Dominik Volland • A Discrete Hilbert Transform with Circle Packings

Dominik Volland • A Discrete Hilbert Transform with Circle Packings

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. Basic knowledge of complex analysis and functional analysis is recommended.

Contents

• Hardy Spaces and Riemann-Hilbert Problems
• The Hilbert Transform in the Classical Setting
• Circle Packings
• Discrete Boundary Value Problems
• Discrete Hilbert Transform
• Numerical Results of Test Computations
• Properties of the Discrete Transform

Target Groups
Lecturers and students of mathematics who are interested in circle packings and/or discrete Riemann-Hilbert problems