Splitting the Riesz basis condition for systems of dilated functions]{Splitting the Riesz basis condition for systems of dilated functions through Dirichlet series

TL;DR

This paper characterizes when systems of dilated functions form Bessel, Riesz, or frame sequences using Dirichlet series and Hardy space multipliers, extending to multivariate cases.

Contribution

It provides new characterizations of dilated systems' properties via Dirichlet series and Hardy space multipliers, including multivariate extensions.

Findings

Characterization of Bessel and Riesz properties via Dirichlet series

Extension of results to multivariate dilated systems

Connections established with Hardy spaces on the polytorus

Abstract

Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different properties of dilations systems of a fixed function $\varphi \in L^2(0,1)$. More precisely, we study when the system $\{\varphi(nx)\}_n$ is a Bessel sequence, a Riesz sequence, or it satisfies the lower frame bound. We are able to characterize these properties in terms of multipliers of the Hardy space $\mathcal{H}^2$ of Dirichtet series and, also, in terms of Hardy spaces on the infinite polytorus. We also address the multivariate case.