Compactly supported Gabor orthonormal bases

Abstract

We characterize all lattices $\hat{I}›\subset {\mathbb{R}}^2$ and all compactly supported functions $g\in L^2(\mathbb{R})$ for which the Gabor system $\left\{e^{2\ddot{I}€isx}g(x-t):(t,s)\in \hat{I}›\right\}$ forms an orthonormal basis for $L^2(\mathbb{R})$. The characterization is given in geometric terms through translation tilings and discreteness properties of lattice projections. In particular, this resolves a conjecture of Han and Wang on the non-existence of Gabor bases along specific irrational lattices. Finally, we construct Gabor bases that cannot be realized by any product set, answering a problem of Iosevich and Mayeli.

17 pages