Completeness of Gabor systems

TL;DR

This paper proves the completeness of certain Gabor systems with specific window functions on rational lattices, highlighting fundamental differences between completeness and frame properties in time-frequency analysis.

Contribution

It establishes new completeness results for Gabor systems with Hermite functions and rational functions times Gaussian, extending classical results and providing insights into coherent states.

Findings

Time-frequency shifts of Hermite functions are complete in L2(R).

Completeness holds for functions factoring into rational functions and Gaussian.

Results reveal a significant difference between completeness and frame properties.

Abstract

We investigate the completeness of Gabor systems with respect to several classes of window functions on rational lattices. Our main results show that the time-frequency shifts of every finite linear combination of Hermite functions with respect to a rational lattice are complete in L2(R), thus generalizing a remark of von Neumann (and proved by Bargmann, Perelomov et al.). An analogous result is proven for functions that factor into certain rational functions and the Gaussian. The results are also interesting from a conceptual point of view since they show a vast difference between the completeness and the frame property of a Gabor system. In the terminology of physics we prove new results about the completeness of coherent state subsystems.