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 ς and two windows functions g₁ and g₂ . We define the short-time Fourier transform on Zⁿ× Tⁿ and modulation spaces on Zⁿ , and present some basic properties. Then, we use modulation spaces on Zⁿ× Tⁿ as appropriate classes for symbols, and study the boundedness and compactness of the localization operators on modulation spaces on Zⁿ . 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 Zⁿ . Finally, under suitable conditions on the symbols, we prove that the localization operators are paracommutators, paraproducts and Fourier multipliers.