The uniformity of non-uniform Gabor bases
It is well known that the 1-dimensional regular Gabor system \(\{e^{2\pi imx}g(x-n) \}_{m,n\in\mathbb{Z}}\) forms an orthonormal basis for \(L^2(\mathbb{R})\) if \(g=\chi_{[0,1]}\). The authors prove some kind of inverse statement related to irregular Gabor systems of the form \(\{e^{2\pi i\lambda x}g(x-p) \}_{\lambda\in F,p\in I}\), where \(F,I\) are discrete subsets of \(\mathbb{R}\): if such a system is an orthonormal basis and the support of \(g\) is an interval \(\Omega\), then the time-frequency shifts necessarily forms a lattice and \(|g|=c\chi_\Omega\). However, for just compactly supported generator functions, irregular Gabor orthonormal bases are proven to exist (also in higher dimensions).
scientific article; zbMATH DE number 1896708