Geodesic
Representation of measurable functions of several variables by multiple trigonometric series
Sbornik. Mathematics, Tome 54 (1986) no. 1, pp. 259-277 
F. G. Arutyunyan. Representation of measurable functions of several variables by multiple trigonometric series. Sbornik. Mathematics, Tome 54 (1986) no. 1, pp. 259-277. http://geodesic.mathdoc.fr/item/SM_1986_54_1_a12/
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     author = {F. G. Arutyunyan},
     title = {Representation of measurable functions of several variables by multiple trigonometric series},
     journal = {Sbornik. Mathematics},
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     year = {1986},
     volume = {54},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1986_54_1_a12/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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