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Convolution theorems for quaternion Fourier transform: properties and applications. (English) Zbl 1297.42015

Summary: 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.

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B37 Harmonic analysis and PDEs
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
35H10 Hypoelliptic equations
35K05 Heat equation
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