$\lambda$-Divergence of the Fourier Series of Continuous Functions of Several Variables

A. N. Bakhvalov

Matematičeskie zametki, Tome 72 (2002) no. 4, pp. 490-501

Résumé

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 $\lambda$-convergent if the limit of rectangular partial sums over all indices $\vec M=(M_1,\dots ,M_n)$, for which $1/\lambda \le M_j/M_k\le \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 $\omega (F,\delta)=\underset {\delta \to +0}\to O(\ln ^{-m}(1/\delta))$ such that its Fourier series is $\lambda$-divergent everywhere for any $\lambda >1$.