Speaker
Description
Let $\varphi \colon \ [0,+\infty ) \to
[0,+\infty )$ be a nondecreasing function, $\omega $ be an arbitrary modulus of continuity. Denote by $\varphi (L)$ the set of all $2\pi$-periodic Lebesgue measurable functions $f$ such that
$\varphi (|f|)$ is summable on $[0, 2\pi )$, and by $H_1^{\omega}$ the set of all $f \in L$ whose $L^1$-modulus of continuity $\omega (f, \delta )_1$ satisfies the condition $\omega (f,\delta )_1 = O(\omega (\delta ))$.
Suppose that $f \in L(\mathbb{T} )$, denote by $S_n(f,x)$ the $n$th partial sum of the trigonometric Fourier series ($n$th Fourier sum) of $f$, and by
$$
M(f,x)= \sup \limits _{n \ge 1} |S_n(f,x) |
$$
the majorant of the Fourier sums of $f$. We consider the problems of conditions for the almost everywhere convergence of the Fourier series and the integrability of the majorant of the Fourier sums of $f$ in terms of the belonging of this function to classes $\varphi (L)$ and $H_1^{\omega}$. We propose to discuss multidimensional analogs of these problems for the case of rectangular partial sums of multiple trigonometric Fourier series.
