Abstract
For an arbitrary open set and an arbitrary function such that on the double Fourier series of with respect to the trigonometric system and the Walsh-Paley system is shown to converge to zero (over rectangles) almost everywhere on . Thus, it is proved that generalized localization almost everywhere holds on arbitrary open subsets of the square for the double trigonometric Fourier series and the Walsh-Fourier series of functions in the class (in the case of summation over rectangles). It is also established that such localization breaks down on arbitrary sets that are not dense in , in the classes for the orthonormal system and an arbitrary function such that as or for , .
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