Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 162769, 10 pages http://dx.doi.org/10.1155/2013/162769 Research Article Convolution Theorems for Quaternion Fourier Transform: Properties and Applications Mawardi Bahri, 1 Ryuichi Ashino, 2 and Rémi Vaillancourt 3 1 Department of Mathematics, Hasanuddin University, Makassar 90245, Indonesia 2 Division of Mathematical Sciences, Osaka Kyoiku University, Osaka 582-8582, Japan 3 Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada K1N 6N5 Correspondence should be addressed to Ryuichi Ashino; ashino@cc.osaka-kyoiku.ac.jp Received 1 June 2013; Revised 1 September 2013; Accepted 7 September 2013 Academic Editor: Narcisa C. Apreutesei Copyright © 2013 Mawardi Bahri et al. his 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 inally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework. 1. Introduction Convolution is a mathematical operation with several appli- cations in pure and applied mathematics such as numeri- cal analysis, numerical linear algebra, and the design and implementation of inite impulse response ilters in signal processing. In [1–3], the authors introduced the Cliford convolution. It is found that some properties of convolution, when generalized to the Cliford Fourier transform (CFT), are very similar to the classical ones. On the other hand, the quaternion Fourier transform (QFT) is a nontrivial generalization of the classical Fourier transform (FT) using quaternion algebra. he QFT has been shown to be related to the other quaternion signal analysis tools such as quaternion wavelet transform, fractional quater- nion Fourier transform, quaternionic windowed Fourier transform, and quaternion Wigner transform [4–9]. A num- ber of already known and useful properties of this extended transform are generalizations of the corresponding properties of the FT with some modiications, but the generalization of convolution theorems of the QFT is still an open problem. In the recent past, several authors [10–13] tried to formulate convolution theorems for the QFT. But they only treated them for real-valued functions which is quite similar to the classical case. In [14], the authors briely introduced, without proof, the QFT of the convolution of two-dimensional quaternion signals. In this paper, we establish general convolutions for QFT. Because quaternion multiplication is not commuta- tive, we ind new properties of the QFT of convolution of two quaternion-valued functions. hese properties describe closely the relationship between the quaternion convolution and its QFT. he generalization of the convolution theorems of the QFT is mainly motivated by the Cliford convolution of general geometric Fourier transform, which has been recently studied in [15, 16]. We further establish the inverse QFT of the product of the QFT, which is very useful in solving partial diferential equations in quaternion algebra framework. his paper consists of the following sections. Section 2 deals with some results on the real quaternion algebra and the deinition of the QFT and its basic properties. We also review some basic properties of QFT, which will be necessary in the next section. Section 3 establishes convolution theorems of QFT and some of their consequences. Section 4 presents an application of QFT to study hypoellipticity and to solve the heat equation in quaternion algebra. Some conclusions are drawn in Section 5.