The existence of tight Gabor duals for Gabor frames and subspace Gabor frames
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 x 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 x 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 x L) <= 1/2 or v(K x L) >= 2. Moreover, if K = alpha Z(d), L = beta Z(d) with alpha beta = 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. (C) 2008 Elsevier Inc. All rights reserved.
Journal of Functional Analysis
"The existence of tight Gabor duals for Gabor frames and subspace Gabor frames" (2009). Faculty Bibliography 2000s. 1615.