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СЕМИНАРЫ |
Seminar on Analysis, Differential Equations and Mathematical Physics
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Dirac-type operators and applications Swanhild Bernstein TU Bergakademie Freiberg, Institute of Applied Analysis |
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Аннотация: Clifford analysis is a higher-dimensional analog of classical complex function theory and refinement of harmonic analysis. The core and center of the theory is the Dirac operator $$ D = \sum_{i=1}^n e_i\frac{\partial}{\partial x_i}, \quad \text{where}\quad e_i,\quad i=1, ...., n, $$ are the generating elements of the Clifford algebra $$ e_ie_j + e_je_i = -2\delta_{ij} . $$ The Dirac operator consists of a radial component and a phase, where [1ex] The zero solutions of the Dirac equation are called monogenic functions. Cauchy integrals can represent monogenic functions. To describe the boundary values of monogenic functions, the Hilbert operators and Hardy spaces are essential. [1ex] We will consider several generalizations of Dirac operators and Hilbert transformations and their applications in optics. An essential tool in these considerations will be the Fourier symbol of these operators and multiplier theorems. Specifically, we will consider Dirac-type operators [1ex] Fractional Dirac and Hilbert operators represent another type of modification. Fractional Hilbert operators |