Decay of coefficients and approximation rates in Gabor Gaussian frames

Gabor frames are a standard tool to decompose functions into a discrete sum of "coherent states", which are localised both in position and Fourier spaces. Such expansions are somehow similar to Fourier expansions, but are more subtle, as Gabor frames do not form orthonormal bases. In this work, we analyze decay properties of the coefficients of functions in these frames in terms of the regularity of the functions and their decay at infinity. These results are analogous to the standard decay properties of Fourier coefficients, and permit to show that a finite number of coherent states provide a good approximation to any smooth rapidly decaying function. Specifically, we provide explicit convergence rates in Sobolev norms, as the number of selected coherent states increases. Our results are especially useful in numerical analysis, when Gabor wavelets are employed to discretize PDE problems.