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# Wigner-Ville distribution function in the framework of linear canonical transform

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## Abstract

In this paper, we define the Wigner-Ville distribution function (WVDF) and corresponding Weyl operator in the linear canonical transform (LCT) domain. Further, we examine Moyle identity for the WVDF and investigate some of its properties. Moreover, we discuss the boundedness and compactness of Weyl operator on the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}$$\end{document} space in the LCT domain.
J. Pseudo-Differ. Oper. Appl. (2022) 13:38
https://doi.org/10.1007/s11868-022-00471-w
Wigner-Ville distribution function in the framework of
linear canonical transform
Amit Kumar1
·Akhilesh Prasad1
Received: 26 May 2022 / Revised: 3 July 2022 / Accepted: 4 July 2022 /
Published online: 13 July 2022
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
Abstract
In this paper, we deﬁne the Wigner-Ville distribution function (WVDF) and corre-
sponding Weyl operator in the linear canonical transform (LCT) domain. Further, we
examine Moyle identity for the WVDF and investigate some of its properties. More-
over, we discuss the boundedness and compactness of Weyl operator on the Lpspace
in the LCT domain.
Keywords Linear canonical transform ·Weyl transform ·Wigner-Ville distribution
function ·Linear canonical-Wigner transform
Mathematics Subject Classiﬁcation 42A38 ·43A32 ·34B20 ·81S30
1 Introduction
In quantum mechanics, the Wigner-Weyl transform is the invertible mapping between
functions in the quantum phase space formulation and Hilbert space operators in
the Schrödinger picture. This mapping was originally devised by Hermann Weyl in
1927 on an attempt to map symmetrized classical phase space functions to operators in
quantum mechanics, a procedure known as Weyl quantization [1]. But the map merely
amounts to a change of representation within quantum mechanics inspite of classical
to quantum quantities [2].
This work is supported by Department of Science & Technology, Govt. of India, under Grant no.
DST/INSPIRE Fellowship/2017/IF170292.
BAkhilesh Prasad
apr_bhu@yahoo.com
Amit Kumar
amit17.iitism@gmail.com
1Department of Mathematics & Computing, Indian Institute of Technology (Indian School of Mines),
Dhanbad 826004, India
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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