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.

翻译：加博框架是一个标准工具,可以将功能分解成“相容状态”的离散总和,这些状态在位置和Fourier空隙中都是本地化的。这种扩张在某种程度上类似于Fourier的扩张,但更微妙,因为加博框架不形成正态基础。在这项工作中,我们分析这些框架中函数系数的衰变特性,从功能的规律性及其无限度的衰变角度看。这些结果类似于Fourier系数的标准衰变特性,并允许显示数量有限的一致状态为任何顺畅的快速衰变功能提供了良好的近似值。具体地说,我们提供了索博列夫规范的明确趋同率,因为所选的一致状态的数量在增加。当加博尔波波波波波波波波波波用于分离PDE问题时,我们的结果在数字分析中特别有用。