  It is shown that the modulation spaces $M_{p}^{w}$ can be characterized by
the approximation behavior of their elements using Local Fourier bases. In
analogy to the Local Fourier bases, we show that the modulation spaces can also
be characterized by the approximation behavior of their elements using Gabor
frames. We derive direct and inverse approximation theorems that describe the
best approximation by linear combinations of N terms of a given function using
its modulates and translates.
