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- W4292079255 abstract "Given a periodic function $f$, we study the convergence almost everywhere and in norm of the series $sum_{k} c_k f(kx)$. Let $f(x)= sum_{m=1}^infty a_m sin {2pi m x}$ where $sum_{m=1}^infty a_{m }^2d(m) <infty$ and $d(m)=sum_{d|m} 1$, and let $f_n(x) = f(nx)$. We show by using a new decomposition of squared sums that for any $Ksubset N$ finite, $ |sum_{kin K} c_k f_k |_2^2 le ( sum_{m=1}^infty a_{m }^2 d(m) ) sum_{kin K } c_{k}^2d(k^2)$. If $f^s (x)= sum_{j=1}^infty frac{sin 2pi jx}{j^s}$, $s>1/2$, by only using elementary Dirichlet convolution calculus, we show that for $0< ele 2s-1$, $zeta(2s)^{-1} |sum_{kin K} c_k f^s_k |_2^2 le frac{1+e}{e } (sum_{k in K} |c_k|^2 s_{ 1+e-2s}(k) )$, where $s_h(n)=sum_{d|n}d^h$. From this we deduce that if $fin {rm BV}(T)$, $langle f,1rangle=0$ and $$sum_{k} c_k^2frac{(loglog k)^4}{(loglog log k)^2} <infty,$$ then the series $ sum_{k } c_kf_k$ converges almost everywhere. This slightly improves a recent result, depending on a fine analysis on the polydisc (cite{ABS}, th.3) ($n_k=k$), where it was assumed that $ sum_{k} c_k^2 , (loglog k)^g $ converges for some $g>4$. We further show that the same conclusion holds under the arithmetical condition $$sum_{ k } c_k^2 (loglog k)^{2 + b} s_{ -1+frac{1}{(loglog k)^{ b/3}} }(k) <infty,$$ for some $b>0$, or if $ sum_{ k} c_k^2 d(k^2) (loglog k)^2 <infty$. We also derive from a recent result of Hilberdink an $O$-result for the Riemann Zeta function involving factor closed sets. We finally prove an important complementary result to Wintner's famous characterization of mean convergence of series $sum_{k=0}^infty c_k f_k $." @default.
- W4292079255 created "2022-08-17" @default.
- W4292079255 creator A5041036701 @default.
- W4292079255 date "2014-07-10" @default.
- W4292079255 modified "2023-09-23" @default.
- W4292079255 title "An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function" @default.
- W4292079255 doi "https://doi.org/10.48550/arxiv.1407.2722" @default.
- W4292079255 hasPublicationYear "2014" @default.
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