Uniform convergence of hyperbolic partial sums of multiple Fourier series. (Q556590)

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)].