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A Discrete Hilbert Transform with Circle Packings

  • Dominik Volland

Part of the BestMasters book series (BEST)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Dominik Volland
    Pages 1-2
  3. Dominik Volland
    Pages 3-7
  4. Dominik Volland
    Pages 9-34
  5. Dominik Volland
    Pages 35-60
  6. Dominik Volland
    Pages 61-88
  7. Dominik Volland
    Pages 89-99
  8. Back Matter
    Pages 101-102

About this book

Introduction

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

The Author
Dominik Volland currently attends his postgraduate studies in the master’s program on computational science and engineering at the Technical University of Munich (TUM). 

Keywords

circle packings Hilbert transform Riemann-Hilbert problem boundary value problem discrete analytic functions discrete complex analysis

Authors and affiliations

  • Dominik Volland
    • 1
  1. 1.Lehrstuhl M3 für Wissenschaftliches RecTU München, Zentrum MathematikGarching near MunichGermany

Bibliographic information