Non-separable lattices, Gabor orthonormal bases and Tilings

TL;DR

This paper explores the conditions under which non-diagonal lattice transformations produce Gabor orthonormal bases, revealing new tiling and spectral properties for sets and providing constructive methods for Gabor windows.

Contribution

It extends the understanding of Gabor bases to non-separable lattices, especially lower block triangular matrices, and characterizes the tiling and spectral properties of the associated sets.

Findings

If the Gabor system forms an orthonormal basis, the set can be expressed as a union of fundamental domains.

When the product of matrices A^t and B is an integer matrix, the set tiles by a lattice and is spectral.

Examples show that unions of multiple fundamental domains are possible, not just single ones.

Abstract

Let $K\subset \Bbb R^d$ be a set with positive and finite Lebesgue measure. Let $\Lambda=M(\Bbb Z^{2d})$ be a lattice in $\Bbb R^{2d}$ with density dens$(\Lambda)=1$. It is well-known that if $M$ is a diagonal block matrix with diagonal matrices $A$ and $B$, then $\mathcal G(|K|^{-1/2}\chi_K, \Lambda)$ is an orthonormal basis for $L^2(\Bbb R^d)$ if and only if $K$ tiles both by $A(\Bbb Z^d)$ and $B^{-t}(\Bbb Z^d)$. However, there has not been any intensive study when $M$ is not a diagonal matrix. We investigate this problem for a large class of important cases of $M$. In particular, if $M$ is any lower block triangular matrix with diagonal matrices $A$ and $B$, we prove that if $\mathcal G(|K|^{-1/2}\chi_K, \Lambda)$ is an orthonormal basis, then $K$ can be written as a finite union of fundamental domains of $A({\mathbb Z}^d)$ and at the same time, as a finite union of fundamental domains of $B^{-t}({\mathbb Z}^d)$. If $A^tB$ is an integer matrix, then there is only one common fundamental domain, which means $K$ tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than one fundamental domains is also possible. We also provide a constructive way for forming a Gabor window functions for a given upper triangular lattice. Our study is related to a Fuglede's type problem in Gabor setting and we give a partial answer to this problem in the case of lattices.