$$\ell ^p$$ ℓ 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.