A Discrete Hilbert Transform with Circle Packings

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. Front Matter ....Pages i-xi Introduction (Dominik Volland)....Pages 1-2 Auxiliary Material and Notation (Dominik Volland)....Pages 3-7 The Continuous Setting (Dominik Volland)....Pages 9-34 Circle Packings (Dominik Volland)....Pages 35-60 Discrete Hilbert Transform (Dominik Volland)....Pages 61-88 Numerical Results and Future Work (Dominik Volland)....Pages 89-99 Back Matter ....Pages 101-102