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This article is cited in 12 scientific papers (total in 12 papers)
Representation of measurable functions of several variables by multiple trigonometric series
F. G. Arutyunyan
Abstract:
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.
Received: 19.10.1983
Citation:
F. G. Arutyunyan, “Representation of measurable functions of several variables by multiple trigonometric series”, Mat. Sb. (N.S.), 126(168):2 (1985), 267–285; Math. USSR-Sb., 54:1 (1986), 259–277
Linking options:
https://www.mathnet.ru/eng/sm1937https://doi.org/10.1070/SM1986v054n01ABEH002970 https://www.mathnet.ru/eng/sm/v168/i2/p267
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Abstract page: | 322 | Russian version PDF: | 112 | English version PDF: | 10 | References: | 40 |
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