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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

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