Invertibility of the Gabor frame operator on some function spaces (Q485029)

Suppose \(g\in L_2(\mathbb R^d)\) is a window function of a Gabor frame and \(\gamma=S_g^{-1}g\) is the canonical dual of \(g\), where \(S_g\) is the Gabor frame operator. The author considers the question, if \(g\) is in a given function space, whether \(\gamma\) is in the same space? The author generalize a number of previous results and show that it is so when \(g\) is in the Wiener amalgam space or in the Lipschitz amalgam space.