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A Discrete Hilbert Transform with Circle Packings (BestMasters) PDF

110 Pages·2017·1.579 MB·English
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Dominik Volland A Discrete Hilbert Transform with Circle Packings BestMasters Springer awards „BestMasters“ to the best master’s theses which have been com­ pleted at renowned Universities in Germany, Austria, and Switzerland. The studies received highest marks and were recommended for publication by supervisors. They address current issues from various fields of research in natural sciences, psychology, technology, and economics. The series addresses practitioners as well as scientists and, in particular, offers guid­ ance for early stage researchers. More information about this series at http://www.springer.com/series/13198 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 Springer Spektrum © Springer Fachmedien Wiesbaden GmbH 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid­free paper This Springer Spektrum imprint is published by Springer Nature The registered company is Springer Fachmedien Wiesbaden GmbH The registered company address is: Abraham­Lincoln­Str. 46, 65189 Wiesbaden, Germany 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.

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