Plancherel theorem and quaternion Fourier transform for square integrable functions
Cheng, Dong; Kou, Kit Ian
AbstractThe quaternion Fourier transform (QFT), a generalization of the classical 2D Fourier transform, plays an increasingly active role in particular signal and colour image processing. There tends to be an inordinate degree of interest placed on the properties of QFT. The classical convolution theorem and multiplication formula are only suitable for 2D Fourier transform of complex-valued signal, and do not hold for QFT of quaternion-valued signal. The purpose of this paper is to overcome these problems and establish the Plancherel and inversion theorems of QFT in the square integrable signals space L-2. First, we investigate the behaviours of QFT in the integrable signals space L-1. Next, we deduce the energy preservation property which extends functions from L-1 to L-2 space. Moreover, some other important properties such as modified multiplication formula are also analyzed for QFT.
KeywordQuaternion Fourier transforms multiplication formula inversion theorem Plancherel theorem linear canonical transform
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Indexed BySCI
WOS Research AreaMathematics
WOS SubjectMathematics
WOS IDWOS:000454929400003
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Cited Times [WOS]:10   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionUniversity of Macau
AffiliationUniv Macau, Fac Sci & Technol, Dept Math, Macau, Peoples R China
First Author AffilicationUniversity of Macau
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GB/T 7714
Cheng, Dong,Kou, Kit Ian. Plancherel theorem and quaternion Fourier transform for square integrable functions[J]. COMPLEX VARIABLES AND ELLIPTIC EQUATIONS,2019,64(2):223-242.
APA Cheng, Dong,&Kou, Kit Ian.(2019).Plancherel theorem and quaternion Fourier transform for square integrable functions.COMPLEX VARIABLES AND ELLIPTIC EQUATIONS,64(2),223-242.
MLA Cheng, Dong,et al."Plancherel theorem and quaternion Fourier transform for square integrable functions".COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 64.2(2019):223-242.
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