Multidimensional distributions and generalized Hilbert transforms in Clifford analysis
The classical Hilbert transform on the real line is a well–known singular integral operator with applications in the theoretical description of many devices and systems. To our knowledge, J. Horváth was the first to introduce a multidimensional vector valued generalization of the Hilbert transform in the framework of Clifford analysis. Clifford analysis is a higher dimensional function theory in the framework of a Clifford algebra which may be seen as a multidimensional generalization of the theory of holomorphic functions in one complex variable and – at the same time – as a refinement of harmonic analysis. In this doctoral thesis we study some specific families of multidimensional distributions in the framework of Euclidean Clifford analysis, meanwhile constructing several generalizations of the Clifford–Hilbert transform, their kernels belonging to one of those families of distributions (Part I). Next, we adopt the idea of an anisotropic (also called metric dependent or metrodynamical) Clifford setting, which offers the possibility of adjusting the co–ordinate system to preferential and not necessarily mutually orthogonal directions. In this area of Clifford analysis, we construct the so–called anisotropic Clifford–Hilbert transform (Part II). Finally, new higher dimensional Hilbert transforms are developed in the framework of Hermitean Clifford analysis, a recent and successful branch of Clifford analysis, offering a refinement of the Euclidean case (Part III).
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Ghent University. Faculty of Engineering and Architecture