Paley_ Wiener Theorems for The Fourier Transform depending on the Heisenberg group

Abstract

In this paper, we talk about Heisenberg group, the most known example from the lie groups. After that, we discuss the representation theory of this group and the relationship between the representation theory of the Heisenberg group and the position and momentum operators and momentum operators relationship between the representation theory of the Heisenberg group and the position and momentum that shows how we will make the connection between the Heisenberg group and physics.

we have considered only the Schrodinger picture. That is, all the representations we considered are realized in the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator-valued function, and other facts and properties.

In our research, we depended on new formulas for some mathematical concepts such as Fourier Transform and Weyl transform. The main aim of our research is to introduce the Paley_ Wiener theorem for the Fourier transform on the Heisenberg group. We will show that the classical Paley_ Wiener theorem for the Euclidean Fourier transform characterizes compactly supported functions in terms of the behaviour of their Fourier transforms and Weyl transform. And we are interested in establishing results for the group Fourier transform and the Weyl transform.