On uniform convergence of Fourier series

Z. A. Chanturiya

Sbornik. Mathematics, Tome 29 (1976) no. 4, pp. 475-495

Résumé

Let $f(x)$ be a continuous $2\pi$-periodic function, $S_n(f, x)$ the $n$th partial sum of its Fourier series, $\omega(\delta,f)$ the modulus of continuity and $v(n,f)$ the modulus of variation of $f(x)$. In this paper the following theorems are proved. Theorem 1. {\it For $f(x)\in C(0,2\pi)$ the estimate $$ \|f(x)-S_n(f, x)\|_{C(0,2\pi)}\leqslant C\min_{1\leqslant m\leqslant[\frac{n-1}2]}\Biggl\{\omega\biggl(\frac1n,f\biggr)\sum_{k=1}^m\frac1k+\sum_{k=m+1}^{[\frac{n-1}2]}\frac{v(k,f)}{k^2}\Biggr\},\quad n\geqslant3, $$ holds, where $C$ is an absolute constant.} From this theorem there follows an estimate of Lebesgue and an estimate of Oskolkov. Theorem 2. {\it In order that all Fourier series of class $H^\omega\cap V[v(n)]$ converge uniformly it is necessary and sufficient that $$ \lim_{n\to\infty}\min_{1\leqslant m\leqslant[\frac{n-1}2]}\Biggl\{\omega\biggl(\frac1n\biggr)\sum_{k=1}^m\frac1k+\sum_{k=m+1}^{[\frac{n-1}2]}\frac{v(k)}{k^2}\Biggr\}=0. $$ } Bibliography: 20 titles.

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