Excesses of Gabor frames

Abstract A Gabor system for L 2 ( R d ) has the form G (g,Λ)={e 2πibx g(x−a)} (a,b)∈Λ , where g∈L 2 ( R d ) and Λ is a sequence of points in R 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.

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