Behavior of Gabor frame operators on Wiener amalgam spaces

It is well known that the Gabor expansions converge to identity operator in weak* sense on the Wiener amalgam spaces as sampling density tends to infinity. In this paper we prove the convergence of Gabor expansions to identity operator in the operator norm as well as weak* sense on $W(L^p, \ell^q)$ as the sampling density tends to infinity. Also we show the validity of the Janssen's representation and the Wexler-Raz biorthogonality condition for Gabor frame operator on $W(L^p, \ell^q)$.