Uniform convergence of hyperbolic partial sums of multiple Fourier series.
type: Item
instance of: scholarly article
title: Uniform convergence of hyperbolic partial sums of multiple Fourier series.
zbMATH Open document ID: 1076.42006
author: M. I. Dyachenko
published in: Mathematical Notes
publication date: 2005-06-21T00:00:00Z
review text: In the paper functions \(f\), \(2\pi\)-periodic in each of \(m\) variables, from the Nikol'skij class \(H_\infty^{m/2-1/2}\) are studied. The latter is defined to be the class of \(L_p\) functions for which the \(k\)th modulus of smoothness \(\omega_k (f;\delta)_p\), in the \(L_p\) metric, is \(O(\delta^{m/2-1/2})\) for small \(\delta\) for all \(k>m/2-1/2.\) Hyperbolic partial sums of \(f\) \[ Y_N(f;x)=\sum\limits_{n\in\Gamma_N}\hat f(n)e^{inx}, \] where \(\hat f(n)\) is the \(n\)th Fourier coefficient of \(f\) and \(nx=n_1x_1+...+ n_mx_m,\) are generated by hyperbolic crosses \[ \Gamma_N=\{n=(n_1,...,n_m)\in{\mathbb Z}^m: | n_1...n_m| \leq N^m\quad\text{ and} \quad \max_{1\leq j\leq m}| n_j| \leq N^m\}. \] The main result of this paper reads as follows. There exists \(f\in H_\infty^{m/2-1/2}\) such that \(Y_N(f;x)\) are divergent, as \(N\to\infty,\) at some point \(x.\) Results on Lebesgue constants of hyperbolic partial sums are used in the proof. The most general one due to the reviewer should be referred to as [``Exact order of the Lebesgue constants of hyperbolic partial sums of multiple Fourier series'', Math. Notes 39, 369--374 (1986); translation from Mat. Zametki 39, No.~5, 674--683 (1986; Zbl 0607.42013)].
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 42B05
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 2177679
zbMATH Keywords: multiple Fourier series
zbMATH Keywords: partial sums
zbMATH Keywords: hyperbolic cross
zbMATH Keywords: Nikol'skij class
zbMATH Keywords: modulus of continuity
MaRDI profile type: Publication
full work available at URL: b:matn.0000049666.00784.9d
OpenAlex ID: W2015402212
DOI: 10.1023/B:MATN.0000049666.00784.9D
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description: scientific article; zbMATH DE number 2177679