Localization operators on discrete modulation spaces

In this paper, we study a class of pseudo-differential operators known as time-frequency localization operators on Zn, which depend on a symbol. and twowindows functions g(1) and g(2). We define the short-time Fourier transform on Z(n) xT(n) and modulation spaces on Z(n), and present some basic properties. Then, we use modulation spaces on Z(n) x T-n as appropriate classes for symbols, and study the boundedness and compactness of the localization operators on modulation spaces on Z(n). Then, we show that these operators are in the Schatten-von Neumann class. Also, we obtain the relation between the Landau-Pollak-Slepian type operator and the localization operator on Z(n). Finally, under suitable conditions on the symbols, we prove that the localization operators are paracommutators, paraproducts and Fourier multipliers.