Geodesic
Everywhere Divergent
Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 50-59

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For a function $f\in L^1({\mathbb T})$, we investigate the sequence $(C,1)$ of mean values $\Phi(|S_k(x,f)-f(x)|)$, where $\Phi (t)\colon [0,+\infty)\to [0,+\nobreak \infty)$, $\Phi (0)=\nobreak 0$, is a continuous increasing function. We prove that if $\Phi $ increases faster than exponentially, then these means can diverge everywhere. Divergence almost everywhere of such means was established earlier.
Keywords: Fourier series, means of Fourier series, the space $L^1({\mathbf T})$. 
@article{MZM_2006_80_1_a6,
     author = {G. A. Karagulian},
     title = {Everywhere {Divergent}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {50--59},
     publisher = {mathdoc},
     volume = {80},
     number = {1},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a6/}
}
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G. A. Karagulian. Everywhere Divergent. Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 50-59. http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a6/