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The aim of this note is to present a self-contained proof of the fact that afunction can be approximated using a linear combination of Gaussian coherentstates, with a number of terms controlled in terms of the smoothness and of thedecay at infinity of the function. This result can easily be obtained usingadvanced results on modulation spaces, but the proof presented here iscompletely elementary and self-contained.

Decay of coefficients and approximation rates in Gabor Gaussian frames