Pointwise convergence of almost periodic Fourier series and associated series of dilates

TL;DR

This paper establishes pointwise convergence and maximal inequalities for Fourier series with almost periodic frequencies in Stepanov spaces, extending classical results and providing new conditions for convergence related to Sidon sequences.

Contribution

It proves maximal inequalities and convergence results for almost periodic Fourier series in Stepanov spaces, generalizing known theorems and introducing new conditions for convergence.

Findings

Maximal inequality for Fourier series in Stepanov space.

Almost everywhere convergence of the series under Wiener's condition.

Convergence criteria linked to Sidon sequences and specific frequency conditions.

Abstract

Let $\mathcal S^2$ be the Stepanov space and let $ \lambda_n\uparrow\infty$. Let $(a_n)_{n\ge 1}$ be satisfying Wiener's condition $A:= \sum_{n\ge 1} \big(\sum_{k\, :\, n\le \lambda_k \le n+1}|a_k|\big)^2 <\infty$. We prove that $\big\| \sup_{N\ge 1} \big|\sum_{n=1}^Na_n{\rm e}^{i\lambda_n t}\big| \, \big\|_{\mathcal S^2}\le C\, A^{1/2} $ where $C>0$ denotes a universal constant. Moreover, the series $\sum_{n\ge 1} a_n{\rm e}^{it\lambda_n }$ converges for $\lambda$-a.e. $t\in \mathbb R$. This contains as a special case Hedenmalm and Saksman result for Dirichlet series. We also obtain maximal inequalities for corresponding series of dilates. Let $1\le p,q\le 2$ be such that $1/p+1/q=3/2$. Then for any sequence $(\alpha_n)_{n\ge 1}$ and $(\beta_n)_{n\ge 1}$ of complex numbers such that $K:=\sum_{n\ge 1} \big(\sum_{k\,:\, n\le \lambda_k< n+1}|\alpha_k|\,\big)^p <\infty$ and $L:=\sum_{n\ge 1} \big(\sum_{k\,:\, n\le \mu_k< n+1} |\beta_k|\,\big)^q <\infty$, we have $$ \Big\|\sup_{N\ge 1} \big|\sum_{n=1}^N \alpha_n D(\lambda_n t)\big|\, \Big\|_{\mathcal S^2} \le C\, K^{1/p}\, L^{1/q } $$ where $D(t)= \sum_{n\ge 1}\beta_n {\rm e}^{i\mu_n t}$ is defined in $\mathcal S^2$. Moreover, the series $\sum_{n\ge 1} \alpha_n D(\lambda_nt)$ converges in $\mathcal S^2$ and for $\lambda$-a.e. $t\in \mathbb R$. We further show that if $\{\lambda_k, k\ge 1\}$ satisfies the following condition $$\sum_{ k\not=\ell\,,\, k'\not=\ell'\atop (k,\ell)\neq(k',\ell')}\big(1-|(\lambda_k-\lambda_\ell)-(\lambda_{k'}-\lambda_{\ell'}) |\big)_+^2 \, <\infty,$$ then the series $\sum_{k} a_k {\rm e}^{i\lambda_kt}$ converges on a set of positive Lebesgue measure, only if the series $\sum_{k=1}^\infty |a_k|^2$ converges. The above condition is in particular fulfilled when $\{\lambda_k, k\ge 1\}$ is a Sidon sequence.