But they only treated them for real-valued functions which is quite similar to the classical case. In the recent past, several authors tried to formulate convolution theorems for the QFT. A number of already known and useful properties of this extended transform are generalizations of the corresponding properties of the FT with some modifications, but the generalization of convolution theorems of the QFT is still an open problem. The QFT has been shown to be related to the other quaternion signal analysis tools such as quaternion wavelet transform, fractional quaternion Fourier transform, quaternionic windowed Fourier transform, and quaternion Wigner transform. On the other hand, the quaternion Fourier transform (QFT) is a nontrivial generalization of the classical Fourier transform (FT) using quaternion algebra. It is found that some properties of convolution, when generalized to the Clifford Fourier transform (CFT), are very similar to the classical ones. In, the authors introduced the Clifford convolution. IntroductionĬonvolution is a mathematical operation with several applications in pure and applied mathematics such as numerical analysis, numerical linear algebra, and the design and implementation of finite impulse response filters in signal processing. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework. ![]() We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. ![]() It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented.
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