Linear transformations of R and problems of convergence of Fourier series of functions which equal zero on some set

Let M be a class of (all) linear transformations of R-N, N >= 1. Let A = A(T-N), T-N = [-pi,pi)(N) be some linear subspace of L-1 (T-N), and let U be an arbitrary set of positive measure U subset of T-N. We consider the problem: how are the sets of convergence and divergence everywhere or almost everywhere (a.e.) of trigonometric Fourier series (in case N >= 2 summed over rectangles) of function (f o m) (x) = f (m(x)), f epsilon A, f (x) = 0 on U, m epsilon M, changed depending on the smoothness of the function f (i.e. on the space A), as well as on the transformation m. In the paper a (wide) class of spaces A is found such that for each A the system of classes (of nonsingular linear transformations) psi(k), psi(k) subset of M (k = 0, 1,..., N), which "change" the sets of convergence and divergence everywhere or a.e. of the indicated Fourier expansions is defined.