A Gabor sampling theorem and some time-bandwidth implications

We address the use of sampling representations for functions expressed in a Gabor expansion. Assuming that the window function of the expansion is bandlimited and that the Gabor coefficients vanish for a sufficiently high index in frequency, one easily obtains a Gabor sampling theorem from the classical Shannon-Kotelnikov (1949) result. This result is then explored to give insight into the relationships between the time bandwidth products of the signal and window, and the number of Gabor coefficients retained in a finite expansion