Decay and Smoothness for Eigenfunctions of Localization Operators

We study decay and smoothness properties for eigenfunctions of compact localization operators. Operators with symbols in the wide modulation space M^{p,infty} (containing the Lebesgue space L^p), p less than infinity, and windows in the Schwartz class S are known to be compact. We show that their L^2-eigenfuctions with non-zero eigenvalues are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols in suitable weighted modulation spaces the L^2-eigenfunctions 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.