Linear Transformations of ℝ^N and Problems of Convergence of Fourier Series of Functions Which Equal Zero on Some Set

Abstract

Let $\mathfrak{M}$ be a class of (all) linear transformations of ℝ^N, N ≥ 1. Let $\mathcal{A} = \mathcal{A}{\text{(}}\mathbb{T}^N {\text{),}}\mathbb{T}^N = [ - \pi ,\pi )^N$ be some linear subspace of $L_{\text{1}} {\text{(}}\mathbb{T}^N {\text{)}}$, and let $\mathfrak{A}$ be an arbitrary set of positive measure $\mathfrak{A} \subset \mathbb{T}^N$.

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 $(f \circ \mathfrak{m})(x) = f(\mathfrak{m}(x)),f \in \mathcal{A}$, $f(x) = 0{\text{ }}on \mathfrak{A}, \mathfrak{m} \in \mathfrak{M}$, changed depending on the smoothness of the function f (i.e. on the space $\mathcal{A}$), as well as on the transformation $\mathfrak{m}$.

In the paper a (wide) class of spaces $\mathcal{A}$ is found such that for each $\mathcal{A}$ the system of classes (of nonsingular linear transformations) $\Psi _k ,\Psi _k \subset \mathfrak{M}$ (k = 0, 1 , . . ., N), which “change” the sets of convergence and divergence everywhere or a.e. of the indicated Fourier expansions is defined.

This work was supported by grant 05-01-00206 of the Russian Foundation for Fundamental Research.