Frames of translates by semiregular sampling

Abstract

In this paper, we consider some kinds of semiregular frames  of translates on the Hilbert space $L^2(\mathbb{R}^d)$. More precisely,  we investigate the frames of the form $\{T_{\mathcal{A}k}f\}_{k\in \mathbb{Z}^d}$, where $\mathcal{A}$ is a real  invertible  $d\times d$ matrix and $f\in L^2(\mathbb{R}^d)$,  and it is a frame for the  closed subspace generated by $\{T_{\mathcal{A}k}f\}_{k\in \mathbb{Z}^d}$.