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

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 _{kin {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.