Decay and smoothness for eigenfunctions of localization operators

Abstract

We study decay and smoothness properties for eigenfunctions of compact localization operators Aaφ1,φ2. Operators Aaφ1,φ2 with symbols a in the wide modulation space Mp,∞ (containing the Lebesgue space Lp), p<∞, and windows φ1,φ2 in the Schwartz class S are known to be compact. We show that their L2-eigenfuctions with non-zero eigenvalues are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols a in the weighted modulation spaces Mvs⊗1∞(R2d), s>0 (subspaces of Mp,∞(R2d), p>2d/s) the L2-eigenfunctions of Aaφ1,φ2 are actually Schwartz functions. An important role is played by quasi-Banach Wiener amalgam and modulation spaces. As a tool, new convolution relations for modulation spaces and multiplication relations for Wiener amalgam spaces in the quasi-Banach setting are exhibited.