Abstract
It follows from results of A. Yudin, V. Yudin, E. Belinskii, and I. Liflyand that if \(m \geqslant 2\) and a \(2\pi \)-periodic (in each variable) function \(f(x) \in C(T^m )\) belongs to the Nikol'skii class \(h_\infty ^{(m - 1)/2} (T^m )\), then its multiple Fourier series is uniformly convergent over hyperbolic crosses. In this paper, we establish the finality of this result. More precisely, there exists a function in the class \(h_\infty ^{(m - 1)/2} (T^m )\) whose Fourier series is divergent over hyperbolic crosses at some point.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
A. A. Yudin and V. A. Yudin, “Discrete embedding theorems and Lebesgue constants,” Mat. Zametki [Math. Notes], 22 (1977), no. 3, 220–226.
E. S. Belinskii, “The behavior of Lebesgue constants for some methods of summation of multiple Fourier series,” in: Metric Questions of the Theory of Functions and Mappings [in Russian], Naukova Dumka, Kiev, 1977, pp. 19–39.
I. R. Liflyand, “The exact order of the Lebesgue constants of hyperbolic partial sums of multiple Fourier series” (Manuscript deposited at VINITI on July 12, 1980; deposition no. 2349–80) [in Russian], Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. (VINITI), Moscow, 1980.
M. I. D'yachenko, “ U-convergence of multiple Fourier series,” Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 59 (1995), no. 2, 128–142.
M. I. D'yachenko, “The order of growth of the Lebesgue constants of Dirichlet kernels of monotone type,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1989), no. 6, 33–37.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
D'yachenko, M.I. Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series. Mathematical Notes 76, 673–681 (2004). https://doi.org/10.1023/B:MATN.0000049666.00784.9d
Issue Date:
DOI: https://doi.org/10.1023/B:MATN.0000049666.00784.9d