Discrete Gabor frames in \(\ell^2(\mathbb Z^d)\) (Q2846909)

Let \(H\) be a separable Hilbert space. A sequence \(\{x_j\}_{j\in\mathcal{J}}\) is called a frame if there exist two positive constants \(C_1\) and \(C_2\) such that, for all \(x\in H\), \(C_1\|x\|^2\leq\sum_{j\in\mathcal J}|\langle x,x_j\rangle|^2 \leq C_2\|x\|^2\), and called a Bessel sequence if the second inequality is satisfied. Given a frame \(\{x_j\}_{j\in\mathcal J}\), a frame \(\{y_j\}_{j\in\mathcal J}\) is called a dual frame for \(\{x_j\}_{j\in\mathcal J}\) if, for any \(x\in H\), \(x=\sum_{j\in\mathcal J}\langle x,x_j\rangle y_j\). Let \(G\) be a subgroup of \(\mathbb Z^d\). A set \(\mathcal D\) is called a complete digit set for \(\mathbb Z^d/G\) if \(\{G+\mathbf{m}:\;\mathbf{m}\in\mathcal D\}\) is a disjoint partition of \(\mathbb Z^d\). Given two invertible integer matrices \(A,\,B\in M_{d\times d}(\mathbb Z)\), let \(\Omega\) be a complete digit set of \(B^\ast\mathbb Z^d\) in \(\mathbb Z^d\), where \(B^\ast\) is the transpose of \(B\) and \(B^\ast\mathbb Z^d:=\{B^\ast\mathbf{m}:\;\mathbf{m}\in\mathbb Z^d\}\), and let, for any \(g\in\ell^2(\mathbb Z^d)\), \(G(A,B,\Omega,g):=\{g_{\mathbf{k},\mathbf {m}}:\;\mathbf{k}\in\Omega, \mathbf{m}\in\mathbb Z^d\}\), where \(g_{\mathbf{k},\mathbf {m}}\) is given by, for all \(\mathbf{n}\in\mathbb Z^d\), \(g_{\mathbf{k},\mathbf {m}}(\mathbf{n}) :=e^{2\pi i\langle\mathbf{k},B^{-1}\mathbf{n}\rangle}g(\mathbf{n}-A\mathbf{m})\) . If \(\{g_{\mathbf{k},\mathbf{m}}\}\) is a frame for \(H\), then it is called a Gabor frame and such \(g\) is called the Gabor atom. If two Gabor atoms \(g\) and \(h\) generate dual frames \(\{g_{\mathbf{k},\mathbf{m}}\}\) and \(\{h_{\mathbf{k},\mathbf{m}}\}\), then \((g,h)\) is called a dual frame pair. In this article, the authors prove: Let \(A,\,B\in M_{d\times d}(\mathbb Z)\) be two invertible matrices, and \(g,\,h\in\ell^2(\mathbb Z^d)\). Assume that two Gabor families \(G(A,B,\Omega,g)=\{g_{\mathbf{k},\mathbf{m}}:\;\mathbf{k}\in\Omega, \mathbf{m}\in\mathbb Z^d\}\) and \(G(A,B,\Omega,h)=\{h_{\mathbf{k},\mathbf{m}}:\;\mathbf{k}\in\Omega, \mathbf{m}\in\mathbb Z^d\}\). Then,NEWLINENEWLINE\quad(i) \((g,h)\) is a dual frame pair if and only if NEWLINE\[NEWLINE\sum_{\mathbf{m}\in\mathbb Z^d}g(\mathbf{n}-A\mathbf{m}) \overline{h(\mathbf{n}-A\mathbf{m}-B\mathbf{j})} =\frac1{|\Omega|}\delta_{\mathbf{0},\mathbf{j}}NEWLINE\]NEWLINE holds for any \(\mathbf{j}\in\mathbb Z^d\) and \(\mathbf{n}\in\mathbb Z^d\), where \(\delta_{\mathbf{0},\mathbf{j}}=1\) if \(\mathbf{j}=\mathbf{0}\) and \(\delta_{\mathbf{0},\mathbf{j}}=0\) otherwise;NEWLINENEWLINE\quad(ii) \(\{g_{\mathbf{k},\mathbf{m}}\}\) and \(\{h_{\mathbf{k},\mathbf{m}}\}\) are orthogonal if and only if NEWLINE\[NEWLINE\sum_{\mathbf{m}\in\mathbb Z^d}g(\mathbf{n}-A\mathbf{m}) \overline{h(\mathbf{n}-A\mathbf{m}-B\mathbf{j})}=0NEWLINE\]NEWLINE holds for any \(\mathbf{j}\in\mathbb Z^d\) and \(\mathbf{n}\in\mathbb Z^d\).NEWLINENEWLINEIn the article, a density theorem for frames and super-frames, and an existence theorem for the tight dual frame of Gabor type are also considered.