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

翻译：本文旨在给出一个自包含的证明：一个函数可通过高斯相干态的线性组合进行逼近，且所需项数受该函数的光滑性和无穷远处衰减性的控制。该结果虽可利用调制空间的先进结论轻松得到，但本文所给出的证明完全基于初等方法且无需借助外部结果。