# Decay and Smoothness for Eigenfunctions of Localization Operators

## TL;DR

This paper investigates the decay and smoothness of eigenfunctions of localization operators, showing they are highly localized or Schwartz functions depending on the symbol class, with new convolution and multiplication relations in quasi-Banach spaces.

## Contribution

It establishes decay and smoothness properties of eigenfunctions for localization operators with symbols in modulation spaces, introducing new convolution and multiplication relations in quasi-Banach spaces.

## Key findings

- Eigenfunctions are highly localized onto few Gabor atoms.
- Eigenfunctions are Schwartz functions for certain symbol classes.
- New convolution and multiplication relations for modulation and Wiener amalgam spaces.

## Abstract

We study decay and smoothness properties for eigenfunctions of compact localization operators. Operators with symbols a in the wide modulation space M^{p,\infty} (containing the Lebesgue space L^p), p<\infty, and windows \f_1,\f_2 in the Schwartz class 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^{\infty}_{v_s\otimes 1} (\rdd), s>0 (subspaces of M^{p,\infty}(\rdd), p>2d/s) the L^2-eigenfunctions of the localization operator 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.