Completeness of Gabor systems

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 L 2 (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.2010 Mathematics Subject Classification. 42C30,42C15,81R30.

The density theorem for discrete series representations restricted to lattices

“…Proofs of the claim were given by Perelomov [57], Bargmann [8], and Neretin [53]. For rational lattices (i.e., αβ ∈ Q), the same claim holds when the Gaussian function is multiplied by a rational function with no real poles [33].…”

“…In fact, a criterion for the completeness of subsystems of coherent states similar to Theorem 1.1 was posed as a question in [58, p.226] 1 . These criteria have been considered for specific systems and vectors in, e.g., [8,33,42,49,53,57,59,61].…”

More about continuous Gabor frames on locally compact abelian groups

“…As another example, suppose that G = R and consider the Gaussian function φ(x) = e −πx 2 . It was already proved in [10] that the Gabor system {E mα T nβ φ} m,n∈Z is complete for αβ ≤ 1 and incomplete for αβ > 1. Moreover, Z αZ φ = 0 a.e.…”

A symmetry property for polyharmonic functions vanishing on equidistant hyperplanes

“…Recall that a 2 N ‐times continuously differentiable function defined on a domain G in R 2 (identified with the complex plane C ) is polyanalytic of order N if Polyharmonic functions in two variables are related to polyanalytic functions in a similar fashion as harmonic functions are related to analytic functions for dimension Polyanalytic functions have been studied intensively in and they found recently very interesting applications in signal analysis , and statistical physics . If G is a simply connected domain then a polyanalytic function of order N can be written in the form where are analytic functions on G .…”

“…Polyharmonic functions in two variables are related to polyanalytic functions in a similar fashion as harmonic functions are related to analytic functions for dimension d = 2. Polyanalytic functions have been studied intensively in [5] and they found recently very interesting applications in signal analysis [1], [15] and statistical physics [17]. If G is a simply connected domain then a polyanalytic function F : G → C of order N can be written in the form…”