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A quantitative subspace Balian-Low theorem
Applied and Computational Harmonic Analysis  (IF3.055),  Pub Date : 2021-06-29, DOI: 10.1016/j.acha.2021.06.005
Andrei Caragea, Dae Gwan Lee, Friedrich Philipp, Felix Voigtlaender

Let GL2(R) be the subspace spanned by a Gabor Riesz sequence (g,Λ) with gL2(R) and a lattice ΛR2 of rational density. It was shown recently that if g is well-localized both in time and frequency, then G cannot contain any time-frequency shift π(z)g of g with zR2Λ. In this paper, we improve the result to the quantitative statement that the L2-distance of π(z)g to the space G is equivalent to the Euclidean distance of z to the lattice Λ, in the sense that the ratio between those two distances is uniformly bounded above and below by positive constants. On the way, we prove several results of independent interest, one of them being closely related to the so-called weak Balian-Low theorem for subspaces.