Localization operators on discrete Orlicz modulation spaces

Abstract

In this paper, we introduce Orlicz spaces on ${\mathbb{Z}}^n\times {\mathbb{T}}^n$ and Orlicz modulation spaces on ${\mathbb{Z}}^n$, and study inclusion relations, convolution relations, and duality of these spaces. We show that the Orlicz modulation space $M^{\Phi }({\mathbb{Z}}^n)$ is close to the modulation space $M^2({\mathbb{Z}}^n)$ for some particular Young function $\Phi$. Then, we study localization operators on ${\mathbb{Z}}^n$. In particular, using appropriate classes for symbols, we prove that these operators are bounded on Orlicz modulation spaces on ${\mathbb{Z}}^n$, compact and in the Schatten--von Neumann classes.