Volland D. A Discrete Hilbert Transform with Circle Packings

Wiesbaden: Spektrum, 2017. — 106 p.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.Auxiliary Material and Notation
The Continuous Setting
Circle Packings
Discrete Hilbert Transform
Numerical Results and Future Work