$\ell ^p$-Linear Independence of the System of Integer Translates

Abstract

Various properties of the system ${\mathcal {B}}_{\psi }$ of integer translates of a square integrable function $\psi \in L^2({\mathbb {R}})$ can be completely described in terms of the periodization function $p_{\psi }(\xi )=\sum _{k\in {\mathbb {Z}}}|\widehat{\psi }(\xi +k)|^2$. In this paper, we consider the problem of $\ell ^p$-linear independence, where $p>2$. The results we present include the method of construction for one type of counterexamples to several naturally taken conjectures, a new sufficient condition for $\ell ^p$-linear independence and a characterization theorem having an additional assumption on ${\mathcal {B}}_{\psi }$. In the latter, we obtain the characterization in terms of the sets of multiplicity of Lebesgue measure zero.