Geodesic
Divergence everywhere of subsequences of partial sums of trigonometric Fourier series
Trudy Instituta matematiki i mehaniki, Function theory, Tome 11 (2005) no. 2, pp. 112-119

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It is proved that for any increasing sequence of natural numbers $\{m_j\}$ and any nondecreasing function $\varphi\colon[0,+\infty)\to[0,+\infty)$ satisfying the condition $\varphi(u)=o(u\ln\ln)$ ($u\to\infty$) there is a function $f\in L[0,2\pi]$ such that $$ \int_0^{2\pi}\varphi(|f(x)|)\,dx\infty, $$ and the Fourier partial sums $S_{m_j}(f)$ diverge unboundedly everywhere. 
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     author = {S. V. Konyagin},
     title = {Divergence everywhere of subsequences of partial sums of trigonometric {Fourier} series},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {112--119},
     publisher = {mathdoc},
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     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2005_11_2_a8/}
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S. V. Konyagin. Divergence everywhere of subsequences of partial sums of trigonometric Fourier series. Trudy Instituta matematiki i mehaniki, Function theory, Tome 11 (2005) no. 2, pp. 112-119. http://geodesic.mathdoc.fr/item/TIMM_2005_11_2_a8/