A Discrete Hilbert Transform with Circle Packings

106 Pages·2017·2.25 MB·German

by  Dominik Volland

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
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
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
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.
1 Introduction
The mathematical branch of complex analysis deals with the behavior of
holomorphic functions, which are di(cid:11)erentiable functions in the complex
plane. Thetheoryoffunctionsofonecomplexvariableisquiteself-contained
nowadays. However, during the last three decades, complex analysis has
found a discrete counterpart in the theory of so-called circle packings.
Thesepackingsarecon(cid:12)gurationsofcirclesthatsatisfyprescribedtangency
relations. The idea of using circles to discretize holomorphic functions is due
to the fact that holomorphic functions are locally conformal transformations
of the complex plane wherever their derivative does not vanish. This means,
a holomorphic function will map an in(cid:12)nitesimally small circle onto an
in(cid:12)nitesimally small circle again.
Circle packings have turned out to be a very faithful discretization of
holomorphic functions indeed. By now, a wide variety of theorems and
concepts from complex analysis have found counterparts in circle packing.
There are still many open problems and the theory of circle packings is only
growing. A broad overview over the theory of circle packings can be found
in Stephenson’s book [27].
This book is devoted to one of the numerous concepts of complex analysis
and the quest for its discrete counterpart. Namely, we discuss a discrete
Hilbert transform in the context of Riemann-Hilbert problems.
Roughly speaking, a function holomorphic in the complex unit disc D is
determined by the real part of its boundary values. The Hilbert transform is
a linear operator that relates the real and imaginary parts of the boundary
values of a holomorphic function.
Riemann-Hilbert problems are boundary value problems for functions
holomorphicinD. ARiemann-Hilbertproblemimposesonerealconditionto
each boundary value of the solution. The Hilbert operator is closely related
totheseproblems. Infact,wewillshowthattheHilberttransformofagiven
function can be obtained from the solution of a suitable Riemann-Hilbert
problem.
TheGermanmathematicianEliasWegert,authorofseveralpapers[31][32]
and a book [30] on Riemann-Hilbert problems, observed that Riemann-
Hilbert problems (cid:12)t nicely into the setting of circle packings. Motivated
2 1 Introduction
by this, he has outlined the de(cid:12)nition of a discrete Hilbert transform. The
strategy of this de(cid:12)nition can be summarized as follows: Find a Riemann-
HilbertproblemfromwhichtheHilberttransformcanbeobtained,discretize
thisproblemwithcirclepackings,solveit,andextractthediscretetransform
from the solution. In this book, we present the discrete operator obtained
from this procedure.
The de(cid:12)nition of the discrete operator rests on the regularity of a certain
Jacobian matrix which arises from the discrete boundary value problem.
The regularity of this matrix has been conjectured by Wegert [34]. In this
book, we give a proof of this conjecture.
Thebookisorganizedasfollows. Chapter2(cid:12)xesnotationconventionsand
statesafewclassicalresultsthatwillbeusedthroughoutthetext. InChapter
3 we will be concerned with the continuous theory of the Hilbert operator.
We will (cid:12)rst introduce the functional analytic basics required to de(cid:12)ne the
Hilbert operator and formulate its most important properties. After that,
we will de(cid:12)ne Riemann-Hilbert problems and discuss their connection to the
Hilbert operator.
In Chapter 4, we will turn to the discrete setting. We will de(cid:12)ne circle
packingsandstatetheirbasicproperties,withanemphasisonthedi(cid:11)erential
geometricstructureofcirclepackings. Chapter5willthenintroducediscrete
boundary value problems. We will give some examples, introduce linearized
discrete boundary value problems and prove the regularity of the matrix
mentioned before. Based on this and the work done in Chapter 3, we will
de(cid:12)ne a discrete Hilbert operator.
Finally, Chapter 6 presents some numerical results and outlines questions
related to the discrete operator that may be addressed in the future.
2 Auxiliary Material and Notation
This chapter introduces our notation conventions and gives a short overview
over some results that will be used without references throughout the book.
Complex analysis. There are many books covering elementary complex
analysis. Our standard reference is the book of Ahlfors [1].
A domain is an open connected set in C. The domain we are dealing with
mostofthetimeistheunit disc fz 2Cjjzj<1gdenotedbyD. Itsboundary
is the unit circle fz 2Cjjzj=1g and will be denoted by T. Other discs will
be denoted by B (z(cid:3)) := fz 2 Cjjz(cid:0)z(cid:3)j < rg. Given a circle C (cid:26) C, the
r
disc bounded by C will be denoted by intC, while extC =Cn(C[intC).
The boundary of a domain U is denoted by @U and the closure U [@U is
denoted by U. Some caution is required here since the complex conjugate of
a complex number z is denoted by z.
By Log we denote the principal branch of the complex logarithm de(cid:12)ned
on the slit domain C(cid:0) :=CnR . By Arg we denote the principal branch
(cid:20)0
of the complex argument function, i.e. Arg(z) = ImLog(z). log resp. arg
denotes any branch of these functions if it is not necessary (or possible) to
specify the branch explicitly.
To keep the overview, we have the following convention for variables: z
always referstoa variablerangingon D orsometimes C, while t isa variable
ranging on T. Arguments of complex numbers are denoted by Greek letters,
mostly (cid:28), sometimes also (cid:30) or (cid:18).
M(cid:127)obius transformations. A Mo(cid:127)bius transformation is a holomorphic func-
tion of the form
az+b
T(z)=
cz+d
where a;b;c;d are complex constants such that ad(cid:0)bc6=0. These functions
play a special role in complex analysis due to their geometric properties. It
is natural to understand them as mappings on the extended complex plane
C[f1g by writing T(1)= a and T((cid:0)d)=1 if c6=0 (or T(1)=1 if
c c
c=0).
4 2 Auxiliary Material and Notation
M(cid:127)obius transformations have the following properties (for proofs, see
Chapter 3, Section 3 of [1]):
(i) T is a bijective map on the extended complex plane C[f1g.
(ii) T isholomorphiconCn(cid:8)T(cid:0)1(1)(cid:9)anditsderivativevanishesnowhere.
(iii) If C (cid:26)C is a M(cid:127)obius circle, i.e. a circle or a line, T(C) is a M(cid:127)obius
circle, too.1
(iv) Tcanbewrittenasacompositionoflinearfunctionsandtheinversion
z 7! 1.
z
(v) Let C (cid:26) C be a circle and T(cid:0)1(1) 2= C. Let z ;z ;z 2 C lie in
1 2 3
counter-clockwise order on C. Then
(cid:15) ifT(cid:0)1(1)2extC,thenT(intC)=intT(C), T(extC)=extT(C),
and T(z );T(z );T(z ) lie in counter-clockwise order on C again.
1 2 3
(cid:15) if T(cid:0)1(1)2intC, then T(intC) = extT(C), T(extC) = intT(C),
and T(z );T(z );T(z ) lie in clockwise order on C.
1 2 3
If c = 0, let d = 1 w.l.o.g. Then T(z) = az+b is a linear function. If in
addition a2T, thenT isanisometryof Candiscalleda plane rigid motion.
Hence, there are 3 real degrees of freedom to construct a plane rigid motion.
Conformal automorphisms. A function f: D ! D is bijective and con-
formal i(cid:11) it is a composition of a rotation z 7!(cid:21)z for (cid:21)2T and a M(cid:127)obius
transformation of the form
z(cid:0)z
z 7! 0
1(cid:0)zz
0
forsomez 2D. Inparticular,thereareexactlythreerealdegreesoffreedom
0
for constructing such a map. For a proof, see [1] (Chapter 4, 3.4).
Blaschke products. A ((cid:12)nite) Blaschke product of degree n is a function
of the form
n
B(z)=cY z(cid:0)zj
1(cid:0)z z
j
j=1
1In this context, a line should be thought of as a circle containing the point 1. From
this,itisclearthatT mapsacircleC toacircleagaini(cid:11)T(cid:0)1(1)2=C.