Geodesic
On the rate of convergence of Ces\`aro means of double Fourier series of functions of generalized bounded variation
Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 38-62.

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In this paper, the rate of convergence of Cesàro means of the double Fourier series of a $2\pi$-periodic function in each variable and of generalized bounded variation, is estimated. The result obtained is a generalization of a result of S. M. Mazhar for a single Fourier series and of our earlier result for a function of two variables.
Keywords: double Fourier series, generalized bounded variation, pointwise convergence, rate of convergence, Cesàro mean. 
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R. K. Bera; B. L. Ghodadra. On the rate of convergence of Ces\`aro means of double Fourier series of functions of generalized bounded variation. Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 38-62. http://geodesic.mathdoc.fr/item/CHEB_2023_24_2_a2/

[1] Bakhvalov A. N., “Divergence everywhere of the Fourier series of continuous functions of several variables”, Sb. Math., 188:8 (1997), 1153–1170 | DOI | MR | Zbl

[2] Bakhvalov A. N., “$\lambda$-divergence of the Fourier series of continuous functions of several variables”, Math. Notes, 72:4 (2002), 454–465 | Zbl

[3] Bakhvalov A. N., “Summation of Fourier series of functions from multidimensional Waterman classes by Cesàro methods”, Dokl. Math., 83:2 (2011), 247–249 | DOI | MR | Zbl

[4] Bojanić R., “An estimate of the rate of convergence of Fourier series of functions of bounded variation”, Publ. Inst. Math. (Beograd), 26:40 (1979), 57–60

[5] Bojanić R., Waterman D., “On the rate of convergence of Fourier series of functions of generalized bounded variation”, Akad. Nauka Umjet. Bosne Hercegov. Rad. Odjelj. Prirod. Mat. Nauka, 22 (1983), 5–11

[6] Bojanić R., Mazhar S. M., “An estimate of the rate of convergence of Cesàro means of Fourier series of functions of bounded variation”, Mathematical analysis and its applications (Kuwait, 1985), KFAS Proc. Ser., 3, Pergamon, Oxford, 1988, 17–22

[7] Bera R. K., Ghodadra B. L., “On the rate of convergence of double Fourier series of functions of generalized bounded variation”, Acta Sci. Math. (Szeged), 88 (2022), 723–737 | DOI | Zbl

[8] Bahduh M., Nikishin E. M., “The convergence of the double Fourier series of continuous functions”, Siberian Math. J., 14:6 (1973), 832–839 | MR

[9] D'yachenko M. I., “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47 (1992) | MR | Zbl

[10] Hardy G. H., “On double Fourier series”, Quart. J. Math., 35 (1906), 53–79

[11] Jordan C., “Sur la series de Fourier”, C. R. Acad. Sci. Paris, 92 (1881), 228–230

[12] Limaye B. V., Ghorpade S. R., A course in multivariable calculus and analysis, Springer, New York–Dordrecht–Heidelberg–London, 2010

[13] Mazhar S. M., “On the rate of convergence of Cesàro means of Fourier series of functions of generalized bounded variation”, Chinese J. Math., 13:3 (1985), 195–202 | Zbl

[14] Móricz F., “A quatitiative version of the Dirichlet-Jordan test for double Fourier series”, Journal of Approx. Theory, 71 (1992), 344–358

[15] Móricz F., “Approximation by rectangular partial sums of double conjugate Fourier series”, Journal of Approx. Theory, 103 (2000), 130–150

[16] Perlman S., “Functions of generalized variation”, Polish Acad. Sci. Inst. Math., 105:3 (1979), 199–211 | MR

[17] Zygmund A., Trigonometric series, v. I, Cambridgee Univ. Press, 1959

[18] Zhizhiashvili L., “Convergence and summability of Fourier series”, Soobšč. Akad. Nauk Gruzin. SSR, 1962, 257–261 | MR | Zbl

[19] Zhizhiashvili L., Conjugate functions and trigonometric series, Izdat. Tbilis. Univ., Tbilisi, 1969 (In Russian)

[20] Zhizhiashvili L. V., “On some problems of the theory of simple and multiple trigonometric series”, Russian Math. Surveys, 28:2 (1973), 65–119 | Zbl