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 Mat. Zametki, 1973, Volume 14, Issue 5, Pages 655–666 (Mi mz9950)

Cantor–Lebesgue theorem for double trignometric series

V. S. Panferov

M. V. Lomonosov Moscow State University

Abstract: Let $||\cdot||$ be a norm in $\mathbf{R}^2$ and let $\Gamma$ be the unit sphere induced by this norm. We call a segment joining points $x, y\in\mathbf{R}^2$ rational if $(x_1-y_1)/(x_2-y_2)$ èëè $(x_2-y_2)/(x_1-y_1)$ is a rational number. Let $\Gamma$ be a convex curve containing no rational segments. Satisfaction of the condition
$$T_\nu(x)=\sum_{||n||=\nu}c_n e^{2\pi i(n_1x_1+n_2x_2)}\to0\qquad (\nu\to\infty)$$
in measure on the set $E\subset[-\frac12, \frac12)\times[-\frac12, \frac12)=T^2$ of positive planar measure implies $||T_\nu||_{L_4}(T^2)\to0$ ($\nu\to\infty$). If, however, $\Gamma$ contains a rational segment, then there exist a sequence of polynomials $\{T_\nu\}$ and a set $E\subset T^2$, $|E|>0$, such that $T_\nu(x)\to0$ ($\nu\to\infty$) on $E$; however, $|c_n|\not\to0$ for $||n||\to\infty$.

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English version:
Mathematical Notes, 1973, 14:5, 936–942

Bibliographic databases:

UDC: 517.5

Citation: V. S. Panferov, “Cantor–Lebesgue theorem for double trignometric series”, Mat. Zametki, 14:5 (1973), 655–666; Math. Notes, 14:5 (1973), 936–942

Citation in format AMSBIB
\Bibitem{Pan73} \by V.~S.~Panferov \paper Cantor--Lebesgue theorem for double trignometric series \jour Mat. Zametki \yr 1973 \vol 14 \issue 5 \pages 655--666 \mathnet{http://mi.mathnet.ru/mz9950} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=330917} \zmath{https://zbmath.org/?q=an:0281.42016} \transl \jour Math. Notes \yr 1973 \vol 14 \issue 5 \pages 936--942 \crossref{https://doi.org/10.1007/BF01462253} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. B. I. Golubov, “On convergence of Riesz spherical means of multiple Fourier series”, Math. USSR-Sb., 25:2 (1975), 177–197
2. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171
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