On the semi-regular frames of translates

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 \(\Phi^{\mathcal{A}}_{f}:=\frac{1}{|\det \mathcal{A}|}\sum_{k\in \mathbb{Z}^d}|\hat{f}(\mathcal{A}^T)^{-1}(\cdot+k)|^2\) be the periodization of \(|\hat{f}|^2\). By using \(\Phi^{\mathcal{A}}_{f}\), 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.