Characterization and perturbation of Gabor frame sequences with rational parameters

Let A⊂L2(R)A⊂L2(R) be at most countable, and p,q∈Np,q∈N. We characterize various frame-properties for Gabor systems of the formG(1,p/q,A)={e2πimxg(x-np/q):m,n∈Zg∈A}in terms of the corresponding frame properties for the row vectors in the Zibulski–Zeevi matrix. This extends work by [Ron and Shen, Weyl–Heisenberg systems and Riesz bases in L2(Rd)L2(Rd). Duke Math. J. 89 (1997) 237–282], who considered the case where AA is finite. As a consequence of the results, we obtain results concerning stability of Gabor frames under perturbation of the generators. We also introduce the concept of rigid frame sequences, which have the property that all sufficiently small perturbations with a lower frame bound above some threshold value, automatically generate the same closed linear span. Finally, we characterize rigid Gabor frame sequences in terms of their Zibulski–Zeevi matrix.