The existence of tight Gabor duals for Gabor frames and subspace Gabor frames

Abstract

Let K and L be two full-rank lattices in R d . We give a complete characterization for all the Gabor frames that admit tight dual of the same type. The characterization is given in terms of the center-valued trace of the von Neumann algebra generated by the left regular projective unitary representations associated with the time–frequency lattice K × L . Two applications of this characterization were obtained: (i) We are able to prove that every Gabor frame has a tight dual if and only if the volume of K × L is less than or equal to 1 2 . (ii) We are able to obtain sufficient or necessary conditions for the existence of tight Gabor pseudo-duals for subspace Gabor frames in various cases. In particular, we prove that every subspace Gabor frame has a tight Gabor pseudo-dual if either the volume v ( K × L ) ⩽ 1 2 or v ( K × L ) ⩾ 2 . Moreover, if K = α Z d , L = β Z d with α β = 1 , then a subspace Gabor frame G ( g , L , K ) has a tight Gabor pseudo-dual only when G ( g , L , K ) itself is already tight.