Localization operators on discrete modulation spaces
Tóm tắt
In this paper, we study a class of pseudo-differential operators known as time-frequency localization operators on
$${\mathbb {Z}}^n$$
, which depend on a symbol
$$\varsigma $$
and two windows functions
$$g_1$$
and
$$g_2$$
. We define the short-time Fourier transform on
$$ {\mathbb {Z}}^n \times {\mathbb {T}}^n $$
and modulation spaces on
$${\mathbb {Z}}^n$$
, and present some basic properties. Then, we use modulation spaces on
$${\mathbb {Z}}^n \times {\mathbb {T}}^n$$
as appropriate classes for symbols, and study the boundedness and compactness of the localization operators on modulation spaces on
$${\mathbb {Z}}^n$$
. 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
$${\mathbb {Z}}^n$$
. Finally, under suitable conditions on the symbols, we prove that the localization operators are paracommutators, paraproducts and Fourier multipliers.
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