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

Let K and L be two full-rank lattices in Rd .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 12 . (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) 12or v(K × L) 2. Moreover, if K = αZd , L = βZd with αβ = 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.