STRUCTURAL AND GEOMETRIC CHARACTERISTICS OF SETS OF CONVERGENCE AND DIVERGENCE OF MULTIPLE FOURIER SERIES OF FUNCTIONS WHICH EQUAL ZERO ON SOME SET

摘要

Let E be an arbitrary set of positive measure in the N-dimensional cube T{sup}N = (-π,π){sup}N (is contained in) R{sup}N, N ≥ 1, and let f(x) = 0 on E. Let A = A(T{sup}N) be some linear subspace of L{sub}1(T{sup}N). We investigate the behavior of rectangular partial sums of multiple trigonometric Fourier series of a function f on the sets E and T{sup}NE depending on smoothness of the function f (i.e. of the space A), and, as well, of structural and geometric characteristics of the set E (SGC(E)). Thus, we are describing pairs (A, SGC(E)). 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 E{sub}1 (is contained in) E, of positive measure (WGL).