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Keywords:
sine series; cosine series; double sine series; sine-cosine series; double cosine series; uniform convergence; regular convergence; general monotone sequence; general monotone double sequence; supremum bounded variation
sine series; cosine series; double sine series; sine-cosine series; double cosine series; uniform convergence; regular convergence; general monotone sequence; general monotone double sequence; supremum bounded variation
Summary:
It is a classical problem in Fourier analysis to give conditions for a single sine or cosine series to be uniformly convergent. Several authors gave conditions for this problem supposing that the coefficients are monotone, non-negative or more recently, general monotone. There are also results for the regular convergence of double sine series to be uniform in case the coefficients are monotone or general monotone double sequences. In this paper we give new sufficient conditions for the uniformity of the regular convergence of sine-cosine and double cosine series, which are necessary as well in case the coefficients are non-negative. The new results also bring necessary and sufficient conditions for the uniform regular convergence of double trigonometric series in complex form.
References:
[1] Chaundy, T. W., Jolliffe, A. E.: The uniform convergence of a certain class of trigonometrical series. Proc. London Math. Soc. 15 (1916), 214-216. MR 1576557
[2] Dyachenko, M., Tikhonov, S.: General monotone sequences and convergence of trigonometric series. Topics in Classical Analysis and Applications in Honor of Daniel Waterman. World Scientific Hackensack, NJ L. De Carli et al. (2008), 88-101. MR 2569380 | Zbl 1167.42303
[3] Dyachenko, M., Tikhonov, S.: Integrability and continuity of functions represented by trigonometric series: coefficients criteria. Stud. Math. 193 (2009), 285-306. DOI 10.4064/sm193-3-5 | MR 2515585 | Zbl 1169.42001
[4] Kórus, P.: Remarks on the uniform and $L^1$-convergence of trigonometric series. Acta Math. Hung. 128 (2010), 369-380. DOI 10.1007/s10474-010-9217-4 | MR 2670995 | Zbl 1240.42015
[5] Kórus, P.: On the uniform convergence of double sine series with generalized monotone coefficients. Period. Math. Hung. 63 (2011), 205-214. DOI 10.1007/s10998-011-8205-y | MR 2853213 | Zbl 1265.42007
[6] Kórus, P., Móricz, F.: On the uniform convergence of double sine series. Stud. Math. 193 (2009), 79-97. DOI 10.4064/sm193-1-4 | MR 2506415 | Zbl 1167.42002
[7] Móricz, F.: Some remarks on the notion of regular convergence of multiple series. Acta Math. Hung. 41 (1983), 161-168. DOI 10.1007/BF01994074 | MR 0704536
[8] Tikhonov, S.: Trigonometric series with general monotone coefficients. J. Math. Anal. Appl. 326 (2007), 721-735. DOI 10.1016/j.jmaa.2006.02.053 | MR 2277815 | Zbl 1141.42004
[9] Tikhonov, S.: Best approximation and moduli of smoothness: Computation and equivalence theorems. J. Approx. Theory 153 (2008), 19-39. DOI 10.1016/j.jat.2007.05.006 | MR 2432551 | Zbl 1215.42002
[10] Zhak, I. E., Shneider, A. A.: Conditions for uniform convergence of double sine series. Russian Izv. Vyssh. Uchebn. Zaved., Mat. 53 (1966), 44-52.
[11] Zhou, S. P., Zhou, P., Yu, D. S.: Ultimate generalization to monotonicity for uniform convergence of trigonometric series. Sci. China, Math. 53 (2010), 1853-1862. DOI 10.1007/s11425-010-3138-0 | MR 2665519 | Zbl 1211.42006
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