On the semi-regular frames of translates

TL;DR

This paper characterizes when translates of a function form various types of frames or bases in $L^2( ^d)$ using a periodization of its Fourier transform, and provides an example of a Parseval frame that is not Riesz.

Contribution

It introduces a characterization of semi-regular translates as frames, Riesz bases, or orthonormal bases using a specific periodization function, and constructs a novel example.

Findings

Characterization of translates as Bessel sequences, frames, Riesz bases, or orthonormal bases.

Conditions involving the periodization function for these properties.

An example of a Parseval frame that is not a Riesz sequence.

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 $\Phi^{\mathcal{A}}_{f}:=\frac{1}{|\det \mathcal{A}|}\sum_{k\in \mathbb{Z}^d}|\hat{f}(\mathcal{A}^T)^{-1}(\cdot+k)|^2$ 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.