English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Given a periodic function $f$, we study the convergence almost everywhere andin norm of the series $\sum_{k} c_k f(kx)$. Let $f(x)= \sum_{m=1}^\infty a_m\sin {2\pi m x}$ 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 newdecomposition of squared sums that for any $K\subset \N$ finite, $ \|\sum_{k\inK} c_k f_k \|_2^2 \le ( \sum_{m=1}^\infty a_{m }^2 d(m)  ) \sum_{k\in K } c_{k}^2d(k^2)$. If $f^s (x)= \sum_{j=1}^\infty \frac{\sin2\pi jx}{j^s}$,  $s>1/2$, by only using  elementary Dirichlet convolution calculus, we show that for $0< \e\le 2s-1$,$\zeta(2s)^{-1} \|\sum_{k\in K} c_k f^s_k \|_2^2 \le \frac{1+\e}{\e } (\sum_{k\in K}  |c_k|^2 \s_{ 1+\e-2s}(k) )$, where $\s_h(n)=\sum_{d|n}d^h$. From this wededuce that if $f\in {\rm BV}(\T)$, $\langle f,1\rangle=0$ and  $$\sum_{k} c_k^2\frac{(\log\log k)^4}{(\log\log \log k)^2} <\infty,$$ thenthe series $ \sum_{k } c_kf_k$ converges almost everywhere.  This slightly improves a recent result, depending on a fine analysis on thepolydisc (\cite{ABS}, th.3) ($n_k=k$), where it was assumed that $ \sum_{k}c_k^2 \, (\log\log k)^\g $ converges for some $\g>4$. We further show that thesame conclusion holds under the arithmetical condition $$\sum_{ k }  c_k^2 (\log\log k)^{2 + b} \s_{ -1+\frac{1}{(\log\log k)^{ b/3}} }(k)<\infty,$$ for some $b>0$, or if $ \sum_{ k} c_k^2 d(k^2) (\log\log k)^2<\infty$. We also derive from a recent result of Hilberdink an $\O$-result forthe Riemann Zeta function involving factor closed sets.  We finally prove an important complementary result to Wintner's famouscharacterization of mean convergence of series $\sum_{k=0}^\infty c_k f_k $.

An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function