A Discrete Hilbert Transform with Circle Packings (BestMasters)

Description

Dominik Volland studies the construction of a discrete counterpart to the Hilbert transform in the realm of a nonlinear discrete complex analysis given by circle packings. The Hilbert transform is closely related to Riemann-Hilbert problems which have been studied in the framework of circle packings by E. Wegert and co-workers since 2009. The author demonstrates that the discrete Hilbert transform is well-defined in this framework by proving a conjecture on discrete problems formulated by Wegert. Moreover, he illustrates its properties by carefully chosen numerical examples. Acknowledgments Contents List of Figures Abstract 1 Introduction 2 Auxiliary Material and Notation 3 The Continuous Setting 3.1 Hardy spaces 3.1.1 Boundary values of holomorphic functions 3.1.2 Integral formulas 3.1.3 Fourier series of the boundary functions 3.2 The Hilbert transform 3.3 Riemann-Hilbert problems 3.3.1 Linear Riemann-Hilbert problems 3.3.2 Nonlinear Riemann-Hilbert problems 3.4 Obtaining the Hilbert transform from a Riemann-Hilbert problem 3.4.1 Choice of the problem 3.4.2 Solutions of the problem 3.4.3 Counterexamples for u not in C1+a 4 Circle Packings 4.1 First examples and ideas 4.2 Basic definitions 4.2.1 Complex 4.2.2 Circle packing 4.3 Manifold structure 4.3.1 Contact function 4.3.2 Angle sums and branch structures 4.3.3 Parametrization of Db 4.3.4 Normalization 4.4 Discrete harmonic functions on circle packings 4.5 Discrete analytic functions 4.6 Maximal packings 4.7 Some results on discrete analytic functions 4.7.1 Discrete maximum principles 4.7.2 Approximation of the Riemann Mapping 5 Discrete Hilbert Transform 5.1 Discrete boundary value problems 5.1.1 Definition and examples 5.1.2 Linearization of boundary value problems 5.2 Proof of the maximal packing conjecture 5.2.1 The transformed packing 5.2.2 Differential of w at the transformed packing 5.2.3 Basis for the kernel of J 5.3 Discrete Hilbert transform 5.3.1 Difficulties of the Schwarz problem 5.3.2 Discretization of the nonlinear problem 5.3.3 Linearization of the discrete operator 6 Numerical Results and Future Work 6.1 Test computations 6.2 Eigenvalues of the discrete transform 6.3 Elimination of constants 6.4 Curvature of the Circle Packing manifold 6.5 Local frames Bibliography