An interplay between parameter \((p, q)\)-Boas transform and linear canonical transform
type: Item
instance of: scholarly article
title: An interplay between parameter \((p, q)\)-Boas transform and linear canonical transform
zbMATH Open document ID: 1484.44003
author: Nikhil Khanna
author: Shiv K. Kaushik
author: Slaviša Djordjevic
published in: Journal of Pseudo-Differential Operators and Applications
publication date: 2021-08-31T00:00:00Z
review text: Exhaustive references document the development of the theory of the linear canonical transform (LCT) and the Boas transform. A brief description of the linear canonical transform which contains four parameters given by Eqn. (1.1) is given. The Hilbert transform and its generalization such as fractional Hilbert transform (FRHT), which play a role for optical analysis and signal processing, are studied. The analytical signals associated with LCT and its applications are explained. The Boas transform (BT) was introduced by \textit{R.~P. Boas jun.} [Trans. Am. Math. Soc. 40, 287--308 (1936; Zbl 0015.21301)], and its relationship with the Hilbert transform is shown in Eqn. (1.7). The introductory part of the present paper also covers the concept of Boas transform and its generalization, the fractional Boas transform (FRBT) related to wavelets and signal processing. In Section~2, the authors employ the generalization of Boas transform known as the parameter \((p,q)\) Boas transform (PBT) which defines complex signals associated with PBT and LCT. The complex signals in the LCT domain are studied, and it is suggested that the resulting signals have no non-zero frequency components in comparison with the case of generalized analytic signals related to PHT. Further, the authors investigate the generalized Boas product transform (Theorem~2.4) which is an analog of Bedrosian's theorem in the LCT domain. Section~3 describes the generalization of BT by combining a rotation parameter and LCT with the classical BT, which is represented as \(\eta\)-LCBT (Definition~3.1) and some properties are studied. The authors point out that the \(\eta\)-LCBT does not satisfy the semigroup property. Then four versions of \(\eta\)-LCCSs are formulated, and their linear canonical transform spectrum and eigenfunction properties are analysed.
reviewed by: Deshna Loonker
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 44A99
zbMATH DE Number: 7388682
zbMATH Keywords: parameter \((p,q)\)-Boas transform
zbMATH Keywords: linear canonical transform
zbMATH Keywords: \(\eta\)-linear canonical Boas transform, Hilbert transform
MaRDI profile type: Publication
full work available at URL: s11868-021-00417-8
OpenAlex ID: W3190470665
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DOI: 10.1007/S11868-021-00417-8
description: scientific article; zbMATH DE number 7388682