Abstract. Let ${\eufm M}$ be the set of (all) linear transformations of ${\msbm R}^N$, and let ${\eufm A}$ be an arbitrary set of positive measure, ${\eufm A} \subset{\msbm T}^N=[-\pi,\pi )^N$, $N\ge2$. We study the problem: how are the sets of convergence and divergence everywhere or almost everywhere (a.e.) of multiple trigonometric Fourier series (summed over rectangles) of the function $(f\circ{\eufm m})(x)=f({\eufm m}(x))$, if $f\in L_p({\msbm T}^N)$, $p\ge1$, $f(x)=0$ on ${\eufm A}$ and ${\eufm m}\in{\eufm M}$, changed (if changed) depending on the transformation ${\eufm m}$. In the paper we give some classes (of nonsingular linear transformations) $\Psi $, $\Psi\subset {\eufm M}$, which ``change" the sets of convergence and divergence everywhere or a.e. of the indicated Fourier series. Such classes are, in particular: {\it a}) the class of transformations consisting of ``almost all" elements of the group of rotations of ${\msbm R}^N$ about the origin; {\it b}) the class of transformations whose inverse transformations have matrices ${\msbm A}=\{a_{j m}\} _{j, m=1}^N$ satisfying the condition: there exists $k$, $1\le k \le N$, such that $\max_{1\le j\le N}|a_{j k}| < 1.$ Let us note that in the paper we consider two settings of the problem under investigation (depending on the way the Fourier series of a function $f\circ{\eufm m}$ is understood).

AMS Subject Classification (1991): 42B05

Keyword(s): multiple trigonometric Fourier series, convergence and divergence everywhere and almost everywhere, linear transformations, rotation group

Received October 26, 2008, and in revised form July 9, 2009. (Registered under 6429/2009.)