Excesses of Gabor frames

Abstract

A Gabor system for $\text{L}{}^2\text{(}\text{R}{}^{\mathrm{d}}\text{)}$ has the form $\text{G}\text{(g,Λ)={e}{}^{\mathrm{2\pi ibx}}\text{g(x−a)}}{}_{\mathrm{(a,b)\in \Lambda }}$, where $\text{g∈L}{}^2\text{(}\text{R}{}^{\mathrm{d}}\text{)}$ and Λ is a sequence of points in $\text{R}{}^{\mathrm{2d}}$. We prove that, with only a mild restriction on the generator g and for nearly arbitrary sets of time–frequency shifts Λ, an overcomplete Gabor frame has infinite excess, and in fact there exists an infinite subset that can be removed yet leave a frame. The proof of this result yields an interesting connection between the density of Λ and the excess of the frame.