Let E be an arbitrary set of positive measure in the N-dimensional cube TN=(-π,π)N⊂ℝN, N≥1, and let f(x)=0 on E. Let [Formula: see text] be some linear subspace of L1(TN). We investigate the behavior of rectangular partial sums of multiple trigonometric Fourier series of a function f on the sets E and TN\E depending on smoothness of the function f (i.e. of the space [Formula: see text]), and, as well, of structural and geometric characteristics of the set E (SGC(E)). Thus, we are describing pairs [Formula: see text]. It is convenient to formulate and investigate the posed question in terms of generalized localization almost everywhere (GL) and weak generalized localization almost everywhere (WGL). This means that for the multiple Fourier series of a function f, that equals zero on the set E, convergence almost everywhere is investigated on the set E (GL), or on some of its subsets E1⊂E, of positive measure (WGL).
On Bochner-Riesz summability almost everywhere of multiple Fourier series
