Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

Question

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is there a result of the form: \begin{align} \left\Vert f - \sum_{m=-M}^{M} \sum_{n=-N}^{N} c_{m,n}\; g_{m,n} \right\Vert \le error(N,M), \end{align} with appropriate restrictions on $f$ and $g$?