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 Mat. Sb. (N.S.), 1985, Volume 126(168), Number 2, Pages 267–285 (Mi msb1937)

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 $f(x_1,…,x_m)$ of $m$ variables, $0\leqslant x\leqslant2\pi$, there exists an $m$-fold trigonometric series
$$\sum_{j_s\in I, 1\leqslant s\leqslant m}\operatorname{Re}(a_{j_1,…,j_m}e^{i(j_1x_1+…+j_mx_m)})$$
(where $I=\bigcup_{k=1}^{+\infty}\{j: N_k\leqslant j\leqslant M_k\}$), which is a.e. summable to $f(x_1,…,x_m)$ 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
$$\sum_{n\in I}\operatorname{Re}(a_ne^{inx})$$
can converge to $+\infty$ on a set of positive measure.
Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Sbornik, 1986, 54:1, 259–277

Bibliographic databases:

UDC: 517.5
MSC: 42B05, 42B99, 28A20

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

Citation in format AMSBIB
\Bibitem{Aru85} \by F.~G.~Arutyunyan \paper Representation of measurable functions of several variables by multiple trigonometric series \jour Mat. Sb. (N.S.) \yr 1985 \vol 126(168) \issue 2 \pages 267--285 \mathnet{http://mi.mathnet.ru/msb1937} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=784357} \zmath{https://zbmath.org/?q=an:0617.42019|0604.42031} \transl \jour Math. USSR-Sb. \yr 1986 \vol 54 \issue 1 \pages 259--277 \crossref{https://doi.org/10.1070/SM1986v054n01ABEH002970} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. A. Talalyan, R. I. Ovsepian, “The representation theorems of D. E. Men'shov and their impact on the development of the metric theory of functions”, Russian Math. Surveys, 47:5 (1992), 13–47
2. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171
3. M. I. Dyachenko, “$u$-convergence of multiple Fourier series”, Izv. Math., 59:2 (1995), 353–366
4. M. I. Dyachenko, “$U$-convergence almost everywhere of double Fourier series”, Sb. Math., 186:1 (1995), 47–64
5. M. I. Dyachenko, “Uniform convergence of double Fourier series for classes of functions with anisotropic smoothness”, Math. Notes, 59:6 (1996), 680–686
6. M. I. Dyachenko, “Two-dimensional Waterman classes and $u$-convergence of Fourier series”, Sb. Math., 190:7 (1999), 955–972
7. M. I. Dyachenko, “$U$-Convergence of Fourier Series with Monotone and with Positive Coefficients”, Math. Notes, 70:3 (2001), 320–328
8. Kozma G., Olevskii A., “Menshov Representation Spectra”, J. Anal. Math., 84 (2001), 361–393
9. A. M. Olevskii, “Representation of functions by exponentials with positive frequencies”, Russian Math. Surveys, 59:1 (2004), 171–180
10. M. A. Nalbandyan, “Representation of measurable functions by series with respect to Walsh subsystems”, Russian Math. (Iz. VUZ), 53:10 (2009), 45–56
11. N. N. Kholshchevnikova, “Countably multiple null series”, Proc. Steklov Inst. Math., 280 (2013), 280–291
12. Kozma G., Olevskii A., “Perturbing Pla”, J. Anal. Math., 121 (2013), 279–298
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