniformly bounded, while if , where , then the double Fourier series of is -convergent everywhere.