Geodesic
On the summability to infinity of trigonometric series and series in the Walsh system
Sbornik. Mathematics, Tome 36 (1980) no. 3, pp. 427-439

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A particular result of this paper is that there exists a trigonometric series $$ \sum_{k=1}^\infty a_k\cos n_kx+b_k\sin n_kx\qquad(n_1\cdots) $$ which is almost everywhere on $(0,2\pi)$ summable to $+\infty$ by all methods $(C,\alpha>0)$ and by the method $A$; moreover $$ \sum_{k=1}^\infty|a_k|^{2+\varepsilon}+|b_k|^{2+\varepsilon}+\infty $$ for any $\varepsilon>0$, and also $\sum_{k=1}^\infty1/n_k+\infty$. An analogous assertion is proved for series in the Walsh system. Bibliography: 13 titles. 
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     author = {L. A. Shaginyan},
     title = {On the summability to infinity of trigonometric series and series in the {Walsh} system},
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L. A. Shaginyan. On the summability to infinity of trigonometric series and series in the Walsh system. Sbornik. Mathematics, Tome 36 (1980) no. 3, pp. 427-439. http://geodesic.mathdoc.fr/item/SM_1980_36_3_a8/