Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them...

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Main Authors: Mawardi Bahri, Ryuichi Ashino, Rémi Vaillancourt
Format: Article
Language:English
Published: Hindawi Limited 2013-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2013/162769
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spelling doaj-f8f2907619be431985433b09c3d419832020-11-24T21:09:28ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/162769162769Convolution Theorems for Quaternion Fourier Transform: Properties and ApplicationsMawardi Bahri0Ryuichi Ashino1Rémi Vaillancourt2Department of Mathematics, Hasanuddin University, Makassar 90245, IndonesiaDivision of Mathematical Sciences, Osaka Kyoiku University, Osaka 582-8582, JapanDepartment of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, CanadaGeneral convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.http://dx.doi.org/10.1155/2013/162769
collection DOAJ
language English
format Article
sources DOAJ
author Mawardi Bahri
Ryuichi Ashino
Rémi Vaillancourt
spellingShingle Mawardi Bahri
Ryuichi Ashino
Rémi Vaillancourt
Convolution Theorems for Quaternion Fourier Transform: Properties and Applications
Abstract and Applied Analysis
author_facet Mawardi Bahri
Ryuichi Ashino
Rémi Vaillancourt
author_sort Mawardi Bahri
title Convolution Theorems for Quaternion Fourier Transform: Properties and Applications
title_short Convolution Theorems for Quaternion Fourier Transform: Properties and Applications
title_full Convolution Theorems for Quaternion Fourier Transform: Properties and Applications
title_fullStr Convolution Theorems for Quaternion Fourier Transform: Properties and Applications
title_full_unstemmed Convolution Theorems for Quaternion Fourier Transform: Properties and Applications
title_sort convolution theorems for quaternion fourier transform: properties and applications
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2013-01-01
description General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.
url http://dx.doi.org/10.1155/2013/162769
work_keys_str_mv AT mawardibahri convolutiontheoremsforquaternionfouriertransformpropertiesandapplications
AT ryuichiashino convolutiontheoremsforquaternionfouriertransformpropertiesandapplications
AT remivaillancourt convolutiontheoremsforquaternionfouriertransformpropertiesandapplications
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