On the summability to infinity of trigonometric series and series in the Walsh system

A particular result of this paper is that there exists a trigonometric series
$$ \sum_{k=1}^\infty a_k\cos n_kx+b_k\sin n_kx\qquad(n_1<n_2<\cdots) $$
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.
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