Divergence everywhere of the~Fourier series of continuous functions of several variables

A. N. Bakhvalov

Sbornik. Mathematics, Tome 188 (1997) no. 8, pp. 1153-1170

Résumé

The Fourier series of a function $f$ of $n$ real variables is said to be $\lambda$-convergent at a point $\vec x$ for $\lambda \geqslant 1$ if there exists the limit $$ \lim _{\min \limits _kM_k\to +\infty}S_{\vec M}(\vec x,f) $$ over all indices $\vec M=(M_1,\dots ,M_n)$ such that $1/\lambda \leqslant M_k/M_j\leqslant \lambda$ for all $k$ and $j$. An example of a continuous function of $2m$ variables with modulus of continuity $$ \omega (F,\delta )=\underset {\delta\to +0}O\Bigl (\ln ^{-m}\frac 1\delta \Bigr) $$ is constructed such that the Fourier series of $F$ with respect to the trigonometric system $\lambda$-diverges everywhere for an arbitrary fixed $\lambda >1$.