On the semi-regular frames of translates

Abstract

In this note, we fix a real invertible $d\times d$ matrix $\mathcal{A}$ and consider $\mathcal{A}{\mathbb{Z}}^d$ as an index set. For $f\in L^2({\mathbb{R}}^d)$, let ${\Phi }_f^{\mathcal{A}}:=\frac{1}{|\det \mathcal{A}|}{\sum }_{k\in {\mathbb{Z}}^d}|\hat{f}({\mathcal{A}}^T{)}^{-1}(\cdot +k){|}^2$ be the periodization of $|\hat{f}{|}^2$. By using ${\Phi }_f^{\mathcal{A}}$, among other things, we characterize when the sequence ${\tau }_{\mathcal{A}}(f):=\{f(\cdot -\mathcal{A}k){\}}_{k\in {\mathbb{Z}}^d}$ is a Bessel sequence, frame of translates, Riesz basis, or orthonormal basis. And finally, we construct an example, in which ${\tau }_{\mathcal{A}}(f)$ is a Parseval frame of translates, but not a Riesz sequence.

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