Representation of measurable functions of several variables by multiple trigonometric series

F. G. Arutyunyan

Sbornik. Mathematics, Tome 54 (1986) no. 1, pp. 259-277

Résumé

Let $\{M_k\}_1^{+\infty}$ and $\{N_k\}_1^{+\infty}$ be sequences of natural numbers satisfying the condition $M_k-N_k\to+\infty$ as $k\to+\infty$. It is proved in this paper that for any a.e. finite measurable function $f(x_1,\dots,x_m)$ of $m$ variables, $0\leqslant x\leqslant2\pi$, there exists an $m$-fold trigonometric series $$ \sum_{j_s\in I,\,1\leqslant s\leqslant m}\operatorname{Re}\bigl(a_{j_1,\dots,j_m}e^{i(j_1x_1+\dots+j_mx_m)}\bigr) $$ (where $I=\bigcup_{k=1}^{+\infty}\{j:\,N_k\leqslant j\leqslant M_k\}$), which is a.e. summable to $f(x_1,\dots,x_m)$ by all the classical summation methods. At the same time examples are exhibited of sequences $\{M_k\}$ and $\{N_k\}$ (with the property mentioned above) such that none of the series $$ \sum_{n\in I}\operatorname{Re}\bigl(a_ne^{inx}\bigr) $$ can converge to $+\infty$ on a set of positive measure. Bibliography: 13 titles.