Two-Sided Fourier Transform in Clifford Analysis and Its Application
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Advances in Applied Clifford Algebras
Two-Sided Fourier Transform in Clifford Analysis and Its Application Haipan Shi, Heju Yang, Zunfeng Li and Yuying Qiao∗ Communicated by Uwe Kaehler Abstract. In this paper, we first define a two-sided Clifford Fourier transform(CFT) and its inverse transformation on L1 space. Then we study the differential of the two-sided CFT, the k-th power of F {h}, Plancherel identity and time-frequency shift of the two-sided CFT. Finally we discuss the uncertainty principle of the two-sided CFT and give an application of the two-sided CFT to a partial differential equation. Mathematics Subject Classification. 30E20, 30E25, 45E05. Keywords. Clifford Fourier transform, Differential property, k-th power of F {h}, Plancherel identity, Time-frequency shift, Uncertainty principle.
1. Introduction In the late 19th century, Clifford [8] created a geometric algebraic structure namely Clifford algebra. After that, Fueter [15] created Clifford analysis. Clifford analysis extends the theory of complex variable functions to high-dimensional spaces, which mainly studies the properties of functions defined on Rn with values in Clifford algebra. The Clifford Fourier transform (CFT) is a nontrivial generalization of the classical Fourier transform (FT) in Clifford algebra. Due to the non-commutativity of multiplication rules in Clifford algebra, there are three different types of CFTs: the left-sided CFT, the right-sided CFT and the two-sided CFT. There are many good properties of the CFT, including linearity, shift, modulation, expansion, energy integration, convolution, etc. The CFT is a very useful tool for applications in non-marginal color image processing, electromagnetism, image processing, etc. ∗ Corresponding
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Adv. Appl. Clifford Algebras
Many scholars have done detailed research on the Fourier transform of one-dimensional space, such as [6,13,23]. In recent years, scholars have set off an upsurge in the study of the Fourier transform in high-dimensional spaces. The quaternion Fourier transform (QFT) and the Clifford Fourier transform (CFT) have become very important and meaningful topics in the field of mathematics. Brackx et al. [7] made definitions of the quaternion Fourier Transform and the Clifford Fourier Transform and studied the twodimensional CFT. Bahri et al. [4] studied some properties of the Clifford windowed Fourier transform (CWFT) and showed the differences between the classical windowed Fourier transform (WFT) and the CWFT. Bahri and Hitzer [2] showed some properties of the CFT and proved the uncertainty principle for Cl(3, 0) multivector functions. Bahri et al. [3] gave an uncertainty principle for the QFT. Then Bahri [1] gave a modified uncertainty principle for the two-sided QFT. Hitzer [16] showed some properties of the QFT. Hitzer and Bahri [18] showed a set of important properties of the CFT on Cln,0 . Hitzer [17] studied some properties of the two-sided CFT with two square roots of − 1 in Cl(p,q). Fu [14] studied the properties and uncert
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