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 Izv. Vyssh. Uchebn. Zaved. Mat., 2008, Number 5, Pages 83–91 (Mi ivm1281)

On the best convergence of multiple trigonometric series

A. I. Rubinshtein

Chair of Higher Mathematics, Faculty of Electronics and System Engineering, Moscow State Forest University, Mytishchi, Moscow region

Abstract: We consider the best convergence of multiple trigonometric series. We indicate essential distinction of the behavior (in this sense) of multiple series from that of simple ones. In particular, the well-known result obtained by S. N. Bernshtein on the best convergence of a series with an odd ratio of frequencies does not hold for a multiple series in the case of the approximation by polynomials with harmonics from rectangles (in the sense of Pringsheim), but it is true for “angular” approximations.

Keywords: the best convergence, summation over rectangles, summation “over angles”.

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2008, 52:5, 72–79

Bibliographic databases:

UDC: 517

Citation: A. I. Rubinshtein, “On the best convergence of multiple trigonometric series”, Izv. Vyssh. Uchebn. Zaved. Mat., 2008, no. 5, 83–91; Russian Math. (Iz. VUZ), 52:5 (2008), 72–79

Citation in format AMSBIB
\Bibitem{Rub08} \by A.~I.~Rubinshtein \paper On the best convergence of multiple trigonometric series \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2008 \issue 5 \pages 83--91 \mathnet{http://mi.mathnet.ru/ivm1281} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2445187} \zmath{https://zbmath.org/?q=an:1158.42305} \elib{http://elibrary.ru/item.asp?id=11034937} \transl \jour Russian Math. (Iz. VUZ) \yr 2008 \vol 52 \issue 5 \pages 72--79 \crossref{https://doi.org/10.3103/S1066369X08050095}