On everywhere divergence of trigonometric Fourier series

S. V. Konyagin

Sbornik. Mathematics, Tome 191 (2000) no. 1, pp. 97-120

Résumé

The following theorem is established. Theorem. {\it Let a function $\varphi\colon[0,+\infty)\to[0,+\infty)$ and a sequence $\{\psi(m)\}$ satisfy the following condition: the function $\varphi(u)/u$ is non-decreasing on $(0,+\infty)$, $\psi(m)\geqslant 1$ $(m=1,2,\dots)$ and $\varphi(m)\psi(m)=o(m\sqrt{\ln m}/\sqrt{\ln\ln m}\,)$ as $m\to\infty$. Then there is a function $f\in L[-\pi,\pi]$ such that $$ \int _{-\pi}^\pi\varphi(|f(x)|)\,dx\infty $$ and $\limsup_{m\to\infty}S_m(f,x)/\psi(m)=\infty$ for all $x\in[-\pi,\pi]$ here $S_m(f)$ is the $m$-th partial sum of the trigonometric Fourier series of $f$}.