Geodesic
On the best convergence of multiple trigonometric series
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2008), pp. 83-91.

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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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A. I. Rubinshtein. On the best convergence of multiple trigonometric series. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2008), pp. 83-91. http://geodesic.mathdoc.fr/item/IVM_2008_5_a9/

[1] Bernshtein S. N., Sobranie sochinenii, T. 24, Izd-vo AN SSSR, M., 1954, S. 178–183. | MR

[2] Rubinshtein A. I., “O nailuchshe skhodyaschikhsya ryadakh”, Dokl. rasshirennykh zasedanii seminara IPM im. I. N. Vekua, T. 1 (23–27 apr. 1985), no. 2, Tbilisi, 1985, 114–117 | MR

[3] Nikolskii S. M., “Priblizhenie funktsii trigonometricheskimi polinomami v srednem”, Izv. AN SSSR. Ser. matem., 10 (1946), 207–256

[4] Singer I., “Caractérisation des éléments de meilleuré approximation dans un espace de Banach quelconque”, Acta Sci. Math. Szeged, 17:3–4 (1956), 181–189 | MR | Zbl

[5] Rubinshtein A. I., “O nailuchshe skhodyaschikhsya v $L_p[0; 2\pi)$ ryadakh”, Matem. zametki., 52:6 (1992), 100–108 | MR

[6] Rubinshtein A. I., “O nailuchshei skhodimosti”, Matem. sb., 192:2 (2001), 119–138 | MR | Zbl