Decay and Smoothness for Eigenfunctions of Localization Operators

Abstract We study decay and smoothness properties for eigenfunctions of compact localization operators A a φ 1 , φ 2 . Operators A a φ 1 , φ 2 with symbols a in the wide modulation space M p , ∞ (containing the Lebesgue space L p ), p ∞ , and windows φ 1 , φ 2 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 a in the weighted modulation spaces M v s ⊗ 1 ∞ ( R 2 d ) , s > 0 (subspaces of M p , ∞ ( R 2 d ) , p > 2 d / s ) the L 2 -eigenfunctions of A a φ 1 , φ 2 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.