Letter to the editor: A new criterion for Gabor frames
The main result gives a condition for a Gabor system \(\{e^{2\pi i \beta n\cdot t}g(t-\alpha k)\}_{k,n\in \mathbb{Z}^d}\) to be a frame for \(L^2(\mathbb{R}^d)\). Assume that \(\alpha \beta\) is irrational and that \(g\) belongs to the Wiener amalgam space \(W(C,\ell^1)\), i.e., \(g\) is continuous and \( \sum_{k\in \mathbb{Z}^d} \sup_{t\in [0,1]^d}|g(t+k)|<\infty\). Let \(G\) denote the bi-infinite matrix-valued function whose \(jl\)-th entry is \(G_{jl}(t)=\sum_{k\in \mathbb{Z}^d} \overline{g}(x-l/\beta -\alpha k)g(x-j/\beta -\alpha k)\). Then \(\{e^{2\pi i \beta n\cdot t}g(t-\alpha k)\}_{k,n\in Z^d}\) is a frame if and only if there exists at least one \(\xi \in [0, \alpha]^d\) such that \(G(\xi)\) is invertible on \(\ell^2\).
scientific article; zbMATH DE number 1890477