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

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 $f(x_1,…,x_m)$ 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.

Full text: PDF file (816 kB)
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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
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

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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. M. I. Dyachenko, “$u$-convergence of multiple Fourier series”, Izv. Math., 59:2 (1995), 353–366  mathnet  crossref  mathscinet  zmath  isi
    4. M. I. Dyachenko, “$U$-convergence almost everywhere of double Fourier series”, Sb. Math., 186:1 (1995), 47–64  mathnet  crossref  mathscinet  zmath  isi
    5. M. I. Dyachenko, “Uniform convergence of double Fourier series for classes of functions with anisotropic smoothness”, Math. Notes, 59:6 (1996), 680–686  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. M. I. Dyachenko, “Two-dimensional Waterman classes and $u$-convergence of Fourier series”, Sb. Math., 190:7 (1999), 955–972  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. M. I. Dyachenko, “$U$-Convergence of Fourier Series with Monotone and with Positive Coefficients”, Math. Notes, 70:3 (2001), 320–328  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. Kozma G., Olevskii A., “Menshov Representation Spectra”, J. Anal. Math., 84 (2001), 361–393  crossref  mathscinet  zmath  isi
    9. A. M. Olevskii, “Representation of functions by exponentials with positive frequencies”, Russian Math. Surveys, 59:1 (2004), 171–180  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    10. M. A. Nalbandyan, “Representation of measurable functions by series with respect to Walsh subsystems”, Russian Math. (Iz. VUZ), 53:10 (2009), 45–56  mathnet  crossref  mathscinet  zmath  elib
    11. N. N. Kholshchevnikova, “Countably multiple null series”, Proc. Steklov Inst. Math., 280 (2013), 280–291  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    12. Kozma G., Olevskii A., “Perturbing Pla”, J. Anal. Math., 121 (2013), 279–298  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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