Geodesic
Subsequences of the Fourier sums of functions with a given modulus of continuity
Sbornik. Mathematics, Tome 17 (1972) no. 3, pp. 441-465 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is proved that for each modulus of continuity $\omega(\delta)$ in the class $H_\omega$ there exists a function $f$ such that for any increasing sequence $\{n_i\}_{i=1}^\infty$ of natural numbers there is a point $x$ at which \begin{gather*} \varlimsup_{t\to\infty}\frac{S_{n_i}(f,x)-f(x)}{\omega(n_i^{-1})\log{n_i}}\geqslant A>0,\\ \varliminf_{t\to\infty}\frac{S_{n_i}(f,x)-f(x)}{\omega (n_i^{-1})\log{n_i}} \leqslant-A<0, \end{gather*} where $A$ is an absolute constant. Also considered is the approximation by sequences of Fourier sums of functions of bounded variation with given modulus of continuity. Bibliography: 7 titles. 
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K. I. Oskolkov. Subsequences of the Fourier sums of functions with a given modulus of continuity. Sbornik. Mathematics, Tome 17 (1972) no. 3, pp. 441-465. http://geodesic.mathdoc.fr/item/SM_1972_17_3_a6/

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