The Density Theorem and the Homogeneous Approximation Property for Gabor Frames

Abstract

Summary. The Density Theorem for Gabor Frames is a fundamental result in timefrequency analysis. Beginning with Baggett's proof that a rectangular lattice Gabor system {e 2πiβnt g(t − αk)} n,k∈Z must be incomplete in L 2 (R) whenever αβ > 1, the necessary conditions for a Gabor system to be complete, a frame, a Riesz basis, or a Riesz sequence have been extended to arbitrary lattices and beyond. The first partial proofs of the Density Theorem for irregular Gabor frames were given by Landau in 1993 and by Ramanathan and Steger in 1995. A key fact proved by Ramanathan and Steger is that irregular Gabor frames possess a certain Homogeneous Approximation Property (HAP), and that the Density Theorem is a consequence of this HAP. This chapter provides a brief history of the Density Theorem and a detailed account of the proofs of Ramanathan and Steger. Furthermore, we show that the techniques of Ramanathan and Steger can be used to give a full proof of a general version of Density Theorem for irregular Gabor frames in higher dimensions and with finitely many generators.

History and Evolution of the Density Theorem for Gabor Frames

“…The Density Theorem for frames and Riesz bases follows from this. A detailed exposition of the arguments of [158] appears in [107]. Gröchenig and Razafinjatovo's modified version of the HAP is essentially the Weak HAP given above, except defined for the setting of systems of translates in the space of bandlimited functions [98], and allowing finitely many generators.…”