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Abstract and Applied Analysis
Volume 2013, Article ID 162769, 10 pages
Research Article

Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

1Department of Mathematics, Hasanuddin University, Makassar 90245, Indonesia
2Division of Mathematical Sciences, Osaka Kyoiku University, Osaka 582-8582, Japan
3Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada K1N 6N5

Received 1 June 2013; Revised 1 September 2013; Accepted 7 September 2013

Academic Editor: Narcisa C. Apreutesei

Copyright © 2013 Mawardi Bahri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.