A characterization of the Riesz family of shifts of functions on locally compact Abelian groups

Abstract

For a locally compact Abelian group $G$, we give a necessary and sufficient condition for shifts of a function $\varphi \in L_2\left(G\right)$ to be a Riesz family. Also, for a finite family $\Phi$ of compactly supported functions in $L_2\left(G\right)$, we show that the shifts of $\Phi$ constitute a Riesz family if and only if the nets $\left(\varphi \right(\xi \eta ){)}_{\eta \in L^{\perp }}$, $\varphi \in \Phi$, are linearly independent for all $\xi \in \hat{G}$.