Abstract
In this paper, we consider the behavior of rectangular partial sums of the Fourier series of continuous functions of several variables with respect to the trigonometric system. The Fourier series is called λ-convergent if the limit of rectangular partial sums over all indices \(\vec M = (M_1 ,...,M_n )\) for which \(1/\lambda \leqslant M_j /M_k \leqslant \lambda \) for all j and k exists. In the space of arbitrary even dimension 2m we construct an example of a continuous function with an estimate of the modulus of continuity \(\omega (F,\delta ) = \mathop O\limits_{\delta \to + 0} ({\text{ln}}^{ - m} (1/\delta ))\) such that its Fourier series is λ-divergent everywhere for any \(\lambda >1\).
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REFERENCES
C. Fefferman, “On the convergence of multiple Fourier series,” Bull. Amer. Math. Soc., 77 (1971), no. 5, 744-745.
C. Fefferman, “On the divergence of multiple Fourier series,” Bull. Amer. Math. Soc., 77 (1971), no. 2, 191-195.
M. Bakhbukh and E. M. Nikishin, “On the convergence of double Fourier series for continuous functions,” Sibirsk. Mat. Zh. [Siberian Math. J.], 14 (1973), no. 6, 1189-1199.
K. I. Oskolkov, “An estimate for the rate of approximation of a continuous function and its conjugate by the Fourier sums on a set of complete measure,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 38 (1974), no. 6, 1373-1407.
L. V. Zhizhiashvili, “On the convergence of multiple trigonometric Fourier series,” Soobshch. Akad. Nauk GSSR, 80 (1975), no. 1, 17-19.
M. I. Dyachenko, “Some problems of the theory of multiple trigonometric series,” Uspekhi Mat. Nauk [Russian Math. Surveys], 47 (1992), no. 5, 97-162.
A. N. Bakhvalov, “On the divergence everywhere of the Fourier series of continuous functions of several variables,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 188 (1997), no. 8, 45-62.
R. Bojanic, “An estimate of the rate of convergence for Fourier series of functions of bounded variation,” Publ. Inst. Math. (Belgrade), 26 (1979), no. 40, 57-60.
L. V. Zhizhiashvili, “On some problems of the theory of simple and multiple trigonometric series” Uspekhi Mat. Nauk [Russian Math. Surveys], 28 (1973), no. 2, 65-119.
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Bakhvalov, A.N. λ-Divergence of the Fourier Series of Continuous Functions of Several Variables. Mathematical Notes 72, 454–465 (2002). https://doi.org/10.1023/A:1020524110083
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DOI: https://doi.org/10.1023/A:1020524110083