A Discrete Hilbert Transform with Circle Packings (BestMasters)

110 Pages·2017·1.579 MB·English

by  Dominik Volland

Dominik Volland
A Discrete Hilbert
Transform with
Circle Packings
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Dominik Volland
A Discrete Hilbert
Transform with
Circle Packings
Dominik Volland
Garching near Munich, Germany
BestMasters
ISBN 978­3­658­20456­3 ISBN 978­3­658­20457­0 (eBook)
https://doi.org/10.1007/978­3­658­20457­0
Library of Congress Control Number: 2017961504
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Acknowledgments
This book originated from a research project of Elias Wegert and Folkmar
Bornemann together with the author in the TUM elite graduate program
TopMath. The work was supported by the DFG-Collaborative Research
Center, TRR 109, \Discretization in Geometry and Dynamics". I would like
to thank both TopMath and DFG for their (cid:12)nancial and ideological support
throughout the project.
I also wish to express my deep gratitude to my supervisors, Elias Wegert
and Folkmar Bornemann, for productive personal communication, for their
valuable support, and also for suggesting me for \Best Masters".
Finally, I’d like to thank my family, my friends, and my colleagues at
TUM for supporting me while I was writing this book.
Contents
1 Introduction 1
2 Auxiliary Material and Notation 3
3 The Continuous Setting 9
3.1 Hardy spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 Boundary values of holomorphic functions . . . . . . . 9
3.1.2 Integral formulas . . . . . . . . . . . . . . . . . . . . . 13
3.1.3 Fourier series of the boundary functions . . . . . . . . 14
3.2 The Hilbert transform . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Riemann-Hilbert problems . . . . . . . . . . . . . . . . . . . . 23
3.3.1 Linear Riemann-Hilbert problems. . . . . . . . . . . . 24
3.3.2 Nonlinear Riemann-Hilbert problems . . . . . . . . . . 25
3.4 Obtaining the Hilbert transform from a Riemann-Hilbert
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.1 Choice of the problem . . . . . . . . . . . . . . . . . . 26
3.4.2 Solutions of the problem . . . . . . . . . . . . . . . . . 27
3.4.3 Counterexamples for u2= C1+(cid:11) . . . . . . . . . . . . . 31
4 Circle Packings 35
4.1 First examples and ideas . . . . . . . . . . . . . . . . . . . . . 35
4.2 Basic de(cid:12)nitions . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.1 Complex. . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.2 Circle packing . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Manifold structure . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.1 Contact function . . . . . . . . . . . . . . . . . . . . . 45
4.3.2 Angle sums and branch structures . . . . . . . . . . . 46
4.3.3 Parametrization of D . . . . . . . . . . . . . . . . . . 49
b
4.3.4 Normalization. . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Discrete harmonic functions on circle packings . . . . . . . . 52
4.5 Discrete analytic functions . . . . . . . . . . . . . . . . . . . . 55
4.6 Maximal packings . . . . . . . . . . . . . . . . . . . . . . . . 56
viii Contents
4.7 Some results on discrete analytic functions . . . . . . . . . . . 57
4.7.1 Discrete maximum principles . . . . . . . . . . . . . . 58
4.7.2 Approximation of the Riemann Mapping. . . . . . . . 58
5 Discrete Hilbert Transform 61
5.1 Discrete boundary value problems . . . . . . . . . . . . . . . 62
5.1.1 De(cid:12)nition and examples . . . . . . . . . . . . . . . . . 62
5.1.2 Linearization of boundary value problems . . . . . . . 67
5.2 Proof of the maximal packing conjecture . . . . . . . . . . . . 68
5.2.1 The transformed packing . . . . . . . . . . . . . . . . 70
5.2.2 Di(cid:11)erential of ! at the transformed packing . . . . . . 71
e
5.2.3 Basis for the kernel of Je . . . . . . . . . . . . . . . . . 76
5.3 Discrete Hilbert transform . . . . . . . . . . . . . . . . . . . . 79
5.3.1 Di(cid:14)culties of the Schwarz problem . . . . . . . . . . . 80
5.3.2 Discretization of the nonlinear problem . . . . . . . . 82
5.3.3 Linearization of the discrete operator. . . . . . . . . . 86
6 Numerical Results and Future Work 89
6.1 Test computations . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Eigenvalues of the discrete transform . . . . . . . . . . . . . . 91
6.3 Elimination of constants . . . . . . . . . . . . . . . . . . . . . 94
6.4 Curvature of the Circle Packing manifold . . . . . . . . . . . 96
6.5 Local frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Bibliography 101
List of Figures
3.1 Construction of a counterexample. . . . . . . . . . . . . . . . 32
4.1 A circle packing . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Packings and complexes . . . . . . . . . . . . . . . . . . . . . 36
4.3 A discrete conformal map . . . . . . . . . . . . . . . . . . . . 37
4.4 Overlapping circles . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5 A branched packing . . . . . . . . . . . . . . . . . . . . . . . 38
4.6 A complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.7 Complexes and hex re(cid:12)nement . . . . . . . . . . . . . . . . . 41
4.8 A face and its face circle . . . . . . . . . . . . . . . . . . . . . 43
4.9 Illustration of the contact equation . . . . . . . . . . . . . . . 45
4.10 Maximal packings . . . . . . . . . . . . . . . . . . . . . . . . 57
4.11 Construction of a discrete Riemann map . . . . . . . . . . . . 59
5.1 A discrete circular Riemann-Hilbert problem . . . . . . . . . 66
5.2 The extended packing . . . . . . . . . . . . . . . . . . . . . . 69
5.3 Proof of Lemma 5.12 . . . . . . . . . . . . . . . . . . . . . . . 71
5.4 Freedoms for maximal packings . . . . . . . . . . . . . . . . . 77
5.5 A di(cid:14)culty of the Schwarz problem . . . . . . . . . . . . . . . 82
5.6 A discrete circular Riemann-Hilbert problem . . . . . . . . . 84
6.1 Numerical results for K[7] . . . . . . . . . . . . . . . . . . . . 90
3
6.2 Numerical results for K0 . . . . . . . . . . . . . . . . . . . . . 90
6.3 Spectra of H . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.4 Eigenfunctions for K0 . . . . . . . . . . . . . . . . . . . . . . 92
6.5 Modi(cid:12)ed Eigenfunctions for K0 . . . . . . . . . . . . . . . . . 93
6.6 Numerical results for Hb . . . . . . . . . . . . . . . . . . . . . 94
6.7 Packings solving instances of (DHRHP) . . . . . . . . . . . . 95
6.8 Results on a di(cid:11)erent complex. . . . . . . . . . . . . . . . . . 96
6.9 Nonlinear versus linearized transform . . . . . . . . . . . . . . 97
Abstract
This book deals with the de(cid:12)nition of a discrete Hilbert transform. The
classical Hilbert transform is a bounded linear operator relating the real-
and imaginary parts of the boundary values of a holomorphic function.
The discretization is based on the theory of circle packings which has been
established as a discrete counterpart of complex analysis in the past three
decades.
The Hilbert transform is closely related to Riemann-Hilbert problems. In
particular, the transform can be computed from the solution of a Riemann-
Hilbert problem. We will therefore de(cid:12)ne the transform based on discrete
Riemann-Hilbert problems which have been investigated by Elias Wegert
since 2009.
The de(cid:12)nition of the discrete operator requires the regularity of a certain
Jacobian matrix. In this book, we present a proof of the regularity of this
matrix.