Document [original]

Thoughts on Harmonic Analysis on

the Sphere: spherical wavelet

frames and kernels

vorgelegt von

BMS Phase II, B.Sc.-Mathematiker

Yizhi Sun

ORCID: 0000-0002-2555-5964

an der Fakultät II - Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr. rer. nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. John M. Sullivan

Gutachter: Prof. Dr. Reinhold Schneider

Gutachter: Prof. Dr. Stephan Dahlke

Tag der wissenschaftlichen Aussprache: 27. November 2019

Berlin 2020

Abstract

In the central spirit of harmonic analysis lies the concept of eectively de-

composing, analyzing and representing functions or functionals. It has lead

to the ourish of Fourier analysis and its modern descendants such as wavelets

and its siblings. Especially, the construction of spherical wavelets and its related

theory is at junior age.

This dissertation gives a brief summary of existing results in this eld, and at

the same time create spherical

-wavelets and further develop spherical kernel

theory. Among various strategies, two types of spherical wavelets are empha-

sized, one constructed in the frequency domain, the other generated through

stereographic projection. In the former one I discuss localized tight frame de-

sign and its directional extension. In the latter one a new anisotropic dilation is

dened, and a representation system generated by it consists of the

spherical

wavelets/shearlets

. Summability properties of those wavelets/shearlets are well

established once they are restricted to certain subspaces of square-integrable

functions newly dened in this dissertation, including the so called

hollow pole

functions

Kernels, though deeply rooted in classical theory, can nd its variation and

application in the frame theory. Indeed,

frame kernel

, a concept which is pro-

posed in this dissertation, is an equivalent formulation to the frame itself. Be-

sides, there exist a variety of kernels which exhibit their own special properties.

For zonal kernels, I give its necessary and sucient conditions to approximate

square integrable functions on the sphere. Multiscale kernel, a recently appeared

concept, will meet its spherical version here and it turns out to have reproducing

property for certain Hilbert space of spherical functions.

One of the climaxes in this work is the invention of two novel frames, based

on the two spherical wavelets constructions. In the zonal kernel approach I give

frame properties inside the multiresolution structure; while for

-wavelets, I

prove that under certain conditions they form tight frames in continuous and

discrete setting respectively, following from which are reproducing formulae that

enable us to reconstruct or approximate numerically an integrable function or

solutions of PDEs. Based on the obtained frames a spherical Galerkin scheme

is proposed afterwards. At the end of this dissertation I prove an inner product

formula with respect to a recently emerged surface-value dependent inner prod-

uct space on a triangular mesh, prove its equivalence to the combinatoric inner

product, and give eigenvalues estimation for a discrete Laplacian.

Zusammenfassung

Ein zentraler Aspekt der Harmonischen Analysis ist die eziente Zerlegung,

Analyse und Repräsentation von Funktionen und Funktionalen. Diese Konzepte

führten zu einem Aufblühen der Fourier Analysis und deren jüngsten Teilgebiete,

wie Wavelets und verwandte Methoden. Dabei bendet sich insbesondere die

Konstruktion von sphärischen Wavelets und die damit verbundene Theorie noch

in den Anfängen ihrer Entwicklung.

Die vorliegende Dissertation gibt einen Überblick über bereits existierende

Resultate in diesem Bereich und entwickelt neuartige Instrumente, wie z.B.

sphärische Wavelets und sphärische Kernels. Ein besonderes Augenmerk liegt

dabei auf zwei Typen von sphärischen Wavelets: Zum einen eine Konstruktion

im Frequenzbereich und zum anderen eine Konstruktion durch stereographis-

che Projektionen. Hinsichtlich der Erzeugung im Frequenzbereich werden ak-

tuelle Fortschritte im Bereich des Localized-tight-Frame und deren richtungsab-

hängige Erweiterungen diskutiert. In den Ausführungen über die Erzeugung

von Wavelets durch stereographische Projektionen wird andererseits ein neues

Konzept von richtungsabhängigen Dilationen eingeführt, was schlieÿlich zu so-

genannten sphärischen

-Wavelets/Shearlets führt. In diesem Zuge können

Summierbarkeitseigenschaften hergeleitet werden, nachdem die konstruierten

Wavelets/Shearlets auf einen Unterraum von

-Funktionen eingeschränkt wur-

den, was insbesondere sogenannte Hollow-Pole Funktionen einschlieÿt.

Es wird sich herausstellen, dass klassische Kernels ebenfalls sehr vielfältige

Anwendungsmöglichkeiten in der modernen Frame-Theorie haben. In diesem

Kontext wird das Konzept von Frame-Kernels eingeführt, welches eine äquiv-

alente Formulierung der Frame-Eigenschaft ermöglicht. Darüber hinaus wer-

den noch weitere Beispiele von Kernels hinsichtlich ihrer speziellen Eigen-

schaften untersucht. Für Zonal-Kernels werden notwendige und hinreichende

Bedingungen gezeigt, sodass diese quadratisch integrierbare Funktionen auf der

Sphäre approximieren. Schlieÿlich wird eine sphärische Version von Multiskalen-

Kernels hergeleitet und es wird gezeigt, dass diese für spezielle Hilberträume von

sphärischen Funktionen die Reproduzierbarkeitseigenschaft besitzen.

Ein Hauptergebnis dieser Arbeit bildet die Erndung von zwei neuarti-

gen Frame-Typen, basierend auf den zwei obengenannten Konstruktionen von

sphärischen Wavelets. Hinsichtlich des Zonal-Kernel-Ansatzes wird die Frame-

Eigenschaft innerhalb der Multiskalenstuktur nachgewiesen. Für

-Wavelets

dahingegen wird bewiesen, dass diese unter bestimmten Annahmen Tight-

Frames sowohl im kontinuierlichen als auch im diskreten Sinne bilden. Daraus

folgt insbesondere die Reproduzierbarkeitseigenschaft, die die exakte Rekon-

struktion von integrierbaren Funktionen oder Lösungen der partiellen Dier-

entialgleichungen ermöglicht. Auf der Grundlage der erhaltenen Frames wird

anschlieÿend ein sphärischer Galerkin-Ansatz vorgeschlagen. Zum Abschluss

der Arbeit wird eine Formel für das innere Produkt eines kürzlich eingeführten

Prä-Hilbertraums bewiesen, der auf einem Dreiecksnetz deniert ist.

Acknowledgements

This dissertation is the fruit of numerous highly ecient working days as well as

dreaming days in the library, half seeking inspiration from the classic, half devel-

oping into new branches. Even though I never intended to write a big volume,

but rather to expose the shining part of an iceberg, both because the full amount

of work in this topic is far beyond a doctoral dissertation and exhausting a eld

that many mathematicians worked for decades is not my purpose, a list of peo-

ple must be thanked by me personally, so that this independent work will not be

solely considered as my work, but rather a natural product out of the support

of many. It is dedicated to my family in Hangzhou for their long-lasting care

and assistance since childhood. I would give special thanks to Prof. Kutyniok

for suggesting me the wonderful dissertation topic about constructing shearlets

on the sphere. Some of her suggestions at the beginning of my doctorate, even

years later, are still helpful. The writing of Sturm-Liouville theory in the last

chapter is inspired during my short visiting at mentor Prof. Fiedler's hospitable

group at Freie Universität. My gratitude goes to Prof. Schneider, who takes

great responsibility to review my dissertation and guarantees the progress of

my doctoral study. I would like to thank Martin and Liselotte for making the

German version "Zusammenfassung" possible. My gratitude also belongs to

those who appeared to me shortly or frequently like ghosts, no matter to dis-

tract me or to encourage me, for they gave me immense inspiration, beauty and

love to carry on the research. Contents of this work are either covered and im-

proved selectively by breaking the shell of previous master pieces and extracting

their essence, or purely created according to my imagination in order to make a

wonderfully integrated hybrid piece; other materials under preparation but not

covered in this piece are left to the future. Last but not least I express my grati-

tude to China Scholarship Council(Grant No.201206320164) and Stipendium des

Präsident von Technische Universität Berlin for several years funding support.

Berlin, März 2020

III

Contents

1 Introduction 1

1.1 Summary of contents . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Spherical representing systems and operators: old and new . . . 4

2 Spherical Dilation Systems 17

2.1 Dilation in frequency domain . . . . . . . . . . . . . . . . . . . . 17

2.2 Dilation through stereographic projection . . . . . . . . . . . . . 26

2.3 Spaces of admissible

-wavelets/shearlets . . . . . . . . . . . . . 31

2.4 Other approaches: a selective review . . . . . . . . . . . . . . . . 39

3 Extension to Miscellaneous Results 49

3.1 Kernel approximation . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Construction of spherical frames . . . . . . . . . . . . . . . . . . 64

3.3 A product formula on simple surfaces . . . . . . . . . . . . . . . 77

4 Supporting topics 85

4.1 Orthogonal polynomials: a dierential

equation point of view . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2 Spectrum of discrete Laplacian . . . . . . . . . . . . . . . . . . . 96

4.3 Other representations on the sphere . . . . . . . . . . . . . . . . 104

Chapter 1

Introduction

The purpose of this dissertation is to build analysis tools to analyze information

distributed on those surfaces that can be identied with the unit sphere either

geometrically or topologically. We shall cover dierent topics from spherical

wavelets, through spherical kernels, to their approximations. However, I am

not ambitious in a way to make an encyclopedia here, but rather to explore

these areas and their interconnection while at the same time make contribution

to some interesting topics along the path, to reect the beauty and colorfulness

lying between harmonic analysis and other mathematical elds and applications.

1.1 Summary of contents

Each chapter is developed systematically and has their own introduction part,

hence here I only briey summarize the contents and contributions. The rst

chapter serves both as an explanation of mathematical concepts that are cru-

cial in later chapters, such as spherical harmonics, Laplace-Beltrami operator,

and as an introduction to spherical operators that emerged recently along with

their properties. Especially I derive some commutativity properties of a newly

introduced spherical Hilbert-transform. Immediately after that is a reminder of

Funk-Hecke formula, integral formulae on the rotation group, as well as the fact

that the rotation operator, which plays fundamental role in later chapters, is an

irreducible representation of the classical rotation group.

The second chapter starts with the construction of spherical wavelets in the

frequency domain, in which a dilation on the half real line using radial basis func-

tions played a key role. In this approach there has been localized tight frames

successfully constructed in recent years and an extended version into anisotropic

case can be achieved by attaching a steerable directional function onto the ra-

dial basis function. In the second section of this chapter, we construct a type

of spherical

-wavelet system through stereographic projection, which incorpo-

rates spherical wavelets and spherical shearlets. This method is geometrically

the most natural, intuitively can be understood as the correspondence between

CHAPTER 1. INTRODUCTION

plane and the sphere; it inherits the advantage of conformal mappings, hence

has great mathematical properties such as preserving frame structure. What

quite dierent from the previous work of others is that I discuss under a setting

incorporating both isotropic and anisotropic dilation. The main new theorem

in this chapter gives a pioneering necessary and sucient condition for a large

subclass of square integrable functions to be admissible, based on which further

frame properties can be achieved. At the end is a selective review of several

other ways of constructing wavelets on the sphere. Each of them has dier-

ent emphasis on one specic aspect, such as orthogonality, restoration, reduced

redundancy, fast computation and so forth.

The third chapter is an extension to dierent topics with multiple new re-

sults that have lines of interconnection beneath. The rst topic concerns kernel

approximation, in which I would like to invite readers to have a tour through

various types of kernels, from the classical reproducing kernel since Hilbert's

time to the recently developed multiscale kernels. The underlying questions

that I am going to answer in this part include: When is a zonal kernel capable

of approximating a square integrable function on the sphere? If such a condition

exists, how well can it approximate the function? Does a sphere version mul-

tiscale kernel and its corresponding multiscale structure exist? If exists, what

properties does it have?

After an armative answer to those questions I would like to draw your

attention to the construction of spherical frames. In this part two dierent types

of new frame systems are built corresponding to two dierent ways of spherical

wavelet construction in the pervious chapter: one is developed in the frequency

domain, where frame property inside the inherited multiresolution structure is

for the rst time explored, which allows us to transform an arbitrary frame or

basis without any good properties into a localized one; while another, technically

much more involved, adapts to the wavelets/shearlets systems coming through

conformal mapping. Notice that mapping a plane wavelet system to the sphere

inevitably suers from defects such as distortion. In other words, a regular grid

on the plane, once we impose on it some translation and dilation operation,

does not necessarily give a regular grid on the sphere. Therefore it becomes

meaningful to develop frame systems directly on the sphere. In fact, I am

going to give both continuous and discrete version localized tight frames on the

sphere and their exact reproducing formulae for the stereographic approach,

which is absent since twenty years. These new frames, unlike the ones obtained

by using energy conservation through conformally projecting planar wavelets

or shearlets frames, have the potential to adapt to any preferable grids on the

sphere. However, I leave the specic choices of grids, that have been or to be

smartly designed and implemented by mathematicians, engineers, scientists and

1.1. SUMMARY OF CONTENTS

artists, as well as redundancy analysis to future work. Based on those spherical

frames and their multiresolution structure a Galerkin scheme which enable us

to do numerical analysis of PDEs on the sphere can be formed.

Afterwards a short discussion is devoted to triangulated surface, as another

step into the discrete surface world. This part is not tightly related to the

previous parts, and by no means I intend to dive deeply here, but rather only

give a avor of how an object living in the continuous world can nd their

discrete partners. We start by giving a formula for a new inner product of

continuous piecewise-linear functions on a given triangulation, comparing it with

the weighted inner product on graphs. Along this way, we see two dierent types

of discrete Laplacian on surfaces, one purely combinatoric, while the other is

dened in a geometrical manner. The eigenvalues of the geometric discrete

Laplacian are less known and I derive their expressions and bounds as a step

forward, while the eigenvalues estimation of the combinatoric Laplacian has

been well established before this dissertation and I summarize some in the nal

chapter as supporting material.

The nal chapter also contains a self-contained section of elegant introduc-

tion to Sturm-Liouville theory is given, from which dierent kinds of orthogonal

polynomials that are essential to the main results in this dissertation are derived

naturally. After that, a small section about quaternions and metrics on rotation

group gives an alternative representation for the spherical points.

Except for the last chapter which consists of reformulation of known results

from personal perspective, most theorems in this dissertation are established

and proved either in original or progressive ways, while prepared by discussions

based on excellent works of others', so that I believe a balance has been achieved

between what are historical and what are innovative in this work.

CHAPTER 1. INTRODUCTION

1.2 Spherical representing systems and operators:

old and new

We live in an era that the perception of the world is reformed into mathematical

simulation and the understanding of its objects is deepened through systematical

deduction using mathematical symbols. Many natural objects like the planet on

which we live can be studied abstractly as a sphere, and representing a sphere

mathematically is the rst step we shall take.

By "representing" I ask two dierent but united questions. The rst ques-

tion is how to represent points on a sphere? It actually asks, how to build a

correspondence relation between the sphere and some parameter space. One

obvious way is using the coordinates in Euclidean space, that we have been fa-

miliar with in our daily life and in the university analysis and geometry course;

another way is to build a group identication, which is partly illustrated in this

chapter and the last chapter for reference.

The second question is what are natural representations on the function

spaces dened on a sphere? The function space in most context of this work

means the

space or subspaces of it, which is Hilbert. We shall encounter

several approaches, including representing a function by the superposition of

eigenfunctions of the spherical Laplace operator, the integral representation with

respect to some wisely chosen kernels, and the representation through expansion

of a specially designed representation system.

Let

(I, dµ)

be a measure space, a family of elements

Φ = {ϕi}i∈I

in a Hilbert

space

is called a dictionary

span{ϕi}=H

, namely it is a complete subset.

Various kinds of dictionaries may be chosen to provide optimal representation of

certain class of function spaces, for instance the spherical harmonics introduced

below constitute an orthogonal basis for the space of square integrable func-

tions on the sphere, in nite element methods multivariable polynomials under

dierent restrictions provide approximation to Sobolev spaces and wavelets are

well adapted to Besov space by its denition. The "optimal" here could either

mean an accurate and unique expression of a signal

with fast convergence

rate, or it could mean the linear expansion of

in terms of an overcomplete

dictionary which has sparse coecients through minimizing certain norms, or

sensitive to special features like high frequency or singularities, depending on

objects or tasks; just like preparing a tasty noodle soup for the new years eve,

you can either choose ingredients like lamb or shes inside if the family members

The name dictionary is borrowed from learning theory, with the underlying meaning that

the vocabulary inside is suciently complete to express any sentences or meanings, in other

context it could be alternatively called atoms or molecules, namely a collection of building

blocks

1.2. SPHERICAL REPRESENTING SYSTEMS AND OPERATORS: OLD

AND NEW

are meat fans, or tofu and seegrass if your lovers coincidentally are vegetarian.

Being overcomplete intuitively means that to prepare a sh noodle soup, you

must at least have sh and noodles, and in addition you can add pepper or soy

to achieve dierent avors. In real computation usually people aim to obtain

a balance between the precision and the sparsity, by minimizing a functional

∥f−Φg∥2

2+αM(g)

, with

a regularization parameter and

some cost

function.

While there is great exibility of choosing dierent ingredients for your soup,

you shall not put in everything so that it spoils or does not remains as a sh

noodle soup anymore; in other words structure stability and compatibility shall

be guaranteed. The concept of frames comes in to provide more exibility than

the orthogonal basis and at the same time requires the basic ingredients that

can stably represent a given function. Precisely, a dictionary

is called a

frame

if there are positive constants

A⩽B

such that

A∥f∥2⩽∫︂I

|⟨f, ϕi⟩|2dµ ⩽B∥f∥2

(1.1)

for any

f∈ H

. In particular, when

is discrete, the above inequality becomes

A∥f∥2⩽∑︂

i∈I|⟨f, ϕi⟩|2⩽B∥f∥2

(1.2)

Meanwhile for any

f∈ H

there exists at least one dual frame(the canonical

dual)

{ϕi}

˜i∈I

with frame bounds

and

such that

f=∑︂

i∈I⟨f, ϕi⟩ϕi

(1.3)

When the frame

{ϕi}i∈I

super tight

, namely when the frame bounds

coincide and equal to one, it holds that

ϕi=ϕi

for all

i∈I

. The canonical

dual may not be equipped with the properties that

has, for instance the dual

system does not necessarily have a single generator when the frame

does,

hence does not inherits a wavelet structure. In fact, there might exist innitely

many (alternate) duals, but how to choose a dual wisely is much depending on

the problems to deal with.

Spherical harmonics form an orthogonal system, hence its dual is itself. It

has become a useful tool to analyze functions on the sphere, since the time of

Laplace and Legendre. A spherical harmonic

is a homogeneous polynomial

of degree

which solves the Laplace equation. After being restricted on the

sphere, they are sometimes called surface spherical harmonics. In this work, if

CHAPTER 1. INTRODUCTION

we don't give additional clarication, it is assumed that spherical harmonics are

already restricted and they satisfy the equations

∆S2Yl=−l(l+ 1)Yl

(1.4)

where

∆S2=1

sin θ(∂

∂θ(sin θ∂

∂θ) + 1

sin θ

∂2

∂φ2)

(1.5)

is the Laplace-Beltrami Operator on

. The

operator of innitesimal rota-

tions

[65] or

operators of angular momentum

[71] up to a change of sign

are

Lx=−i(y∂

∂z −z∂

∂y ) = i(sin φ∂

∂θ + cot θcos φ∂

∂φ)

Ly=−i(z∂

∂x −x∂

∂z ) = i(−cos φ∂

∂θ + cot θsin φ∂

∂φ)

Lz=−i(x∂

∂y −y∂

∂x) = −i∂

∂φ

(1.6)

It can be immediately veried that

−∆S2=L2

x+L2

y+L2

(1.7)

If we denote by

(η1, η2, η3)T

the eigenvector of

σ∈SO(3)

with its length

equal to the rotation angle

and the identity matrix of rotation group by

σ0

when

is mall, there is

R(σ) = R(η1, η2, η3) = R(σ0)−iLxη1−iLyη2−iLzη3+O(ϕ2)

where

R(σ0)

is just the identity operator.

Since for

s, s′>0

there is the obvious relation

R(sη1, sη2, sη3)R(s′η1, s′η2, s′η3) = R((s+s′)η1,(s+s′)η2,(s+s′)η3)

it follows that

dR(sη1, sη2, sη3)

ds =−iR(sη1, sη2, sη3)Lη

(1.8)

where

Lη =Lxη1+Lyη2+Lzη3

, hence

e−iLη =R(η1, η2, η3)

(1.9)

and the unitarity of representation

R(η1, η2, η3)

is equivalent to the hermiticity

of the operators

and

L2(S2)

1.2. SPHERICAL REPRESENTING SYSTEMS AND OPERATORS: OLD

AND NEW

Notice that the element

σ˜ = σ1σσ−1

leaves

σ1(η1, η2, η3)T

invariant. Fur-

thermore, if

η⊥

is a unit vector perpendicular to

η= (η1, η2, η3)T

such that

(ση⊥, η⊥) = cos ϕ

, then

(σ˜σ1η⊥, σ1η⊥) = cos ϕ

. Thus the rotation angle of

σ˜

the same as

. Assume that

σ1

is a small rotation, namely

R(σ1) = R(σ0)−i(Lxζ1+Lyζ2+Lzζ3) + O(|ζ|2)

with

|ζ|

small. Let

η= (|η|,0,0)T

and

ζ= (0,|ζ|,0)T

, then as a consequence of

the identity

R(σ˜) = R(σ1)R(σ)R(σ−1

there is

R(σ˜) = R(σ0)−i(Lxη˜1+Lyη˜2+Lzη˜3) + O(|η|2)

= (R(σ0)−i|ζ|Ly)(R(σ0)−i|η|Lx)(R(σ0)−i|ζ|Ly)

=R(σ0)−i|η|Lx+|η|·[Lx, Ly]|ζ|+O(|ζ|2)

(1.10)

Identity

σ1ζ=ζ

implies

σ1=⎛

⎝cos ϕ10 sin ϕ1

0 1 0

−sin ϕ10 cos ϕ1⎞

⎠

where

ϕ1=|ζ|

and

η˜ = σ1η= (|η|cos ϕ1,0,−|η|sin ϕ1)T

, hence by comparing

the rst order term of

|ζ|

(1.10)

we obtain

[Lx, Ly] = iLz

. By exchanging the

role of

x, y

and

, we arrive at the following elegant commutation rules

[Lx, Ly] = iLz,[Ly, Lz] = iLx,[Lz, Lx] = iLy

(1.11)

By dening

L+=Lx+iLy, L−=Lx−iLy

relations

(1.11)

become

[L+, Lz] = −L+,[L−, Lz] = L−,[L+, L−]=2Lz

(1.12)

Notice that

L+

L−

and

are self-adjoint and if

m∈R

nonzero is an

eigenvalue of

with eigenfunction

, then

L+f

is an eigenvector corresponding

to eigenvalue

m+ 1

while

L−f

is one corresponding to eigenvalue

m−1

. Since

the total number of distinct eigenvalues is nite, assume that the largest of them

and write its normalized eigenfunction as

, similarly the smallest of them

is written as

l∗

with eigenfunction

Yl∗

Dene by induction

αmYm−1=L−Ym

where

αm

are chosen so that

⟨Ym−1, Y m−1⟩= 1

CHAPTER 1. INTRODUCTION

Since

L+Yl= 0

, that is

L+Yl−1=1

αl

L+L−Yl=2

αl

LzYl=2l

αl

Yl=αlYl

(1.13)

an induction argument leads to the claim that

L+Ym

is proportional to

Ym+1

namely

L+Ym=αm+1Ym+1

. Moreover, the fact that

L†

+=L−

gives

αm⟨Ym, Y m⟩=αm⟨Ym−1, Y m−1⟩

namely

αm=αm=αm

. Thus from the observation that

α2

m+1Ym+1 =L+L−Ym+1 = (2Lz+L−L+)Ym+1 = (2m+2+α2

m+2)Ym+1

we obtain the relation

α2

m+1 −α2

m=−2m

. From this relation, our observation

that

α2

l= 2l

leads to the expression

α2

m= (l+m)(l−m+ 1)

(1.14)

L−Yl∗= 0

gives

αl∗= 0

, hence

l∗=−l

. Therefore

is either an integer or half

an odd number and every irreducible representation is uniquely determined by

with its dimension equal to

2l+ 1

In real calculation and approximation, frequently used is the normalized

expression

l(θ, φ)=(−1)m√︄(2l+ 1)(l−m)!

4π(l+m)! eimφPm

l(cos θ)

(1.15)

which naturally appears when one solves the Laplace equation through separa-

tion of variables, where

l∈N

and

|m|⩽l

. Those spherical harmonics of degree

form an orthonormal basis for the space of homogeneous polynomials of degree

, denoted by

From the property that

−i∂

∂φ Ym

l=mY m

we already see that

l(φ, θ) =

eimφFm

l(θ)

for some function

l(θ) =: F

˜m

l(cos θ)

. Let

x= cos θ

, from

(1.5)

deduce that

(1 −x2)d2

dx2F

˜m

l(x)−2xd

dxF

˜m

l(x) + [︃l(l+ 1) −m2

1−x2]︃F

˜m

l(x)=0

(1.16)

when

m= 0

it is exactly dierential equation

(4.41)

, thus

˜m

are nothing else

1.2. SPHERICAL REPRESENTING SYSTEMS AND OPERATORS: OLD

AND NEW

but the associated Legendre Polynomials

l(x) := (−1)m(1 −x2)m/2dm

dxmPl(x)

=(−1)m(1 −x2)m/2

2ll!

dl+m

dxl+m(x2−1)l

= (−1)m(l+m)!

(l−m)!P−m

l(x)

(1.17)

which satisfy the orthogonality relation that

∫︂1

−1

l(x)Pm

l′(x)dx =(l+m)!

(l−m)!

2l+ 1δl,l′

(1.18)

where the last equality of

(1.17)

follows from Rodrigues formula

dl−m(x2−1)l

dxl−m=(l−m)!

(l+m)!(x2−1)mdl+m(x2−1)l

dxl+m

Sometimes Laplace's integral expression for Legendre polynomials is as useful

as the dierential expression. It reads equivalently as

Pl(x) = 1

π∫︂π

0(︂x±√︁x2−1 cos θ)︂ldθ

(1.19a)

Pl(x) = 1

π∫︂π

0(︂x∓√︁x2−1 cos θ)︂−l−1dθ

(1.19b)

More generally there is

l(x) = γ+

l,m ∫︂π

0(︂x∓√︁x2−1 cos θ)︂lcos(mθ)dθ

l(x) = γ−

l,m ∫︂π

0(︂x±√︁x2−1 cos θ)︂−l−1cos(mθ)dθ

where

γ+

l,m = (±1)m(l+m)!

πl!e−m

2πi

and

γ−

l,m = (±1)ml!

π(l−m)! e−m

2πi

The spherical Hilbert transform, as an analogue of the plane situation, rstly

appears in [85](but no mathematical properties are given there)

ˆ︁

(Hf)lm =⎧

⎨

⎩

−iˆ︁

flm m > 0

0m= 0

iˆ︁

flm m < 0

(1.21)

CHAPTER 1. INTRODUCTION

where

ˆ︃

flm

is the Fourier coecient of a square integrable function

with respect

to the spherical harmonics

. Let us derive some new commutativity relations

and formulae of those operators in the next theorem.

Theorem 1.1.

(i)

is self-adjoint and

(iH)2f=−f

for any

f∈L2(S2)

When

ˆ︁

fl,m =ˆ︁

fl,−m

for all

and

|m|⩽l

and

iHf

are orthogonal.

(ii) The dierential operator

dened in

(1.6)

commute with

In particular

[∆S2,H]≡0

(iii) Given any bounded operator

A:L2(S2)→L2(S2)

, there is

[∆S2, A]Hl⊥Hl

and

[∆S2, A]Mm⊥Mm

(1.22)

where for each

m∈Z

Mm=

span

{Ym

l:l⩾0}

(1.23)

(iv) For each

, there holds the identity

{ˆ︁

(Hf)l,m}m⊛{ˆ︁

(Hf)l,m}m(n)−{ˆ︁

fl,m}m⊛{ˆ︁

fl,m}m(n)

=−i

sgn

([n]) (︂{ˆ︁

(Hf)l,m}m⊛{ˆ︁

fl,m}m(n) + {ˆ︁

fl,m}m⊛{ˆ︁

(Hf)l,m}m(n))︂

(1.24)

where

({am}⊛{bm}) (n) = ∑︁

|m|⩽l

a[n−m]bm

and

[n]

taking the module

Proof.

The claim that

is self-adjoint and

(iH)2f=−f

is obvious. The rest

of (i) is a result of the observation that

⟨f, iHf⟩L2=∑︂

l∑︂

|m|≤l

sgn(m)|f

ˆlm|2

(1.25)

is zero since each

term cancels the

−m

term.

It can be checked through calculation that the identity

LzHYm

l=⎧

⎨

⎩

−imY m

m > 0

m= 0

imY m

m < 0

=−i|m|Ym

=HLzYm

Here I make the convention that

[n] = −(|n|mod l+ 1)

when

is a negative integer;

[n] = l+ 1

when

(n−mmod l+ 1) = 0

1.2. SPHERICAL REPRESENTING SYSTEMS AND OPERATORS: OLD

AND NEW

holds for any spherical harmonics, hence for square integrable functions. Ob-

serve that

HL+Ym

l=−iαm+1

sgn

(m)Ym

l=L+HYm

for

m=l

HL+Yl

l=0=L+HYl

and

[H, L−]Ym

l= 0

for any

. As linear combination of

L+

and

L−

, operators

and

commute with

respectively. The assertion that

∆S2

commutes

with

is thus an immediate result of

(1.7)

Besides, there is

⟨[∆S2, A]Ym

l, Y m′

l′⟩=⟨AY m

l,∆S2Ym′

l′⟩−⟨A∆S2Ym

l, Y m′

l′⟩

= [−l′(l′+ 1) + l(l+ 1)]⟨AY m

l, Y m′

l′⟩

(1.26)

Similarly,

⟨[Lz, A]Ym

l, Y m′

l′⟩=−⟨AY m

l, LzYm′

l′⟩−⟨ALzYm

l, Y m′

l′⟩

= (m′−m)⟨AY m

l, Y m′

l′⟩

(1.27)

hence (iii) is veried.

Finally, the left hand side of

(1.24)

is equal to

−∑︂

|m|⩽l

(

sgn

([n−m])

sgn

(m) + 1) f

ˆl,[n−m]f

ˆl,m

while the right hand side is equal to

−

sgn([n])

∑︂

|m|⩽l

(

sgn

([n−m]) +

sgn

(m)) f

ˆl,[n−m]f

ˆl,m

hence identical to each other. Indeed, under our convention if

[n−m]

and

are

of opposite sign, both left and right hand side vanish; if sgn

([n−m]) =

sgn

(m) =

, then sgn

[n] = 1

; if sgn

([n−m]) =

sgn

(m) = −1

, then sgn

[n] = −1

The renormalized Poisson kernel has the expression

Qr(t) = 1

4π

1−r2

(1 −2rt +r2)3/2=∞

∑︂

l=0

2l+ 1

4πrlPl(t)

(1.28)

with

r∈(0,1)

and

t∈[−1,1]

. It has the following approximation property for

continuous functions, which shall be used later. Its proof can be found in [40]

for instance, but for completeness we give a greatly simplied version.

CHAPTER 1. INTRODUCTION

Lemma 1.2.

For any continuous function

on the two sphere, there is

lim

r→1−sup

y∈S2|∫︂S2

Qr(x·y)f(x)dΩ(x)−f(y)|= 0

Proof.

Firstly notice that for any

ϵ > 0

there exists

δ∈(0,1)

such that

|f(x)−

f(y)|< ϵ

whenever

|x·y−1|< δ

∫︂S2

Qr(x·y)dy =∞

∑︂

n=0

2n+ 1

2rn∫︂1

−1

Pn(t)dt = 1

hence

∫︂S2

Qr(x·y)f(y)dy −f(x)

⩽∫︂S2

Qr(x·y)|f(y)−f(x)|dy

⩽2∥f∥C∫︂x·y⩽1−δ

Qr(x·y)dy +ϵ

2∫︂x·y>1−δ

Qr(x·y)dy

⩽∥f∥C∫︂1−δ

−1

1−r2

(1 + r2−2rt)3/2dt +ϵ

2∫︂1

1−δ

1−r2

(1 + r2−2rt)3/2dt

=1−r2

r∥f∥C

√1 + r2−2rt

1−δ

−1

+1−r2

2√1 + r2−2rt

1−δ

→ϵ

approaches

1−

. Due to the arbitrariness of

, the claim follows.

Spherical harmonics in (1.15) have the important property that they form

an orthonormal basis for the Hilbert space

L2(S2, dΩ)

. Therefore

Zl(ξ, η) = ∑︂

|m|≤l

l(ξ)Ym

l(η)

(1.29)

gives a unique reproducing kernel of the space

. This denition does not

depend on the orthonormal system that we choose. In fact, the well known

addition theorem says that

Zl(ξ, η) = 2l+ 1

4πPl(ξ·η).

(1.30)

1.2. SPHERICAL REPRESENTING SYSTEMS AND OPERATORS: OLD

AND NEW

where

are Legendre Polynomials. For curious readers, I refer to [10] or [36]

for its proof. In a general dimension,

can be dened in the same way, with

l=l+λd

|Sd−1|λd

Pλd

(1.31)

where

λd= (d−2)/2

and

Pλd

is the Gegenbauer polynomials. Dene projection

operators

Pl:L2(S2)→ Hl

lf(ξ) := f∗Zd

l(ξ) := ∫︂Sd−1

l(ξ·η)f(η)dΩ(η)

(1.32)

In most of the discussion below we assume

d= 3

and briey denote it by

Every function on

[−1,1]

which is integrable with respect to the weight

function

(1 −t2)d−3

satises the so called Funk-Hecke formula,

∫︂S2

f(ξ·η)Y(η)dΩ(η) = clY(ξ)

(1.33)

with

cl=|Sd−1|∫︁1

−1f(t)(1 −t2)d−3

2Pλd

l(t)

Pλd

l(1) dt

and

Y∈ Hl

. Furthermore, as an

immediate consequence,

∫︂S2

Zl(ξ, ω)Zl(η, ω)dΩ(ω) = Zl(ξ, η)

(1.34)

Let

SO(3)

be the rotation group, consisting of matrices that are orthogonal

and of determinant one. This group or its representation plays an essential role

in the analysis on spheres. For instance, the Fourier transform of

f∈L1(S2)

can be dened alternatively as

𭟋f(γ) = ∫︂SO(3)/SO(2)

f(x)(−x, γ)dµ(x), γ ∈Γ

(1.35)

where

is the dual group of

SO(3)/SO(2)

, annihilator of

SO(2)

Since

SO(3)/SO(2)

is compact,

is discrete. Let

q:SO(3) →SO(3)/SO(2)

be the natural homomorphism, then

f→∫︂SO(3)

f◦q dσ

(1.36)

is a bounded linear functional on

C(SO(3)/SO(2))

, hence induces a unique

measure

µ∈M(SO(3)/SO(2))

with

∥µ∥≤∥σ∥

and

∫︂SO(3)

f◦q dσ =∫︂SO(3)/SO(2)

fdµ

(1.37)

CHAPTER 1. INTRODUCTION

The operator dened by

Sf([x]) = ∫︁SO(2) f(xy)dµ(y)

projects functions in

C(SO(3))

onto

C(SO(3)/SO(2))

Besides, the integral on

SO(3)

can be expressed explicitly in the form

∫︂SO(3)

fdσ =1

8π2∫︂2π

0∫︂π

0∫︂2π

f(φ¯1, θ

¯, φ¯2) sin θ

¯dφ¯1dθ

¯dφ¯2

(1.38)

where (

φ¯1

φ¯2

) is the Euler angle. When

depends only on the rotation angle

it has the expression

∫︂SO(3)

fdσ =2

π∫︂π

f(α) sin2α

2dα

(1.39)

where

α(g1gT

2) = arccos(1

2(Trg1gT

2−1))

is the rotation angle.

What is important to later chapters is the left regular unitary representation

SO(3)

on the Hilbert space

L2(S2)

dened by

(R(σ)f)(ω) = f(σ−1ω)

(1.40)

where

σ∈SO(3)

R(σ)

is called

rotation operator

. [71] is a brilliant reference for

some of its interesting properties and applications in angular momentum theory

of quantum mechanics. [16] is a classic in both group representation theory

and orthogonal polynomials. The rotation operator, together with the spherical

dilation operators that are going to be dened and discussed intensively in the

next chapter, are building blocks for the frame systems in this dissertation.

We have mentioned that for nonnegative integer

{Y−l

l,··· , Y l

form an

orthonormal basis for

. In this situation

is irreducible, for otherwise sup-

pose there is an invariant subspace

H′

, then it is invariant under

L+

L−

and

as well. Take any

h′=

∑︁

m=−l

cmYm

l∈ H′

, let

m′

be the smallest index such that

cm′= 0

. There is

Ll−m′

+h′=cm′αl···αm′+1Yl

l∈ H′

by invariance under

L+

Thus

l∈ H′

and

Ll−m

−Yl

l=αlαl−1···αm+1Ym

l∈ H′

implies that

l∈ H′

for arbitrary

|m|⩽l

, hence

H′=Hl

Similarly we can dene the regular representation of

SO(d)

and denote by

the

Rl,d

its restriction on

Hd,l

. In fact, the next classic theorem generalize

the irreducible property to

Rd,l

SO(d)

. Its proof is quite instructive in

decomposing polynomial spaces aspect, hence I include it here. However, it is

like a cherry on the cake, not every child likes or needs to eat it.

Theorem 1.3.

Rd,l

is an irreducible representation of

SO(d)

1.2. SPHERICAL REPRESENTING SYSTEMS AND OPERATORS: OLD

AND NEW

Proof.

For

σd−1∈SO(d−1)

, there is

Rl,d(σd−1) = ⨁︂

l′⩽lRl′,d−1(σd−1)

(1.41)

Indeed, for

f:= xl−l′

dhl′

∈xl−l′

dHd−1,l′

assume with out loss of generality that

σd−1

leaves

unchanged, then there is

Rl,d(σd−1)f=xl−l′

dhl′

(σ−1

d−1(x1,··· , xd−1))

hence

xl−l′

dHd−1,l′

is an invariant subspace of

Pd,l

, the space of homogeneous

polynomials of degree

. Since

Pd,l =Hd,l ⊕r2Pd,l−2

, there is

Pd,l =

⌞l

2⌟

⨁︂

k=0

r2kHd,l−2k

(1.42)

Let

q:Pd,l ↦→ Pd,l/r2Pd,l−2

be the canonical map and

l,l′=q(xl−l′

dHd−1,l′)

which is invariant under

Rl,d(σd−1)

. By

(1.42)

there is

Pd,l =r2Pd,l−2+xdPd−1,l−1|x′+Pd−1,l|x′

=r2Pd,l−2+xd

⌞l−1

2⌟

∑︂

k=0

(r2−x2

d)kHd−1,l−1−2k+

⌞l

2⌟

∑︂

k=0

(r2−x2

d)kHd−1,l−2k

⊂r2Pd,l−2+

⨁︂

k=0

dHd−1,l−k

where

x′= (x1,··· , xd−1)

, and the converse inclusion is obvious. Thus we get

Pd,l =r2Pd,l−2+

⨁︂

k=0

dHd−1,l−k

(1.43)

hence

q(Pd,l) = Pd,l/r2Pd,l−2=

⨁︁

k=0

l,k

and

(1.41)

follows.

As a consequence is the irreducibility of

Rl,d

of the group

SO(d)

for

d⩾3

Suppose this has been proved for

d⩽n−1

. For

d=n

, notice in

(1.41)

that

the restriction of

Rl,d

SO(d−1)

is the the direct sum of irreducible

Rl′,d−1

l,l′

. Therefore if

is a non-trivial invariant subspace under

Rl,d

, it must

CHAPTER 1. INTRODUCTION

be of the form

W=⨁︁l′∈L Vd

l,l′

for some subset

{0,··· , l}

. Thus it only

remains to prove

L={0,··· , l}

Let

be the smallest index in

and

biggest and assume

l,l′⊂W

Since each rotation can be decomposed as a series of rotations in planes

xj, xk

(j=k)

, innitesimal operators

Lj,k =xk∂f

xj−xj∂f

∂xk

must leave

l,l′

invariant

and choose some

such that

∂hd−1,lb

∂xj= 0

. If neither

nor

are equal to

then

Ljk(xl−l′

dhd−1,l′

(x′)) ∈xl−l′

dHd−1,l′

. If

k=d

for instance(the same to the

situation

j=d

Lj,d(xl−l′

dhd−1,l′

(x′)) = xl−l′+1

∂hd−1,l′

∂xj−(l−l′)xl−l′−1

dxjhd−1,l′

∈xl−l′+1

dHd−1,l′−1+xl−l′−1

dHd−1,l′+1

Thus if

lb< l

, then

Lj,d(xl−lb

dHd−1,lb)∩xl−lb−1

dHd−1,lb+1 =∅

and

xl−lb−1

dHd−1,lb+1 ⊂W

since

is invariant, contradictory to the assumption that

is the biggest

index. Similarly, if

ls>0

, then

∂hd−1,ls

∂xj= 0

implies

xl−ls+1

dHd−1,ls−1⊂W

contradictory to the smallest assumption on

. Therefore we can conclude that

W=Hd,l

On the one hand we have been immersed in the exhilarating success of spher-

ical harmonics which not only expand the

space, but also form rotational

invariant subspaces with hierarchical structure; on the other hand we have to

admit that, the nowadays most widely used planar wavelets, which was origi-

nally based on the idea of Gabor functions, have replaced traditional Fourier

transform in dealing with shock waves in seismology, acoustic or image signals

characterized with singularities, and spherical harmonics face the same awk-

wardness in this aspect due to its global feature. Furthermore, after witnessing

the fast development of wavelets' planar descendants like ridgelets, curvelets,

brushlets, contourlets and cone-adapted shearlets in an attempt to complement

the ineciency of the wavelets in detecting anisotropic structures, it is very

natural to ask for the generalization of those "-lets" adapting to an arbitrary

surface.

Chapter 2

Spherical Dilation Systems

In the past decade emerged multiple methods to dene wavelets on the sphere

or a general manifold, and each of them bears dierent merits. Furthermore,

like those ane-like systems on the plane, whenever possible, it is convenient

to introduce operations like translation and dilation. Not only because through

them a system can be generated and implemented in a simple manner, but also

pertaining to them there are wonderful properties. Thus in this section let us

pay attention to those special wavelet systems equipped with various means of

dilation operation.

2.1 Dilation in frequency domain

One attempt of this, the so called the curvelets on the sphere, is given in [22].

The idea behind this is similar to that of dierential geometry, namely the

surface is divided smoothly into charts for multiple scales and then apply the

curvelets on each of the building block. The scaling function there is proposed

as follows. Let

L= 2J

for some

J∈N

and

ϕL=

∑︂

l=0 ˆ︁

ϕL(l, 0)Yl,0

(2.1)

to form a sequence of functions of multi-scale

ϕL

ϕ2−1L

···

ϕ2−jL

With the low pass lter

ˆ︁hj(l, m) = {︄ˆ︁

ϕ2−j−1L(l,m)

ˆ︁

ϕ2−jL(l,m)l < 2−j−1L

and

m= 0

otherwise (2.2)

CHAPTER 2. SPHERICAL DILATION SYSTEMS

and the high pass lter

ˆ︁gj(l, m) = ⎧

⎪

⎨

⎪

⎩

ˆ︁

ψ2−j−1L(l,m)

ˆ︁

ϕ2−jL(l,m)l < 2−j−1L

and

m= 0

1l⩾2−j−1L

and

m= 0

otherwise

(2.3)

where one of the simplest choice of

ˆ︁

ψ2−jL(l, m) = ˆ︁

ϕ2−j+1L(l, m)−ˆ︁

ϕ2−jL(l, m)

Clearly in this case

ˆ︁gj= 1 −ˆ︁hj

The isotropic wavelet coecients function for a

function

are

wj=ϕ2−jL∗f∗gj

ˆ︁wj=ˆ︁

ϕ2−jLˆ︁

f(1 −ˆ︁hj) = ˆ︁

ψ2−jLˆ︁

(2.4)

One possible choice of

ˆ︁

ϕL(l, m)

2B3(︁2l

L)︁

, the B-spline of order

, but in

general it is not specied except for being bandlimited. However, a class of

choices using radial basis functions can be found in the earlier works in [66][67],

namely if we rstly ignore the bandlimit and simply set the scaling function as

ˆ︁

Φj(l) = γ(2−jl)

, where

γ: [0,∞)→R

monotonously decreasing satises the

following conditions

⎧

⎪

⎨

⎪

⎩

continuous at zero and

γ(0) = 1

∑︁

2l+1

4π(︄sup

x∈[l,l+1) γ(2−jx))︄2

<∞

(2.5)

In particular the rst condition implies

lim

j→∞∥f−f∗Φj∥L2= 0

, namely

Φj

forms an approximate identity. Examples of such functions include

(1 + x)−s

with

x∈[0,∞)

, which is natural for the Sobolev setting, the linear construction

γ(x) = ⎧

⎨

⎩

for

x∈[0, τ)

1−x

1−τ

for

x∈[τ, 1)

for

x∈[1,∞)

(2.6)

and the cubic construction

γ(x) = {︃(1 −x)2(1 + 2x)

for

x∈[0,1)

for

x∈[1,∞)

(2.7)

Wavelets and its dual are chosen in this setting to meet the simple equality

Ψj∗Ψ

˜j= Φj+1 ∗Φj+1 −Φj∗Φj

(2.8)

2.1. DILATION IN FREQUENCY DOMAIN

and consequently there is

Φ0∗Φ0+∞

∑︂

j=0

Ψj∗Ψj= 1

(2.9)

Two typical choices are thus

ˆ︁

Ψj=ˆ︁

˜j=√︂ˆ︁

Φjˆ︁

Φj−ˆ︁

Φj+1 ˆ︁

Φj+1

and

{︄ˆ︁

Ψj=ˆ︁

Φj−ˆ︁

Φj+1

ˆ︁

˜j=ˆ︁

Φj+ˆ︁

Φj+1

(2.10)

The advantage of this setting is that reconstruction of zonal functions follows

immediately from the hierarchical property of radial basis functions without

extra eorts. Indeed, by set

Vj={Φj∗Φj∗f:f∈L2}

where

Φj∗Φj

is the

low-pass and

Wj={Ψj∗Ψ

˜j∗f:f∈L2}

where

Ψj∗Ψ

˜j

is the band-pass, a

natural multi-resolution structure appears, namely

⎧

⎪

⎨

⎪

⎩

V0⊂ ··· ⊂ Vj⊂Vj+1 ⊂ ··· ⊂ L2

∞

⋃︁

j=0

dense in

Φj∗Φj∗f∈Vj,

then

Φj+1 ∗Φj+1 ∗f∈Vj+1

(2.11)

As a result any square-integrable function can be approximated by adding de-

tailed terms from

level-wise. It is desirable that band-pass vanishes for lower

orders, one possible way is to dene

γL0

equal to

[0, L0]

being continuous

x=L0

and set

ˆ︁

ΦL0

j(l) = DL0

jγL0(l) = γL0(L0+ 2−j(l−L0))

(2.12)

which has the properties that

ˆ︁

ΦL0

j(l) = 1

for

l= 0,··· , L0

and

lim

j→∞ ˆ︁

ΦL0

j(l) = 1

for any

. Thus for arbitrarily given

f∈L2

ΨL0

j∗Ψ

˜L0

j∗f

is orthogonal to

for

l⩽L0

. Meanwhile for bandlimited function, say

ˆl= 0

for

l > L

, we could

instead set

ˆ︁

ΦL0

j(l) = γL0(2−jl)

, then the corresponding wavelets have the good

property that

ΨL0

j∗Ψ

˜L0

j∗f= 0

whenever

2−j⩽L0/L

In the continuous setting, however, the admissibility conditions imposed on

CHAPTER 2. SPHERICAL DILATION SYSTEMS

the wavelets are

ˆ︁

Ψρ(0) = 0

and

∫︂∞

0ˆ︁

Ψρ(l)α(ρ)dρ = 1

for

l⩾1

∞

∑︂

2l+ 1

4π(︃∫︂∞

aˆ︁

Ψρ(l)α(ρ)dρ)︃2

<∞

∫︂1

−1∫︂∞

Ψρ(t)α(ρ)dρdt < T

(2.13)

where

α(ρ)dρ

is an arbitrary positive measure. Under the rst condition there

∫︂∞

a∫︂S2

Ψρ(ξ·η)f(η)dηα(ρ)dρ =∑︂

lˆ︁

Φa(l)Pl(f)→f

a→0

in the sense of

, where

Φa(t) = ∫︁∞

aΨρ(t)α(ρ)dρ

or equivalently

ˆ︁

Φa(l) =

∫︁∞

aˆ︁

Ψρ(l)α(ρ)dρ

is well dened in

since the second condition of

(2.13)

im-

plies that

Φa∈L2([−1,1])

. The convergence holds because of the uniform

boundedness

ˆ︁

Φa(l)⩽∫︁1

−1∫︁∞

aΨρ(t)α(ρ)dρdt < T

and the Banach-Steinhaus

theorem.

To deal with the anisotropic situation, one strategy is to add an additional

directional function

h∈L2(S2)

with

∑︁|m|≤l|hlm|2= 0

to form the directional

wavelet

ˆlm =ˆ︁

Ψ(l)hlm

(2.14)

is bandlimited, one can assume without loss of generality that

preserves

the energy on each degree

, namely

∑︁|m|≤l|hlm|2= 1

, hence

∥ψ∥2

2=∑︁∞

0ˆ︁

Ψ(l)2

The dilation operation in this formulation is dened by

ˆ︂

D2(a)ψlm =ˆ︁

Ψ(al)hlm

(2.15)

By choosing a

such that

suppˆ︁

Ψ⊂(a, a−1)

with

a∈(0,1)

and

the

smallest integer such that

aJL⩽1

, one can dene in the same way as in

(2.10)

that

ˆ︁

Ψj(L−1l) = ˆ︁

Ψ(a−jL−1l) = √︂γ2

a(a−(j−1)L−1l)−γ2

a(a−jL−1l)

(2.16)

with

γ2

a(t) = ∫︁1

tr2

a(u)du

∫︁1

ar2

a(u)du

and

ra(t) = r(︂2

1−a(t−a)−1)︂

for some Schwartz func-

tion

[−1,1]

2.1. DILATION IN FREQUENCY DOMAIN

Note that

supp(ra)⊂[a, 1]

and

γa(t)=1

when

t⩽a

, hence there is

suppγa(a−jL−1l)⊂(−∞, ajL]

and

ˆ︁

ψj

lm =ˆ︁

Ψj(l)hlm

is supported in

[aj+1L, aj−1L]

, which together with the

scaling function

ˆ︁φlm =γa(a−JL−1l)δm,0

(2.17)

give the reconstruction formula for each

within the bandlimit of

that

ˆ︁φ2

l0+

∑︂

j=0 ∑︂

|m|⩽lˆ︁

ψ2

lm(a−jL−1l)=1

(2.18)

in a similar manner as

(2.9)

One choice for the directional function

, as proposed in [41], is based on the

concept of steerability. The class of steerable functions on

under the steering

constrain that

fθ(r, ϕ) =

∑︂

m=1

km(θ)gm(r, ϕ)

(2.19)

for some

M∈N

and basis functions

were discussed rstly in [69]. In [9]

this denition was applied to spherical functions with respect to the third Euler

angle, a quantity that can be understood as the direction in the tangent plane.

Let us, however, generalize the denition in [41] slightly and prove a stronger

result on the sphere. Denote by

R(t)

the rotation operation around axis

ξ0

an angle

, we call a function

f∈L2(S2)

steerable

, if there exists some

tm∈S1

m= 1,··· , M

such that

R(t)f=

∑︂

m=1

km(t)R(tm)f

(2.20)

holds for almost every

Proposition 2.1.

Steerability is equivalent to the existence of an azimuthal

band limit in

for

L2(S2)

functions.

Proof.

In fact, if a function

f(θ, φ) = ∞

∑︁

n=−∞

an(θ)einφ ∈L2(S2)

satises

(2.20)

for some

and almost every

t∈[0,2π]

, then

∞

∑︂

n=−∞

einφ [︄an(θ)(︄e−int −

∑︂

m=1

km(t)e−intm)︄]︄= 0 a.e.

(2.21)

CHAPTER 2. SPHERICAL DILATION SYSTEMS

Since Fourier series forms a Schauder basis for

L2(S1)

, for any

an(θ)= 0

, there

e−int −

∑︁

m=1

km(t)e−intm= 0 a.e.

Without loss of generality, assume for some

L > M

there is

⎛

⎜

⎝

e−i(t1−t)e−i(t2−t)··· e−i(tM−t)

e−i2(t1−t)e−i2(t2−t)··· e−i2(tM−t)

e−iL(t1−t)e−iL(t2−t)··· e−iL(tM−t)

⎞

⎟

⎠⎛

⎜

⎝

k1(t)

k2(t)

kM(t)

⎞

⎟

⎠=⎛

⎜

⎝

⎞

⎟

⎠

(2.22)

or simply

Gtkt=E

for almost every

, which implies that rank

Gt≤M a.e.

contradictory to the linear independence of set

{eijt :j= 1,··· , L}

. Therefore

the number of non-zero Fourier coecients of

with respect to variable

is at

most

. The other direction of the proof of the equivalence is an immediate

consequence of the next Lemma.

Lemma 2.2.

f∈L2(S2)

and

ˆlm = 0

for any

|m|⩾L+ 1 ∈N0

, then for

any

t∈[0,2π]

∑︂

n=−L

k(t−tn)R(tn)f=R(t)f

(2.23)

with

tn=2πn

2L+1

and

k∈L2(S1)

given by

k(t) =

∑︁

n=−L

2L+1 eint

Proof.

Since

2L+ 1 ∑︂

|n|⩽L

e−imtn={︃1

for

m= 0

for

m= 0

(2.24)

the Fourier coecients of both sides of (2.23) coincide, namely

ˆ︂

(R(t)f)lm′=eim′tf

ˆlm′

2L+ 1 ∑︂

|m|⩽L∑︂

|n|⩽L

ei(m′−n)tmeintf

ˆlm′

2L+ 1 ∑︂

|m|⩽L∑︂

|n|⩽L

ein(t−tm)ˆ︂

(R(tm)f)lm′

=∑︂

|m|⩽L

k(t−tm)ˆ︂

(R(tm)f)lm′

Therefore the equality (2.23) holds.

2.1. DILATION IN FREQUENCY DOMAIN

Under the assumption that

is steerable, namely

R(t)h=∑︁

|m′|⩽L

km′(t)R(tm′)h

for some nite

, there is

R(t)ψ=∑︂

lˆ︁

Ψ(l)R(t)Plh

=∑︂

|m′|⩽L

km′(t)∑︂

lˆ︁

Ψ(l)Pl(R(tm′)h)

=∑︂

|m′|⩽L

km′(t)R(tm′)ψ

(2.25)

namely

is steerable too. In this case there is

⟨R(t)ψj,R(t′)ψj⟩L2=∑︂

l∑︂

|m′|⩽min{l,L}

e−im′(t−t′)ψ

ˆj

lm′

=∑︂

∆l(t−t′)Ψ

ˆj(l)

(2.26)

where

∆l(t) = ∑︁

|m′|⩽L|hlm′|2e−im′(t)

. Intuitively speaking, if we choose

0 = t1<

t2<··· < tN<2π

and form a

N×N

matrix

M=⟨R(ts)ψj,R(tu)ψj⟩L2

(2.27)

with

1⩽s, u ⩽N

, the diagonal elements are 1's, while the smaller the o

diagonal elements are, the more directional

ψj

is.

Within this approach, the best known work in frame properties aspect is

probably [30], where a localized tight frame is constructed in the following del-

icate way. By taking a continuous function

supported in

2,2]

, for instance

a(t) = m0(πlog2(t))

with

the standard orthogonal wavelet mask on real line,

such that

|a(t)|2+|a(2t)|2= 1

2,1]

(2.28)

We have obviously for any

J∈Z+

that

b(2−Jt) :=

∑︂

j=−∞a(2−jt)2

={︄a(2−jt)2+a(2−j+1t)2= 1

for

t∈2j[1

2,1], j ⩽J

a(2−Jt)2

for

t∈2J+1[1

2,1]

(2.29)

CHAPTER 2. SPHERICAL DILATION SYSTEMS

In particular, when

J→ ∞

∞

∑︁

j=−∞a(2−jt)2= 1

(0,∞)

. Thus, if we dene

for

j⩽J

that

j(t) = ⎧

⎪

⎨

⎪

⎩

∞

∑︁

l=1

a(2−jl) cos(larccos t)

for

d= 2

∞

∑︁

l=0

a(2−(j+jd)(l+λd))Zd

l(t)

for

d⩾3

(2.30)

and

J(t) = ⎧

⎪

⎨

⎪

⎩

2π+1

∞

∑︁

l=1

b(2−Jl) cos(larccos t)

for

d= 2

∞

∑︁

l=0

b(2−(J+jd)(l+λd))Zd

l(t)

for

d⩾3

(2.31)

where

jd= log2[λd]

for

d⩾3

and

j1= 0

are so that the minimal eigenvalue of

the operator

Ln=√︁λd−∆Sd−1=∞

∑︂

l=0

(l+λd)Pd

(2.32)

lies in interval

[1,2]

, then for any

f∈L2(Sd−1)

∑︂

j=−∞

j∗Ad

j∗f={︃(Bd

J−P0)∗f

for

d= 2

J∗f

for

d⩾3

(2.33)

and by Funk-Hecke formula and the support of

⟨Ad

j∗Ad

j′∗f, f⟩=⟨∑︂

a(2−(j+jd)(l+λd))a(2−(j′+jd)(l+λd))Pd

lf, f⟩

= 0

(2.34)

for any

|j−j′|⩾2

d⩾3

Suppose

Sd−1

has a subdivision into

{Ωi}i∈I

labeled by a set of points

{pi}i∈I

. For a mesh with good uniformity, namely the

mesh ratio

ρV=hV

sup

x∈Sd−1

inf

pi∈V d(x, pi)

2min

pi=pj

d(pi, pj)

(2.35)

is larger than

, where

is called

mesh norm

and

called

separation radius

respectively, there exists a nested sequence

Vk⊂ Vk+1

such that

4hVk< hVk+1 <1

2hVk

(2.36)

2.1. DILATION IN FREQUENCY DOMAIN

It is proved in [30] as main theorems, here we cite as lemmas without proof,

that

Lemma 2.3.

There exist constants

depending solely on dimension

such

that, for any

δ∈(0,1

and integer

L∈Z+

, whenever

∥V∥ = max

p∈V

diam

(Ωi)⩽

δs−1

d(L+λn)−1

(1 −δ)∥f∥1⩽∑︂

p∈V |f(p)||Ωp|⩽(1 + δ)∥f∥1

(2.37)

holds for spherical harmonics in

Πd

L=⨁︁

l⩽LHd

; there exist positive weights

w(p)

such that

∫︂Sd−1

f(η)dη =∑︂

p∈V

w(p)f(p)

(2.38)

and

w(p)

have the following bounds

1−2δ

1−δ|Ωp|⩽w(p)⩽|Sd−1|

dim

(Πd

[L/2])

(2.39)

Lemma 2.4.

For

a∈Ck(R)

with

k > max{d−1,2}

, if

f∈Lq(Sd−1)

with

1⩽q⩽∞

, then there exists constant

Cb,k,d

such that

∥f−Bd

J∗f∥q⩽Cb,k,d

dist

Lq(f, Πd

(2.40)

As a result, on

j−

th level mesh

, by dening

ψj,p(η) = √wj,pAd

j(η·p)

(2.41)

with

a∈Ck

, we have for

f∈C(Sd−1)

f∈Lq(Sd−1)

with

1⩽q < ∞

f=∞

∑︂

j=0 ∑︂

p∈Vj⟨f, ψj,p⟩ψj,p

(2.42)

with convergence in the corresponding space norms. For

f∈L2

, it is equivalent

∥f∥2=⎧

⎪

⎨

⎪

⎩

2π|⟨f, 1⟩|2+∞

∑︁

j=0 ∑︁

p∈Vj|⟨f, ψj,p⟩|2

for

d= 2

∞

∑︁

j=0 ∑︁

p∈Vj|⟨f, ψj,p⟩|2

for

d⩾3

(2.43)

CHAPTER 2. SPHERICAL DILATION SYSTEMS

namely

{ψj,p}

forms a tight frame. Meanwhile, when

2−j< ϵ

, there is the

integral expression of

ϵ(t) = ⎧

⎪

⎨

⎪

⎩

∞

∑︁

l=1

a(ϵ) cos(larccos t)

for

d= 2

∞

∑︁

l=0

a(ϵ(l+λd))Zd

l(t)

for

d⩾3

that

ϵ(cos θ) = γd

sind−1θ∫︂π

Cϵ,d(ϕ)

(cos θ−cos ϕ)dϕ

(2.44)

where

γd=2λdΓ(λd+1/2)

√π|Sd−1|Γ(λd)Γ(2λd)

and

Cϵ,d(ϕ) = 1

2ϵ∑︂

n∈Z

(−1)(d−2)nQ

˜d−2(id

dϕ)a

ˆ(ϕ+wπn

ϵ)

(2.45)

with

˜d−2(z) = ⎧

⎪

⎨

⎪

⎩

[d−2

∏︁

j=1 (︁z2−(λd−j)2)︁zsin(λdπ)

even

[d−2

∏︁

j=1 (︁z2−(λd−j)2)︁cos(λdπ)

odd

(2.46)

It leads to an estimation for

a∈Ck(R)

that

|Ad

ϵ(cos θ)|⩽βd,k,aϵ−d+1

1+(θ

ϵ)k

(2.47)

with some constants

βd,k,a

. Thus we see that when

and

increase,

|ψj,p|

decrease in scale value as

2−(j+jd)k

, hence is localized.

2.2 Dilation through stereographic projection

In contrast to doing dilation in frequency domain, this section we discuss dilation

through geometric approach. As we have encountered at the beginning of the

chapter and we shall see in the review part later, one common strategy is to

wisely cover the whole manifold by local patches, and then construct a dictionary

as it was done on the plane followed by lifting it back. This method has the

advantage that it can be applied to any manifolds and it has localized nature.

2.2. DILATION THROUGH STEREOGRAPHIC PROJECTION

On the sphere there is much more to say than on a general manifold, not only

because the stereographic projection is obviously ideal for building a one-to-one

correspondence with the plane, but also that the Euclidean sphere has exquisite

group structures that has been introduced above. Stereographic map is confor-

mal and can be used to generate the whole system globally, while by choosing

localized generators it allows us to construct frame systems with ideal local prop-

erties. Although at the pole it may cause distortion it avoids the consistency

problem arising from dierent patches, hence much easier to implement in my

opinion. The rst work in this approach probably dates back to [92][93], where

admissibility condition is proposed without successfully formulating frames.

In this section, however, dilation is done in both isotropic and parabolic

manner. Especially the second brings much dierence when one deals with

anisotropic problems, exactly for this reason I shall call the corresponding gen-

erators "spherical shearlets" or "spherical

-wavelets". A long lasting unsolved

problem in the stereographic projection approach before the writing of this dis-

sertation is the lack of a constructible tight frame. Discussion about that is

going to be delayed till the next chapter, where exact design of a class of tight

frames is given.

By identifying a 2-sphere with the homogeneous space

SO(3)/SO(2)

, the

sphere is embedded into the rotation group and group operations can be applied

naturally. Therefore the translation on the sphere can be achieved by rotation.

However, in comparison with translation in the Euclidean plane, it obviously

brings much more complicated work since parameters are located in the rotation

group instead of

. Let

dΩ = sin θdθdφ

be the rotation invariant measure on the

unit sphere, and

be the Lebesgue measure on

. Denote by

an element

in the group

SO(3)

and by

dσ

the Haar measure on it. The Haar measure is

normalized so that the whole group

SO(3)

has volume one. Recall that the left

regular unitary representation dened by

(R(σ)f)(ω) = f(σ−1ω)

where

σ∈SO(3)

. Related to the rotation operator

R(σ)

there is the Wigner

D-matrix which consists of coecients with respect to the normalized orthogo-

nal spherical harmonics. Peter-Weyl Theorem tells us that these coecients are

dense in

L2(S2)

and

can be written as the direct sum of nite-dimensional irre-

ducible representations.

L2(S2)

is correspondingly decomposed into

-invariant

vector subspaces, those are exactly the eigenspaces

corresponding to the dif-

ferent eigenvalues of the Laplace-Beltrami operator on the sphere, as we have

seen in the introduction.

CHAPTER 2. SPHERICAL DILATION SYSTEMS

Schulr's orthogonality relations in this case are

∫︂SO(3)

mn(σ)Dl′

m′n′(σ)dσ =1

δll′δmm′δnn′

(2.48)

where indexes

and

m′

n′

are integers of absolute value no larger than

and

l′

respectively.

Let

ω= (θ, φ)

denote the polar coordinate of a point on the unit sphere.

In particular,

θ=π

is the north pole

ξ0

. The stereographic projection

π:

S2\{ξ0} ↦→ R2

, by

π(θ, φ)=(rcos φ, r sin φ)

(2.49)

with

r= 2 tan(θ

, gives an isomorphism, and its inverse is denoted by

π−1

One strategy of doing dilation on the sphere is to utilize the dilation operator

d(a)

L2(R2, dx)

dened as

(d(a)h)(x, y) = a−(1+α)/2h(a−1x, a−αy)

(2.50)

with

α∈[0,1]

Denition 2.5.

Given

a∈R+

and

f∈L2(S2, dΩ)

, dene

D(a) : L2(S2, dΩ) ↦→

L2(S2, dΩ)

D(a)f(ω) = {︃U−1d(a)Uf(ω)ω=ξ0

a−1−α

2f(ξ0)ω=ξ0

where

is the operator such that

(Uf)(x) = ν(π−1x)f(π−1x)

with

ν(θ, φ) = cos2(θ/2)

guaranteeing the unitarity of the operator.

Let us make the convention that throughout this chapter

is chosen as a

positive number no larger than

. In the special case

α=1

D(a)

will be called

parabolic spherical dilation operator

A simple calculation shows that

ω= (θ, φ)

and the point after dilation

ω1

a=d

ˆ(a)(θ, φ) = (θ1

a, φ1

, are related by

⎧

⎪

⎨

⎪

⎩

tan(θ1

a/2) = √︁γ(a, φ) tan(θ/2)

tan φ1

a=a1−αtan φ

when

φ=π

and

φ=3π

φ1

a=φ

when

φ=π

3π

(2.51)

where

√︁γ(a, φ)=(a−2cos2φ+a−2αsin2φ)1/2

2.2. DILATION THROUGH STEREOGRAPHIC PROJECTION

Let

θ′=θ1/a

φ′=φ1/a

. By (2.51), it is easy to see the following relations

that are going to facilitate our mathematical deduction.

sin2φ=a2α−2tan2φ′

1 + a2α−2tan2φ′,cos2φ=1

1 + a2α−2tan2φ′

sin θ=2 tan(θ′

2) cos φ′√︁a2+a2αtan2φ′

1 + tan2(θ′

2) cos2φ′(a2+a2αtan2φ′)

cos θ=1−tan2(θ′

2) cos2φ′(a2+a2αtan2φ′)

1 + tan2(θ′

2) cos2φ′(a2+a2αtan2φ′)

(2.52)

With these, we arrive at an explicit formulation of the spherical dilation

operator.

Proposition 2.6.

For

f∈L2(S2, dΩ)

∥D(a)f∥2=∥f∥2

(2.53)

(D(a)f)(ω) = {︄√︁λ(a, θ, φ)f(ω1

a)ω=ξ0

a−1−α

2f(ξ0)ω=ξ0

(2.54)

where

√︁λ(a, θ, φ) = 2a3−α

Φ−

acos θ+ Φ+

=a−(1+α)/2(1 + J2) cos2(θ′

(2.55)

Φ±

a=a2[1 ±γ(a, φ)]

and

J= tan θ

2= tan(θ′

2) cos φ′√︁(a2+a2αtan2φ′)

Proof.

∥D(a)f∥2=∥f∥2

comes from the fact that both

d(a)

and

are unitary

operators. From the denition (2.5) it follows immediately that

U−1g(θ, φ) = 1

ν(θ, φ)g(π(θ, φ))

(2.56)

and from (2.49) that

π−1(x1, x2) = (2 arctan(√︁x2

1+x2

2),arctan(x2

))

(2.57)

CHAPTER 2. SPHERICAL DILATION SYSTEMS

for any

g∈L2(S2, dΩ)

and

x= (x1, x2)

on the plane. Therefore by the deni-

tions (2.50) and (2.5) for

θ=π

D(a)f(θ, φ)=[U−1d(a)Uf](θ, φ)

ν(θ, φ)(d(a)Uf)(π(θ, φ))

=a−(1+α)/2

ν(θ, φ)ν(π−1(2 tan θ

2cos φ

a,2 tan θ

2sin φ

aα))f(ω1/a)

=a−(1+α)/2

cos2θ

cos2(arctan[(cos2φ

a2+sin2φ

a2α)1/2tan θ

2])f(ω1/a)

=2a(3−α)/2f(ω1/a)

(a2−a2−2αsin2φ−cos2φ) cos θ+a2+a2−2αsin2φ+ cos2φ

which is exactly the expression

(2.54)

Meanwhile, from

(2.52)

we deduce that

Φ±

a=a2±1

cos2φ′(1 + a2α−2tan2φ′)

hence

Φ−

acos θ+ Φ+

a=a2(cos θ+ 1) + a2tan2(θ′

2)(1 −cos θ)

=2a2

1 + J2+2a2tan2(θ′

1 + J2

=2a2sec2(θ′

1 + J2

and

(2.55)

follows.

Remark 2.7.

The coecient

λ(a, θ, φ)

can be alternatively dened as the

Radon-Nikodym derivative

dΩ(ω1/a)

dΩ(ω)

, and it is easy to check these two dier-

ent ways of denition give the same result. In this sense,

can be interpreted

as the change of measure caused by dilation operation.

With the rotation operator as well as the dilation operator, the continuous

spherical wavelets on the sphere is generated by a single function.

2.3. SPACES OF ADMISSIBLE

-WAVELETS/SHEARLETS

Denition 2.8.

For

ψ∈L2(S2, dΩ)

, dene

spherical

-wavelet system

{ψσ,a(ω) = R(σ)D(a)ψ(ω) = √︁λ(a, θ, φ)ψ((σ−1ω)1/a) : σ∈SO(3), a ∈R+}

in particular when

α= 1

, let us call it a continuous

spherical shearlet system

Dene

spherical

-wavelets/shearlet transform

Sf(σ, a) = ⟨f, ψσ,a⟩.

It is natural to ask if we can exchange the order of the rotation and the

dilation as dened above, namely if

D(a)

is a rotation commutative operator,

unfortunately the answer is no in general. In fact, there are the following simple

counter examples.

Example 2.9.

(i) Let

[D(a),R(σ)] = D(a)R(σ)−R(σ)D(a)

σ3(φ) = ⎛

⎝cos φsin φ0

−sin φcos φ0

0 0 1 ⎞

⎠

α < 1

in (2.51), then as

a→0

[D(a),R(σ3(φ))]f(θ0, φ0)−→ f(π, φ)−f(π, 0)

f(π, π +φ)−f(π, π)

(ii)

σ2(θ) = ⎛

⎝cos θ0 sin θ

0 1 0

−sin θ0 cos θ⎞

⎠

α < 1

, then as

a→0

[D(a),R(σ2(θ))]f(θ0, φ0)−→ f(π+θ, 0) −f(π, 0)

f(π+θ, π)−f(π, π)

, depending on whether

tan φ⩾0

tan φ < 0

2.3 Spaces of admissible

-wavelets/shearlets

Like the Fourier basis in

, spherical harmonics have the draw back that they

are not sensitive to local behavior in the spatial domain, or more precisely, a

perturbation of the function value at a point may lead to the change of all

coecients and we are forced to do integration on whole sphere. Consequently

they are insucient in representing functions of high frequency. This motivates

us to construct localized generators.

CHAPTER 2. SPHERICAL DILATION SYSTEMS

Denition 2.10.

A spherical

-wavelet

admissible

, if

0< Cl

ψ=1

2l+ 1 ∑︂

|m|≤l∫︂∞

a3|(ˆ︁

ψa)lm|2<∞

where

(ˆ︁

ψa)lm

is the Fourier coecients of the dilated function

ψa=D(a)ψ

with

respect to spherical harmonics. Denote by

the set of admissible functions.

Without loss of generality we use spherical shearlet for the development of

the theory in the rest of this section.

Proposition 2.11.

When a spherical shearlet is admissible, the following re-

construction formula holds

∫︂∞

a3∫︂SO(3) Sf(σ, a)ψσ,a(ω)dσ =∑︂

l∑︂

|m|≤l

ψˆ︃

flmYm

l(ω).

Proof.

By denition

mn(σ) = ⟨R(σ)Ym

l, Y n

l⟩

(2.58)

Since

are invariant subspaces under

R(σ)

, it follows immediately that

ˆ︂

[R(σ)f]lm =∑︂

|n|≤l

mn(σ)ˆ︁

fln

(2.59)

Therefore (2.48) and (2.59) together give us

∫︂∞

a3∫︂SO(3)⟨f, R(σ)ψa⟩[R(σ)ψa](ω)

=∫︂∞

a3∫︂SO(3)⟨∑︂

l∑︂

|m|≤lˆ︃

flmYm

l,R(σ)ψa⟩∑︂

l′∑︂

|n|≤l′

ˆ︂

[R(σ)ψa]l′,nYn

l′(ω)

=∫︂∞

a3∫︂SO(3) ∑︂

l,l′∑︂

|m|≤lˆ︃

flm ∑︂

|n|≤l

mn(σ)ˆ︂

(ψa)ln ∑︂

|m′|≤l′∑︂

|n′|≤l′

Dl′

m′n′(σ)ˆ︃

(ψa)l′n′Ym′

l′(ω)

=∑︂

2l+ 1 ∫︂∞

a3∑︂

|m|≤l∑︂

|n|≤l|ˆ︃

(ψa)ln|2ˆ︃

flmYm

l(ω)

=∑︂

l∑︂

|m|≤l

ψˆ︃

flmYm

l(ω)

2.3. SPACES OF ADMISSIBLE

-WAVELETS/SHEARLETS

Thus the reconstruction formula holds when

in Denition 2.10 has nite

value for every

In particular, if

is positive and independent of

, identity in Proposition

2.11 becomes

f(ω) = C∫︂∞

a3∫︂SO(3) Sf(σ, a)ψσ,a(ω)dσ

(2.60)

for some constant

C > 0

. The question concerning the existence of such con-

struction arises, but an armative answer comes after much eorts.

Furthermore, the observation

∑︂

|m|≤l|(ˆ︁

ψa)lm|2

2=∥Plψa∥2

2⩽∥ψa∥2

2=∥ψ∥2

2<∞

(2.61)

and

∑︂

l∥Plψ∥2

2=∥ψ∥2

(2.62)

show that

ψ= 0

for all

only if

vanishes identically, and that under the

assumption

ψ=1

2l+ 1 ∫︂∞

∥Plψa∥2

a3da < ∞

the space of admissible functions is closed under certain algebraic rules.

Lemma 2.12.

For

ψ∈L2(S2)

, the two conditions:

(i)ψ∈ A

(ii)∥Plψa∥2=o(a)

a→0

for

l∈Z+

are equivalent. In particular, it indicates that

(i′)f1

f2∈ A ⇒ c1f1+c2f2∈ A

for any pair

1+c2

2>0

(ii′)

p+1

q= 1

and

p, q > 0

f2∈A⇒f

2∈ A

(iii′){fn}⊂A

fn⇒f⇒f∈ A

Let us dene some important function spaces that will appear frequently

through out this section and later.

Denition 2.13.

Let

be the subset of square integrable functions with the

property that

lim

θ→π|f(θ, φ)·tannθ

exists and being bounded. Dene

N={f∈L2(S2) : ∫︂2π

0∫︂π

f(θ, φ) tan(θ

2)dθdφ = 0}

CHAPTER 2. SPHERICAL DILATION SYSTEMS

and for

ξ∈S2

, let

KΘ(ξ) = {f∈L∞(S2) : f(ξ·η)=0 a.e.

for

ξ·η⩾cos Θ}.

In particular,

is called a

hollow pole function

, if

f∈ K(ξ0) = ⋃︁ΘKΘ(ξ0)

By denition it is clear that

KΘ⊂ Bn

, hence we have the inclusion relation

Bn⊃ K

. Furthermore, there is

K=Bn

, where the closure can be taken both in

L∞

and

norm.

Let us take the following formula from [60]

Pl(cos θ) = (︃θ

sin θ)︃1

J0(︃(l+1

2)θ)︃

+{︃θ1/2O(l−3

c/l ⩽θ⩽π−ϵ

θ2O(1)

0< θ ⩽c/l

(2.63)

where

Jν

are the Bessel functions

(4.52)

, but here we need its integral form

Jν(z) = (z/2)ν

Γ(ν+1

2)Γ(1

2)∫︂1

−1

(1 −t2)ν−1

2eiztdt

(2.64)

Taking derivative leads to the following asymptotic result.

Lemma 2.14.

P(m)

l(cos θ) = Γ(l+m+ 1)

(l+1

2)m(l−m)! sinmθ(︃θ

sin θ)︃1

Jm(︃(l+1

2)θ)︃

+{︃θ1/2O(l−3

c/l ⩽θ⩽π−ϵ

θm+2O(lm)

0< θ ⩽c/l

With the preparation above we are standing at the point to prove the main

result of this section. It indicates how to construct a shearlet system meeting

the admissibility condition. Without loss of generality let us take

α=1

, the

parabolic case, for the simplicity of the proof.

Theorem 2.15.

Bn∩N =Bn∩A

for

n⩾3

Proof.

We rstly prove that

K ∩ N =K ∩ A

. Set

θ′=θ1/a

φ′=φ1/a

dΩ′= sin θ′dθ′dφ′

and suppose

|ψ(θ, φ)·tan θ

2|⩽M a.e.

. By Lemma 2.12, in

order to prove mutual inclusion we only have to check that

ψ∈ N

if and only

∥Plψa∥2

2=∑︁|m|≤l|(ˆ︂

ψa)lm|2=o(a2)

2.3. SPACES OF ADMISSIBLE

-WAVELETS/SHEARLETS

By (1.15),

l(θ, φ) = cl,meimφ sinm(θ)P(m)

l(cos θ)

with

cl,m = (−1)m√︂(2l+1)(l−m)!

4π(l+m)!

Consider

ψΘ=χ[0,2π]×[0,Θ]ψ∈ K

. Using the fact that

λ(a, θ, φ) = dΩ′

dΩ

, we

get

(ˆ︂

(ψΘ)a)lm =∫︂S2

l(ω)(D(a)ψΘ)(ω)dΩ

=∫︂S2

l(θ, φ)√︁λ(a, θ, φ)ψΘ(θ1/a, φ1/a)dΩ

=∫︂S2

l(θ, φ)λ−1/2(a, θ, φ)ψΘ(θ′, φ′)dΩ′

(2.65)

To avoid confusion of the notation in (2.65) and also in later part, we clarify here

that

φ(θ′, φ′)

are now taken as new variables while

and

shall be understood

θ(θ′, φ′)

and

φ(θ′, φ′)

Insert (2.51) into (2.65) and replace

√︁λ(a, θ, φ)

by the expression

a−1+α

2(1+

J2) cos2(θ′

(2.55)

, we get

(ˆ︂

(ψΘ)a)lm = 2a1+α

2cl,m ∫︂2π

0∫︂π

e−imφ sinm(θ)P(m)

l(cos θ)ψΘ(θ′, φ′) tan(θ′

2)dθ′dφ′

1 + J2

(2.66)

where

J= tan(θ′

2) cos φ′√︁(a2+a2αtan2φ′)

Let us examine the integrand in (2.66) in detail. Since

θ′∈(0,Θ)

tan θ′

2⩽

tan Θ

bounded almost everywhere. What's more, Lemma 2.14 and the obser-

vation that

θ→0

a→0

tell us

sinmθP (m)

l(cos θ) = O(aαml2m)

. Indeed,

according to (2.52) there is

|sin θ|=|2 tan(θ′

2) cos φ′√︁a2+a2αtan2φ′

1 + tan2(θ′

2) cos2φ′(a2+a2αtan2φ′)|

=|2 tan(θ′

2) cos φ′√︂a2+a2αtan2φ′|+O(a3α)

=O(aα)

we conclude that for xed

l(θ, φ) = O(aα|m|+(1+α)/2)

(2.67)

CHAPTER 2. SPHERICAL DILATION SYSTEMS

for all

m⩾0

, and hence for all

|m| ≤ l

due to the fact that

Y−m

l= (−1)mYm

(2.68)

Thus

|(ˆ︂

(ψΘ)a)lm|=O(aαm+(1+α)/2)

(ˆ︂

(ψΘ)a)lmYm

l∈ A

for all

|m|⩾1

, and

∥Pl(ψΘ)a∥2

2=∑︁|m|≤l|(ˆ︂

(ψΘ)a)lm|2=o(a2)

if and only if we impose the require-

ment that

0 = lim

a→0a−1(ˆ︂

(ψΘ)a)l0

= 2 lim

a→0a−1/4cl,0∫︂2π

0∫︂Θ

Pl(cos θ)ψ(θ′, φ′) tan(θ′

1 + J2dθ′dφ′

= 2 lim

a→0a−1/4cl,0∫︂2π

0∫︂Θ

ψ(θ′, φ′) tan(θ′

2)dθ′dφ′

namely that

∫︁2π

0∫︁Θ

0ψ(θ′, φ′) tan(θ′

2)dθ′dφ′= 0

. The last step used Lebesgue

dominated Theorem with the observation that

|Pl(cos θ)|⩽Pl(1) = 1

J→0

a→0

. Although we used a specially designed

ψΘ

, the whole argument we

used above applies to any function in

, namely

K∩A =K∩N

Due to our assumption that

ψ∈ Bn

|ψ(θ′, φ′) tan3(θ′

2)|

is bounded by some

M > 0

almost everywhere whenever

|Θ−π|< δ

, we see that if

∥Pl(ψ)a∥2

limΘ→π∥Pl(ψΘ)a∥2

2=o(a2)

, we necessarily have

0 = lim

Θ→π∫︂2π

0∫︂Θ

ψ(θ′, φ′) tan(θ′

2)dθ′dφ′=∫︂2π

0∫︂π

ψ(θ′, φ′) tan(θ′

2)dθ′dφ′

Conversely, assume

∫︁2π

0∫︁π

0ψ(θ, φ) tan(θ

2)dθdφ = 0

. Let

˜︂

ψΘ(θ, φ) = {︃ψΘ(θ, φ)−TψΘ(θ, φ)

for

θ≤Θ

for

θ > Θ

(2.69)

where

TψΘ(θ, φ) = (tan θ

2)−1∫︂2π

0∫︂π

ψΘ(θ, φ) tan(θ

2)dθdφ

(2.70)

Since

˜︂

ψΘ(θ, φ)∈ KΘ∩N

, by our former conclusion we get

˜︂

ψΘ(θ, φ)∈ A

. Hence

it follows from Lemma 2.12 that

∥Pl(˜︂

ψΘ)a∥2

2=o(a2)

. Fix

θ < π

, it's easy to

see that

˜︂

ψΘ(θ, φ)→ψ(θ, φ)

Θ→π

, and that

|(˜︂

ψΘ(θ, φ)−ψΘ(θ, φ)) tan3θ

2.3. SPACES OF ADMISSIBLE

-WAVELETS/SHEARLETS

is bounded for any

Θ< π

. In fact, there is

|˜︂

ψΘ(θ, φ)−ψ(θ, φ)|tan3(θ

⩽(︂|˜︂

ψΘ(θ, φ)−ψΘ(θ, φ)|+|ψΘ(θ, φ)−ψ(θ, φ)|)︂tan3(θ

⩽∫︂2π

0∫︂π

(ψΘ(θ, φ)−ψ(θ, φ)) tan(θ

2)dθdφtan2(θ

+ max

θ>Θψ(θ, φ) tan3(θ

2)

⩽tan2(︃θ

2)︃∫︂2π

0∫︂π

Θψ(θ, φ) tan(θ

2)dθdφ + max

θ>Θψ(θ, φ) tan3(θ

2)

which is bounded almost everywhere as

approaches

, since

ψ∈ B

. By

replacing

˜︂

ψΘ

in (2.66), we use Lebesgue dominated Theorem again to get

lim

a→0a−2∥Pl(ψa)∥2

2= lim

a→0lim

Θ→πa−2∥Pl(˜︂

ψΘ)a∥2

2= 0

In the last step the exchange of order of limits is allowable since the convergence

is uniform. Indeed, for

m= 0

and any

a∈(0,1]

, there is

a−1

ˆ︂

(︂(˜︂

ψΘ)a)︂l0−ˆ︃

(ψa)l0=a−1

4∫︂2π

0∫︂π

Pl(cos θ)

1 + J2(︂˜︂

ψΘ−ψ)︂(θ′, φ′) tan θ′

2dθ′dφ′

Since

˜︂

ψΘ−ψ∈ N

, the latter goes to

4a3

4∫︂2π

0∫︂π

da (︃Pl(cos θ)

1 + J2)︃(︂˜︂

ψΘ−ψ)︂(θ′, φ′) tan θ′

2dθ′dφ′

a→0

, which further converges to zero uniformly due to the fact that

|˜︂

ψΘ(θ, φ)−ψ(θ, φ)|tan3(︁θ

2)︁

is bounded almost everywhere. For

m= 0

it can

be veried similarly. Thus we have proved that

ψ∈ A

This main result reduces the complicated admissibility condition into the

easy-to-check condition in Denition 2.13, hence greatly helpful for selecting

candidate shearlets. Another natural question is whether an admissible shearlet

remains admissible after dilation. In other words, we need to check whether our

system dened in (2.8) is closed under the dilation operation. The following

lemma gives us a positive answer.

CHAPTER 2. SPHERICAL DILATION SYSTEMS

Proposition 2.16.

ψ∈ N ∩ Bn

, then

ψa∈ N ∩ Bn

for any

a∈R+

and

n⩾1

Proof.

Let

θ′=θ1/a

and

φ′=φ1/a

. Suppose

|ψ(θ, φ) tannθ

2|⩽M

for some

M > 0

, then by (2.51) and Proposition 2.6,

|ψa(θ, φ) tannθ

2|=|√︁λ(a, θ, φ)ψ(θ′, φ′) tannθ

⩽2a3−α

2+nM

|Φ−

acos θ+ Φ+

a|(cos2φ+a2−2αsin2φ)n/2

Further, we observe that

∫︂2π

0∫︂π

ψa(θ, φ) tan(θ

2)dθdφ

=∫︂2π

0∫︂π

0√︁λ(a, θ, φ)ψ(θ1/a, φ1/a) tan(θ

2)dθdφ

=∫︂2π

0∫︂π

sec2θ

2λ−1/2(a, θ, φ)ψ(θ′, φ′) sin θ′dθ′dφ′

=∫︂2π

0∫︂π

1 + J2

a(α+1)/2sec2(θ′

1 + J2ψ(θ′, φ′) sin θ′dθ′dφ′

=a(α+1)/2∫︂2π

0∫︂π

ψ(θ′, φ′) tan(θ′

2)dθ′dφ′

therefore

ψa(θ, φ)∈ Bn∩N

Remark 2.17.

Once we have a square integrable function

on the sphere

which fullls the requirements in Theorem 2.15, then taking the dierence of

two sides of (2.3), we see a natural candidate of admissible shearlets is the

function

ψa(θ, φ)−a3/4ψ(θ, φ)

. Besides, for hollow pole functions, operator

D(a)

preserves regularity on the whole sphere. Indeed, given

f∈Ck(S2)

with

k⩾0

, it is obvious that

D(a)f

-times continuously dierentiable away from

the pole according to

(2.54)

; while if

is a hollow pole function, then its dilated

version keeps regularity at

θ=π

, hence

D(a)f∈Ck(S2)

The relationship between functions on

and shearlets on

is described by

the next proposition. More precisely, we prove that every zero mean function on

after being projected inversely by

is an admissible shearlet on the sphere.

2.4. OTHER APPROACHES: A SELECTIVE REVIEW

Proposition 2.18.

Let

ψ∈L2(R2)

be a function such that

∫︂R2

ψ(x1, x2)dx= 0

and

|ψ(x1, x2)rn+2|⩽M∀n⩽N

for some

M > 0

and

N∈Z+

. Then

U−1ψ∈ BN∩N ∈ A

Proof.

Let

r= 2 tan(θ

compactly supported implies

U−1ψ

is a hollow pole

function.

∫︂2π

0∫︂π

U−1ψ(θ, φ) tan(θ

2)dθdφ =∫︂2π

0∫︂π

ψ(rcos φ, r sin φ) sec2(θ

2) tan(θ

2)dθdφ

2∫︂2π

0∫︂∞

ψ(rcos φ, r sin φ)rdrdφ

=∫︂R2

ψ(x1, x2)dx

Since

∫︁R2ψ(x1, x2)dx= 0

, by denition, we have

U−1ψ∈ N

. Furthermore,

|ψ(x1, x2)rn+2|⩽M

for all

n⩽N

implies that

U−1ψ∈ BN

. Therefore,

U−1ψ

is an element in

2.4 Other approaches: a selective review

In this section let us have a tour and make some comments on other represen-

tative and creative approaches of constructing spherical wavelets that exhibit

certain merits as well as insuciency in dierent applications.

Firstly, I want to comment that, in the dilation aspect, the anisotropic spher-

ical Gaussian(ASG) is probably the most widely used tool by engineers, for in-

stance in the description of directional dependence radio waves from antenna.

Mathematically, without loss of generality, if we denote by

a set of mutu-

ally perpendicular unit eigenvectors of a

3×3

symmetric matrix

, correspond-

ing to eigenvalues

λ1⩾λ2⩾λ3

respectively, the ASG(as a function of

) is

dened to be

G(ξ, A) = eξTAξ ·max{z·ξ, 0}

Note that the matrix

can be rewritten in the form

A= (λ1−λ3)xxT+

(λ2−λ3)yyT+λ3I

, hence ASG has another expression

e−[(λ1−λ3)(ξ·x)2+(λ2−λ3)(ξ·y)2+λ3]·max{z·ξ, 0}

(2.71)

CHAPTER 2. SPHERICAL DILATION SYSTEMS

Comparing it with the traditional von Mises-Fisher distribution

e2λ(ξ·z−1) =e−λ[(ξ·x)2+(ξ·y)2]·e−λ(ξ·z−1)2

=e−λ[(ξ·x)2+(ξ·y)2])+1]·e−λ(ξ·z)2+2λ(ξ·z)

(2.72)

where

is the bandwidth and

is chosen to be the lobe axis, the axis of

the smallest eigenvalue which corresponds to the smallest radiation amplitude,

we see that the last

term is replaced by a smoothing term, which is used

to constrain value in upper hemisphere while preserving smoothness. It is

demonstrated[7] that ASG have approximate closed-form solutions for product

and convolution operators.

A subdivision scheme is used in the 1995 work by Schröder and Swelden[25]

to build the orthogonal Haar wavelet transform on arbitrary manifolds. This

method starts with an initial quasi-uniform triangulation of the surface and

builds ner level mesh by connecting the midpoint of each edge. Whenever

there is a multiresolution analysis on the surface either based on vertex basis or

face basis, one can use the lifting scheme to obtain wavelets of better properties

like improved smoothness or vanishing moments. That is to say, if we have a

biorthogonal wavelet basis system such that for any

f∈L2(S2)

there is

f=∑︂

j,m⟨f, ψ

˜j,m⟩ψj,m

ψj+1,k =∑︂

k′

˜j,k,k′φj,k′+∑︂

g˜j,k,mψj,m

(2.73)

with renement relations

∑︁

k′

˜j,k,k′φ˜j+1,k′=φ˜j,k

and

∑︁

g˜j,k,mφ˜j+1,m =ψ

˜j,k

then a new wavelet system can be generated through the following design of

lters

⎧

⎪

⎨

⎪

⎩

hnew

j,k,k′=hj,k,k′

˜new

j,k,k′=h

˜j,k,k′+∑︁

cj,k,mg˜j,k′,m

g˜new

j,k,m =g˜j,k,m

gnew

j,k,m =gj,k,m −∑︁

k′

cj,k,k′hj,m,k′

(2.74)

namely the scaling function remains the same while its dual and the wavelet

function are lifted, where

gj,k,m

and

hj,m,k′

are similarly dened and

cj,k,k′

and

cj,k,m

are coecients to be chosen.

There are in general two types of schemes of subdivision, the face splitting

scheme and the vertex splitting scheme. In the face setting, the scaling function

2.4. OTHER APPROACHES: A SELECTIVE REVIEW

and its bi-orthogonal dual at dierent levels supported in a local triangle can

be simply formed by taking the characteristic functions, namely setting

φj,k =

χTj,k

and

φ˜j,k =1

|Tj,k|χTj,k

. If the subdivision triangles of

Tj,k

are labeled as

{T0, T1, T2, T3}

with

indicating the middle triangle, the wavelet dual pairs

can be chosen as

ψj,m = 2 (︃φj+1,m −∫︁S2φj+1,mdΩ

∫︁S2φj,0dΩφj,0)︃

(2.75a)

˜j,m =1

2(φ˜j+1,m −φ˜j,k)

(2.75b)

where

m= 1,2,3

, so that the wavelet here has vanishing integral. In order to

achieve higher vanishing moments in the sense that there exist

linearly inde-

pendent polynomials restricted on the sphere such that their wavelet coecients

vanish for all

j⩾0

and all

in index set at level

, one can propose the lifted

dual as

˜j,m =1

2(φ˜j+1,m −φ˜j,k)−∑︂

k′∈N(k)

cj,k′,mφ˜j,k′

(2.76)

where

k′

are index of the neighboring triangles of

Tj,k

In the vertex setting, the scaling functions are often chosen as delta functions.

If the vertices are labeled by

at level

and the midpoint of an edge(or the newly

generated vertex) is labeled by

, then one could simply subsample followed by

upsampling the scaling coecients and let the renement relation be

ψj,m =φj+1,m −∑︂

k∈N(m)

cj,k,mφj,k

A special choice would be

cj,k,m ={︄∫︁S2φj+1,mdΩ

2∫︁S2φj,kdΩ

for

k=v1, v2

otherwise

where

and

are endpoints corresponding to

Suppose now we have a convex polyhedron

with triangular surfaces and

having all its vertices located on the

-sphere. Through certain subdivision

scheme we obtain rened triangulations

with the set of vertices

and

nodal functions

ϕv

at vertex

v∈ Vj

. The space

of piecewise linear continuous

function on

is a subspace of

j+1

. In fact, if only neighboring vertices are

involved, there is the relation

ϕj

v=ϕj+1

v+1

2∑︁

v′∈N(v)

ϕj+1

v′

; while in the buttery

CHAPTER 2. SPHERICAL DILATION SYSTEMS

scheme more vertices could be involved. Thus the composition of

ϕj

and inverse

projection

p−1:S2→∂Γ

, denoted by

φj

, satisfy the renement equation

φj

v=φj+1

v+1

2∑︂

v′∈N(v)

φj+1

v′

(2.77)

and if

u∈ Vj+1\Vj

be the newly added midpoint on an edge

[v1, v2]

, the next

step is to dene a wavelet space such that

Wj⨁︁P1

j=P1

j+1

. One strategy used

in [70] is to assume that the wavelet function at

is a linear combination of

the nodal functions of the previous level in the neighborhood of

and

, and

vanishes

. It is proved there if one dene

ψj,u =⎛

⎝c1ϕj+1

v1+∑︂

v′∈Nj+1(v1)

cv′ϕj+1

v′⎞

⎠+⎛

⎝c2ϕj+1

v2+∑︂

v′∈Nj+1(v2)

cv′ϕj+1

v′⎞

⎠

=ψ(1)

j,u +ψ(2)

j,u

(2.78)

by requiring that for any

v∈ Vj

⟨ψ(1)

j,u, ϕj

v⟩∂Γ=⎧

⎨

⎩

(−1)2−2jγ v =v1

2−2jγ v =v2

0else

and

⟨ψ(2)

j,u, ϕj

v⟩∂Γ=⎧

⎨

⎩

(−1)2−2jγ v =v2

2−2jγ v =v1

0else

where

is a given nonzero constant, so that

⟨ψj,u, ϕj

v⟩∂Γ= 0

, then

ψ(1)

j,u

and

ψ(2)

j,u

are uniquely determined, hence

ψj,u

Spherical Haar wavelets are later improved into a both orthogonal and sym-

metric basis in the work [46], where instead of using the geodesic bisectors they

smartly designed the subdivision by employing a spherical trigonometry for-

mula from a college and school book a century ago that is not well-known to

the people nowadays, so that areas of the children triangles on the ner level

are equal.

The authors of [82] project from the plane to the sphere the Mexican hat

wavelet

√2π(2 −x2)e−x2/2

, which is almost the Laplacian of a Gaussian, to

form the so called spherical Mexican wavelet. Noticed by computer scientists

that spherical Haar wavelets are constructed after subdivision and the rening

2.4. OTHER APPROACHES: A SELECTIVE REVIEW

scheme depending heavily on the connectivity of the mesh hence computation-

ally expensive, the authors from [59] recently introduced a Mexican hat wavelet

formulated in the frequency domain as an alternative choice, namely by den-

ing the continuous wavelet on an a compact manifold as the derivative of the

heat kernel

ht(x, y)

and the discrete version wavelet as its dierence. It has the

advantage being localized in both space and frequency domains. Precisely, let

ψt(x, y) = ∞

∑︂

k=0

λke−λktϕk(x)ϕk(y)

(2.79)

with

{ϕk}

are eigenfunctions of the Laplace-Baltrami operator on the manifold

and

λk

the corresponding distinct eigenvalues, then the continuous wavelets

transform of a square integrable function

Wψf(x, t) = ∫︂M

ψt(x, y)f(y)dy

and it has the inverse transform

f=∫︂∞

0Wψf(x, t)dt +f

ˆ(0)ϕ0(x)

(2.80)

By denition heat kernel and the associated wavelets have the properties that

for all

and

⎧

⎪

⎨

⎪

⎩

lim

t→0ht(x, y) = δx(y)

lim

t→∞ht(x, y) = 1/ν(M)

lim

t→∞ψt(x, y)=0

lim

t→0+ψt(x, y) = ∑︁

λkϕk(x)ϕk(y)

(2.81)

In particular the last term is the kernel of

∆M

Similarly, if we divide the time line into a sequence

[t0,··· , tN]

, the discrete

Mexican hat wavelet transformation and its inverse are

ˆ︁

W†

ψf(k) = ˆ︂

ψtj(k)ˆ︁

f(k)

f(x) =

∑︂

j=1 W†

ψf(x, tj) + ∫︂M

htN(x, y)f(y)dy

(2.82)

where

ˆ︂

ψtj(k) = e−λktj−1−e−λktj

, using the fact that

is the reproducing

kernel of the Hilbert space

L2(M)

. Since the heat kernel preserves the integral

CHAPTER 2. SPHERICAL DILATION SYSTEMS

∫︁Mht(x, y)dy = 1

for all

t⩾0

, the wavelet formulated in this way automatically

has zero mean. I would like to briey mention that a comparable construction of

the Mexican hat wavelet is the Poisson wavelet from [51], where frame properties

are discussed in a more theoretical way.

Another relatively new method is the Geometric Multi-Resolution Analy-

sis(GMRA) adapted to a

-dimensional smooth compact Riemannian manifold

embedded in

was built in [97]. This method consists of the following

steps. Firstly decompose the

-dimensional manifold in each scale

j∈Z

into

subsets

Uj={Uj,k}k∈Kj

, such that:

(i) They completely cover

and

ν(Uj,k′⋂︁Uj,k)=0

for any

k′=k∈ Kj

(ii)

Uj′,k′

is included in one and only one

Uj,k

whenever

j < j′

(iii) comparison principle: for each level

Bρ(cj,k,2−j′

)⊂Uj⊂Bρ(cj,k,2−j)

holds for some

j′> j

, where the Riemannian metric is denoted by

and the

Borel measure by

. Such a decomposition for instance can be achieved through

intersection of the manifold with dyadic cubes in

The second step is to nd the minimizer of the functional

min

P∫︂Uj,k

d(x, P)2dν(x)

(2.83)

where

measures the Euclidean distance to an ane plane

. The solution

of this minimization problem is the ane space spanned by the

eigenvectors

corresponding to the maximum eigenvalues of the covariance matrix

E[(x−Ex)(x−Ex)†|x∈Uj,k] = Bj,kΣj,kB†

j,k

(2.84)

with

Σjk

dim P×dim P

diagonal matrix and

dim P

equals the number of

vectors in

Bj,k

, centered at

cj,k =E[x∈Uj,k] = 1

ν(Uj,k)∫︂Uj,k

xdν(x)∈Rn

denoted by

cj,k +Vj,k

. Indeed, let

ξi(i= 1,··· , n −m)

be orthonormal unit

2.4. OTHER APPROACHES: A SELECTIVE REVIEW

vectors in

, then

(2.83)

equals to

min

λ∈Rn−m

ξi∈Sn−1

ν(Uj,k)∫︂Uj,k

n−m

∑︂

i=1 |x·ξi−λi|2dν(x)

= min

ξi∈Sn−1

ν(Uj,k)∫︂Uj,k

n−m

∑︂

i=1 |⟨x−Ex, ξi⟩|2dν(x)

= min

ξi∈Sn−1

n−m

∑︂

i=1

ξ†

iE[(x−Ex)(x−Ex)†|x∈Uj,k]ξi

(2.85)

therefore the minimum value is achieved by choosing arbitrary

n−m

eigenvec-

tors corresponding to the minimum eigenvalues. We point out that in the paper

[97] the assumption that

E[(x−Ex)(x−Ex)†|x∈Uj,k]

has rank

m≪n

would

be not right when the manifold is of poor regularity or having large curvature.

Covariance matrix in general could have full rank when its curvature is not iden-

tically zero. In fact, assume without loss of generality that

Bj,k

form the last

coordinates and points in

have the expression

x= (h1(x′),··· , hn−m(x′), x′)

where

x′

-tuple

(xn−m+1,··· , xn)

. If at

the curvature is large, then

U⊂P

and for any

i⩽n

and

j > m

, since

xj= 0

there is

cij =1

|U|∫︂U

(xj−Exj)(xi−Exi)dx = 0

hence the rank of

E[(x−Ex)(x−Ex)†|x∈U]

is at most

. However, when

the curvature at

is not zero, say

hi>0

for all

i⩽n

, then

cii >0

and

det E[(x−Ex)(x−Ex)†|x∈U]

does not have to vanish. In other words co-

variance matrix can have full rank

regardless of the fact that

is a

-dim

manifold. Nevertheless if we know in advance that the surface consists of points

distributed around a hyperplane or its sectional curvatures are small every-

where, namely

hi≪

diam

(U)

, then the low rank assumption holds as a rst

order approximation.

Let

Πj,k := Bj,kB†

j,k

, the third step is to compare the dierence between

the ane projection

xj,k = Πj,k(x−cj,k) + cj,k

to the plane centered at

cj,k

and the projection

Πj+1,k′(x−cj+1,k′) + cj+1,k′

to the next level plane, where

CHAPTER 2. SPHERICAL DILATION SYSTEMS

x∈Uj,k ∩Uj+1,k′

. Without confusion we omit the index

and

k′

. Then

∆jx:= Πj+1(x−cj+1)−Πj(x−cj)

= (I−Πj)Πj+1(x−cj+1)−Πj[(x−cj)−Πj+1(x−cj+1)]

= (I−Πj)Πj+1(x−cj+1)−Πj[(x−cj)−ΠJ+1(x−cJ+1)]

∑︂

s=j+1

Πj(Πs+1(x−cs+1)−Πs(x−cs))

=wj−Πj[(x−cj)−ΠJ+1(x−cJ+1)] + Πj

∑︂

s=j+1

∆sx

(2.86)

for any

J⩾j+ 1

, where

wj∈Wj+1 := (I−Πj)Vj+1

. Thus we have the

decomposition of

Πj+1(x−cj+1)∈Vj+1

into two orthogonal parts, namely the

detail part

and the coarse part that belongs to

. An obvious fact in

this method is that the centers change at dierent levels and eigenvectors must

be recalculated, as a result there is no simple operations like translation and

dilation directly applicable.

In my humble opinion one of the main obstacles in local projection approach

is how to smartly build an adaptive mesh on an arbitrary surface so that the

transition functions have good regularity properties. Even for the special case

of the unit sphere, it is not a trivial question. Best performance in regularity

aspect is achieved interestingly, however, by the earliest works in Germany. In

[74] the authors divide the whole domain into quad mesh and utilize tensor

product of exponential splines and B-wavelets to form the wavelets on a square

and join smoothly the pull-back of those wavelets which are able to obtain

regularity at the pole and

C∞

elsewhere. In [79] arbitrary smoothness of

the wavelets is realized on the whole sphere in theory, although it had the

drawback in implementation according to Prof. Dahmen and Prof. Schneider.

An optimized and implementable version is given by Kunoth and Sahner[48].

A similar strategy using local plane approximation for arbitrary manifold is

adopted recently in [94]. The representation system formed in this way in general

could be highly redundant and it is not always clear whether frame properties

hold globally.

In our projection means for constructing spherical

-wavelets/shearlets, how-

ever, as I have mentioned in Remark 2.17, hollow pole functions preserve reg-

ularity globally under dilation and as you shall see in the next chapter, they

generate a frame system under suitable assumptions, so that they recover a

function completely and stably even though there might exist overlaps.

Those methods mentioned above as well as ours have potentially many col-

2.4. OTHER APPROACHES: A SELECTIVE REVIEW

laborators in dierent scientic elds from astrophysics to medical imaging. For

instance, as a strong support for the big bang theory, the cosmic microwave back-

ground was originally assumed to be perfectly uniform or isotropic[32]. However,

detection of small uctuation of its temperature leads physicists to search for

the anisotropic structure behind it. Thus precise measurement of small param-

eters of the cosmological model is crucial according to physicists, and the scale

property of wavelets naturally perfectly ts into this needs. In this aspect I

would like to refer the readers to [24][58] for physics background, [42] for real

observation data based on wavelet tools and [23] for a recent survey. In [90]

spherical wavelets are used for analyzing the lithosphere structure of terrestrial

planets including the Earth, Venus, Mars and Moon, where the admittance and

correlation functions of given wavelet degree possess negative values for lowland

basins and positive values for highlands. Besides, spherical wavelets are also

used in image segmentation[84], and other applications.

Chapter 3

Extension to Miscellaneous

Results

3.1 Kernel approximation

A real continuous kernel

K:M×M→R

for some subset

M⊂Rn

is said to be

positive denite

if for any integer

the quadratic form

∑︂

i,j=0

K(xi, xj)cicj⩾0

(3.1)

holds for arbitrary set of points

{xi}N

i=1

in the set

and coecient vector

(c1, c2, . . . , cN)∈RN

. When

is symmetric, according to Theorem

4.7

, integral

operator

∫︁MK(x, y)f(y)dy

has eigenfunctions

{φn}n∈N

forming an orthonormal

basis of

L2(M)

, hence

K(x, y) = ∑︁

cn(x)φ(y)

for some vector of functions

depending on

. Replacing this into the equation

∫︂M

K(x, y)φn(y)dy =λnφn(x)

and using the linear independence of

{φn}n∈N

, we see that

cn(x) = λnφn(x)

and hence obtain the representation

K(x, y) = ∞

∑︂

n=0

λnφn(x)φn(y)

(3.2)

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

Conversely if

has such an expression with respect to an orthonormal basis,

then each basis element

φn

is an eigenfunction corresponding to

λn

. Especially

when

is equipped with a norm and

K(x, y) = ˜︁

K(∥x−y∥)

for some continuous

function

˜︁

and norm

∥·∥

, it is called a

radial kernel

On a compact domain, Mercers' Theorem says that a continuous symmetric

kernel

is a positive denite if and only if

∫︂M×M

K(x, y)w(x)w(y)dxdy ⩾0

(3.3)

for all

w∈L1(M)

We remark here that both the kernel and the coecient vector

c∈Rn

can

be complex, and the denition can obviously be rewritten in the matrix form

c†Kc≥0

. Nevertheless the denition (

3.1

) shall serve our current purpose.

Let us list some important examples of positive denite kernels:

(i)

(

Stationary kernel and its multivariate version

)

The name "stationary" comes from the translation-invariant property. Let

x, y ∈Rn

, a stationary kernel is of the form

K(x, y) = k

˜(x−y)

for some

continuous function

˜:Rn→R

. Bochner showed that

∫︂Rn

˜(x−y)w(x)w(y)dxdy ⩾0

for all

w∈L1(R)

is equivalent to the existence of a positive, nite Borel measure

such that

˜=µˆ

In the multi-dimension situation tensor product of one dimension kernels

˜i:R→R

with

i= 1,··· , n

leads to the multivariate kernel

K(x, y) =

∏︁

i=1

˜i(xi−yi)

. This kind of kernels, since it involves dierent dimensions, could

certainly embrace anisotropic traits, and is mostly used in statistics as a spe-

cial nonparametric regression method. In that context, given

-variate random

vectors

xi(i= 1,··· , n)

with a common density function

, one needs to wisely

choose symmetric and positive denite

d×d

smoothing matrix

, such that the

kernel density estimate

ρH(x) = 1

∑︂

i=1

KH(x−xi)

with

KH(x) = |H|−1/2K(H−1/2x)

minimizing the mean integrated squared

error MISE

(ρH) = ∫︁E[︁(ρH(x)−ρ(x))2]︁dx

. The latter can be decomposed

3.1. KERNEL APPROXIMATION

and written as

∫︂[︂

Var

(ρH(x)) + (EρH(x)−ρ(x))2]︂dx

=∫︂[︃1

n(︁K2

H∗ρ(x)−E2(ρH(x)))︁+ (KH∗ρ(x)−ρ(x))2]︃dx

where we used the fact that

EρH(x) = ∫︂KH(x−y)ρ(y)dy

=∫︂KH(y)[ρ(x)−H−1/2y·∇ρ(x) + 1

2(H1/2y)T

Hess.

ρ(x)H1/2y

+o((H1/2y)T(H1/2y))]dy.

and

Var

(ρH(x)) = 1

n2E⎡

⎣(︄n

∑︂

i=1

KH(x−xi))︄2⎤

⎦−EρH(x)EρH(x)

n[︁K2

H∗ρ(x)−(EρH(x))2]︁

An example is that

chosen to be diag

(h2

1,··· , h2

to be the Gaussian

|H|−1/21

(2π)d/2exp(−1

2xTH−1x)

, and

ρH(x) = 1

n∏︂

h−1

∑︂

i=1

K(x1−X1

,··· ,xd−Xd

)

Under the moments assumption that

∫︂K(x)dx= 1,∫︂xK(x)dx= 0

and

∫︂K(x)xxTdx=µI

where

µ=∫︁x2

iK(x)dx

independent of

, and the restriction that entries of

and

n−1|H|−1/2

both go to zero as

goes to innity, as well as that each term

of Hess.

is piecewise continuous and square integrable, the above expressions

become

EρH(x) = ρ(x) + µ

2tr(H.

Hess.

ρ) + o(|H|)

Var

ρH(x) = 1

n|H|−1/2∫︂K(y)2dyρ(x) + o(|H|−1/2·1

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

thus

MISE

(ρH) =

AMISE

(ρH) + o(1

n|H|−1/2+

2H)

where the leading term

AMISE

(ρH) = 1

n|H|−1/2∫︂K2(x)dx+µ2

4∫︂

2(H

Hess.

ρ)dx

is called asymptotic mean integrated squared estimate. Certainly one can adopt

an adaptive strategy and locate dierent smoothing matrices at dierent points,

but in general a linear transformation enables one to consider random data of

zero mean and unit covariant matrix. To maintain the structure of this chap-

ter I do not intend to stretch out to full details, but rather refer the interested

readers to monographs [15][72] among many excellent literatures. However, I do

want to mention that the spirit of this method is very close to that of wavelets

approximation, and it has the advantage that data distribution can be expressed

graphically and perceived by human cognition very well, similar to the isotherm

on a temperature distribution map.

(ii)

(

Power series kernel and zonal kernel

)

For

y∈(−1,1)n

, the kernel

K(x, y) = ∑︂

α∈Nn

cα

xα

α!

yα

α!

with

∑︂

α∈Nn

cα

(α!)2<∞

(3.4)

is rstly introduced in [1] as a generalization of the innite product kernel

∞

∑︁

n=0

cn(x·y)n

given in [45], where the coecients

are often assumed to be

positive to guarantee the positive deniteness of the kernel, and the nonlinearly

factorizable kernel of the form

∏︁

j=1

∞

∑︁

n=0

cn(xj·yj)n

. Among the simplest and

most popular examples there are the exponential kernel

exp(x·y) = ∞

∑︁

n=0

x·y

n!=

∑︁

α∈Zn

|α|!xαyα

and

∏︁

j=1

1−cxj·yj

with

c∈(0,1)

. It has the expansion

(3.2)

with

the eigenfunctions being some modication of the Hermite polynomials

(see

Example 4.13).

Zonal kernel adapts the stationary kernel in Euclidean spaces to spheres, in

the sense that it uses the geodesic distance between

and

Sn−1

instead of

the Euclidean distance. Note that when the kernel function is analytic, it re-

duces to an innite product kernel. A special example is the spherical Gaussian

3.1. KERNEL APPROXIMATION

exp(−2ϵ(1 −x·y))

. We shall discuss properties of this kind of kernels further

in this chapter.

(iii)

(

Multiscale kernel

)

Sometimes the function spaces arising from PDEs or analysis have multi-

scale structure, hence a class of kernels combined with multiscale property is of

particular interest. As was introduced in

[

][

]

, the multiscale kernel on the

plane has the form

K(x, y) = ∑︂

j⩾0

λj∑︂

k∈Zd

φ(2jx−k)φ(2jy−k)

with

x, y ∈Rd

(3.5)

It is clearly a special example of

(3.2)

and being particularly interesting in the

sense that it is endowed with wavelet-like properties, where

is either compactly

supported or of decay rate

O(1+∥x∥−(d+1)

. However, in the light of radial basis

construction of spherical wavelets in section

2.1

, I would like to propose a new

type of kernels of the form

KΦ,λ(x, y) = ∑︂

l⩾0∑︂

j⩾0

µjˆ︁

Φ4

j(l)Pl(x·ξ0)Pl(y·ξ0)

(3.6)

which can be viewed as spherical version of the multiscale kernel. As we shall

see soon this kernel is a reproducing kernel for a certain Hilbert space.

(iv)

(

Reproducing kernel

)

Suppose

is a function space which is Hilbert under certain norm, for

instance a subspace of

L2(M,R)

C(M,R)

. It is well known that

has a

reproducing kernel

if and only if the evaluation operator is bounded, namely

there exists

C > 0

such that

|v(x)|⩽C∥v∥H

for all

and any

v∈H

Reproducing kernel is positive denite due to the observation that

∑︂

i,j=1

cicjK(xi, xj) =

∑︂

i=1 ∥ciKxi∥2⩾0

(3.7)

where we adopt the notation

Kx=K(x, ·)

. In particular, under the further

assumption that

∑︂

i=1

civ(xi)=0

for every

and

{xi}N

implies

ci= 0

(3.8)

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

is strictly positive denite. Conversely, if

is symmetric positive denite

on a set

, there exists a unique Hilbert space in which

is the reproducing

kernel. In fact,

can be dened as the completion of

{Kx}x∈M

with respect

to inner product

⟨∑︁

aiKxi,∑︁

bjKxj⟩=∑︁

i∑︁

aibjK(xi, xj)

and uniqueness of

follows from a mutual inclusion argument.

Many questions can be solved or approximated once they are restricted to a

nite dimensional subspace which has a reproducing kernel. In applied mathe-

matics, a useful example is that the minimization problem

min

g∈X{∥Ag −f∥2

Y+αM(g)}

(3.9)

in some Hilbert space

provided that

is a dierentiable functional, can be

solved explicitly by taking derivative if the solution is of the form

∑︁

i=1

ciKxi

This kind of minimization is widely applicable in inverse problems[37]. In

complex analysis, an important example is the Bermann kernel

KB(z, w¯) =

∞

∑︁

k=0

λkφk(z)φk(w)

with

{φk}k⩾0

an orthonormal basis such that

∫︂C

φk(z)φk′(z)dz =δkk′

and

∫︂Ω

φk(z)φk′(z)dz =λkδkk′

which is the reproducing kernel for Bergmann space which consists of ana-

lytic functions on a bounded domain

Ω⊂Cd

with nite

norm; while the

Hua-Poisson kernel

H(z, w) = P(z,w¯ )P(w,z¯)

P(z,z¯)

dened by Hua[57] with

P(z, w¯) =

∞

∑︁

n=0

φn(z)φn(w)

, is the reproducing kernel for the class of functions

v(z) =

∑︁

k⩾0

ckφk(z)/P(z, z¯)

on the characteristic manifold

C⊂∂Ω

Ω

, namely

v(z) = ∫︂C

H(z, w)v(w)dw

(3.10)

In particular, for Laplacian

∆BC= 4 ∑︁

j,k

1−|z|2

3(δj,k −z¯jzk)∂2

∂zk∂z¯j

on the complex

ball

BC⊂Cn

with

C=Sd−1

u(z) = ∫︂Sd−1

H(z, w)f(w)dw

(3.11)

solves the Cauchy-Dirichlet problem

{︄∆Bu(z)=0 z∈BC

u|Sd−1

C(z) = f(z)z∈Sd−1

(3.12)

3.1. KERNEL APPROXIMATION

with

PBC(z, w) = (d−1)!

2πd(1−z·w¯)d

. Nevertheless, we shall not extend further into

those examples.

In spherical context, note that

are RKHS themselves with kernel

(1.29)

. Suppose

is a reproducing kernel of a subspace

H′

of the square

integrable functions on the sphere and every

is an element of

H′

, then

meets the condition

(3.8)

, hence is strictly positive denite. Indeed, if we assume

that

∑︁

i=1

civ(xi)=0

for every

v∈H′

, in particular

∑︁

i=1

ciPl(xi·x)=0

, then

there is

∑︂

i=1

ciQr(xi·x) = ∞

∑︂

l=0

2l+ 1

4πrl

∑︂

i=1

ciPl(xi·x)=0

where

Qr(t)

is the Poisson kernel. Let

θi(x) = {︃x·xi−1+h

x·xi<1−h

else (3.13)

with

h⩽min

i=jxi·xj

. Since

θi

is continuous and

θi(xj) = δij

, by Lemma

1.2

follows that

ci=

∑︂

j=1

cjθxi(xj) =

∑︂

j=1

cjlim

r→1−⟨Qr(xj·x), θxi(x)⟩L2= 0

A function

ψ: [0, π]→R

is called

conditionally strictly positive denite

order

Sn−1

ψ(xi·xj)

is positive denite with respect to

{(c1, . . . , cN) :

∑︂

i=1

ciY(xi)=0

for all

Y∈ Hl−1}

a concept facilitates the discussion of positiveness on subspaces like

⨁︁

l′∈L′Hl′

Clearly a radial function

that is conditionally strictly positive denite on

conditionally strictly positive denite on

Sn−1

. Furthermore,

∑︁

l′∈L′

al′Pl′(xi·yj)

being strictly positive denite for

al′>0

is obviously equivalent to that of

∑︁

l′∈L′

Pl′(xi·yj)

, which holds if and only if

L′

contains innitely many odd

terms and even terms, which was proved in [6].

Before we proceed, I would like to give the following lemma which bridges two

dierent reproducing kernel spaces. Let

and

be two reproducing kernel

Hilbert spaces of real(or complex)-valued functions on

and

respectively,

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

and

be the reproducing kernel of

. The lemma says that under certain

isometries

is uniquely determined by

Lemma 3.1.

is the reproducing kernel of

and

F:H1→H2

is an

isometry such that

FIvh=IvFh

for any integrable function

h:M1×M1→R

where

Iv(·) := ⟨v, ·⟩

is the evaluation operator at

v∈H1

, then

FFyK1(x)

is the

reproducing kernel of

Proof.

For xed point

in the domain

of the functions in

(FyK1)(z) :=

F(K1(·, z))(y)

is a function in

with respect to

x∈M1

⟨Fv, F (FyK1)⟩=⟨v, FyK1⟩

=Iv(FyK1)

= (FIvK1)(y)

=Fv(y)

(3.14)

By uniqueness of the reproducing kernel we arrive at our conclusion.

Remark 3.2.

Whenever there is a linear isometry

between

L2(S2)

into

L2(M)

, we are able to use the the above lemma to obtain the reproducing

kernel of

L2(M)

. In fact, the linearity of

implies that

commutes with

the integral operator.

For sampling points

x1,··· , xN

, it is a classical result that

v∗

from a

RKHS

achieves

sup

∥v∥2⩽E

∑︁

i=1

v2(xi)

when

v∗(x) =

∑︂

i=1

ξM(xi)Kxi

(3.15)

with

ξM

the eigenfunction corresponding to the maximum eigenvalue

λM

of the

matrix

(K(xi, xj))i,j=1,···,N

. In fact,

∑︂

i=1

v2(xi) =

∑︂

i=1

v(xi)⟨v, Kxi⟩

⩽∥v∥⌜



⎷

∑︂

i,j=1

v(xi)v(xj)K(xi, xj)

⩽E⌜



⎷λM

∑︂

i=1

ξ2

M(xi)

(3.16)

3.1. KERNEL APPROXIMATION

with equality holds if and only if

v=c

∑︁

i=1

v(xi)Kxi

and

∑︁

j=1

v(xj)K(xi, xj) =

λMv(xi)

, namely

v=v∗

Another question that is often asked is whether there exist

v∈H

of mini-

mum norm that solves the equations

v(xi) = ai

(3.17)

where

ai∈R

are given. The answer is armative at the presence of a reproduc-

ing kernel that satises

(3.8)

. Indeed, if

(3.17)

holds for

, then its projection

into the space

HN=

span

{Kxi,··· , KxN}

, written as

, solves the set of

equations as well, since

⟨v−vN, Kxi⟩= 0

for any

i∈ {1,··· , N}

. Furthermore,

∥vN∥⩽∥v∥

, thus the minimum norm solution, if exists at all, lives in

. Un-

der assumption

(3.8)

, the matrix

A= (K(xi, xj))

is invertible and the unique

minimum solution is given by

(a1,··· , aN)A−T(Kx1,··· , KxN)T

(3.18)

Our next theorem gives the kernel condition under which kernel integral

expression can be used to approximate an arbitrary

integrable function. I

also refer to the coming work [20] for a generalization of this result to higher

dimension situation.

Theorem 3.3.

Given

K∈L1([−1,1])

and a dense subset

Γ = {σi}i

of the

rotation group, the span of functions

{R(σi)K(ξ0·y)}

is dense in

L1(S2)

if and

only if

ˆl= 0

for all

l∈N

Proof.

Firstly observe that for any

g∈L∞(S2)

there is

(ˆ︂

K∗g)m

l=∫︂S2∫︂S2∫︂S2

K(x·y)Pl(x·z)g(y)dxdyY m

l(z)dz

ˆl∫︂S2∫︂S2

Pl(y·z)Ym

l(z)g(y)dzdy

ˆlgˆm

(3.19)

hence it is easy to see that

Kl= 0

for all

l∈N

, if and only if the equation

K∗g(x)=0 a.e.

(3.20)

does not have any bounded solution other than the zero function.

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

Let

V={

all nite linear combinations

∑︁

ciK(xi·y)}

and denote by

its

closure. For every

S∈V

and

ϵ > 0

, suppose

∥S−SN∥L1⩽1

with

SN(y) =

∑︁

ciK(xi·y)

, we see immediately from the fact that

∫︂S2|(S−SN)(y)g(y)|dy ⩽∥g∥∞·1

the equation

(3.20)

holds if and only if

∫︂S2

S(y)g(y)dy = 0

(3.21)

for any

S∈V

, which is further equivalent to the condition that

¯=L1(S2)

. In-

deed, Hahn-Banach Theorem implies that if

¯=L1(S2)

, we can nd a nonzero

continuous linear functional

L1(S2)

which is vanishing on

. Since

is of

nite measure, by Riesz-representation Theorem there exists a bounded nonzero

function

such that

v(S) = ∫︁S2S(y)g(y)dy = 0

for any

S∈V

, a contradiction.

Thus we have proved that the span of all rotated functions

{R(σi)K(ξ0·y)}

being dense in

L1(S2)

is equivalent to the condition

ˆl= 0

for

∀l∈N

Finally, by the density of continuous functions in

space, for any

f∈L1

there exist

Kc∈C([−1,1])

and some set

{σN

i}i

such that

∥SN−Sc

N∥L1<1

and

∥SN−f∥<1

, where

N=∑︁

ciR(σN

i)Kc(η0·y)

. Now by the density

assumption of

SO(3)

, for arbitrary

ϵ > 0

, we can nd some

{σi′} ⊂ Γ

such

that

∥f−∑︁

i′

ci′R(σi′)Kc(η0·y)∥L1< ϵ

, hence the claimed result follows.

Similarly, for a family of kernels

{Kj}j∈J

we have the following extension

Corollary 3.4.

The equations

∫︂S2

Kj(x·y)g(y)dy = 0 ∀x∈S2, j ∈J

have a nonzero bounded solution

when and only when for some

l∈N

ˆj

l= 0

for all

j∈J

Proof.

ˆj

l= 0

for all

j∈J

, then

∫︁S2Kj(x·y)Ym

l(y)dy = 0

for all

and

Conversely, if

∫︁S2Kj(x·y)g(y)dy = 0

has a nonzero bounded solution

for all

, then

0 = Zl∗(Kj∗g) = K

ˆj

lPlg

. Since

is not identically zero, there exists

some

Plg= 0

, hence

ˆj

l= 0

for all

j∈J

3.1. KERNEL APPROXIMATION

Remark 3.5.

In the proof we already use the fact that

g∈L∞(S2)⊂L2(S2)

otherwise

(3.19)

does not make sense. In fact, the theorem is still valid even if

we set

g∈L2(S2)

, and the corresponding proof is not much changed.

Let

be an arbitrary linear continuous functional on

L1([−1,1])

. Due to

the fact that

ν(K) = ∫︁1

−1K(t)gν(t)dt

for some

gν∈L∞([−1,1])

and that

∥ν∥= sup

∥gν∥∞=1 ∫︁1

−1K(t)gν(t)dt

, as an immediate consequence of theorem 3.3 is

following expansion result for linear continuous functionals on

L1(S2)

in terms

of translations of

Corollary 3.6.

Given a dense subset

Γ = {σi}i

of the rotation group and

ν∈(L1(S2))∗

such that

ker ν={0}

, there is

(L1(S2))∗=

span

{νi}

where

νi(f) = ∫︂S2

f(y)R(σi)gν(η0·y)dy

for any

f∈L1(S2)

Proof.

We only need to prove

(L1(S2))∗⊂

span

{νi}

. For any

µ∈(L1(S2))∗

and

f∈L1(S2)

, there exists

h∈L∞(S2)

, such that

µ(f) = ∫︁S2f(y)h(y)dy

Due to our assumption that

ker ν={0}

(3.20)

implies that

ˆ︁gνl= 0

for any

. Consequently, given any

ϵ > 0

there exist some

and

{σi}i⊂Γ

such that

∥h(y)−∑︁

ciR(σi)gν(η0·y)∥L1(S2)< ϵ

, hence we have

µ(f)−∑︂

ciνi(f)< ϵ∥f∥∞

for any

f∈L∞(S2)

such that

∥f∥L1= 1

, since

L∞

is dense in

on compact

set. The claim follows from the arbitrariness of

Let

Qa∗g:= ∫︁SK(x·ya)g(y)dy

, the observation that

ˆ︂

Qa∗g(l, m) = ∫︂ ∫︂ K(x·ya)Ym

l(x)g(y)dydx

=∫︂K

ˆ(l)Ym

l(ya)g(y)dy

(3.22)

leads to

lim

a→0+

ˆ︂

Qa∗g(l, m) = 1

√4πK

ˆ(l)δm,0∫︂S2

g(x)dx

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

lim

a→0+Qa∗g(y) = ∑︂

√2l+ 1

4πK

ˆ(l)∫︂S2

g(x)dxPl(cos θy)

(3.23)

and

lim

a→1

ˆ︂

Qa∗g(l, m) = K

ˆ(l)gˆlm

or equivalently

lim

a→1∫︂S2|Qa∗g(y)−g(y)|2dy =∑︂

l∑︂

|m|⩽l|K

ˆ(l)−1|2|gˆlm|2

(3.24)

Proposition 3.7.

{Qa}

is an approximate identity at

a= 1

i

ˆ(l) = 1

for any

l∈N

Let

be the parameter set of frames

{ϕi}i∈I

and

{ϕ†

i}i∈I

. Consider kernels

of the form

Kt(x, y) = ∑︂

i∈I

λi(t)ϕi(x)ϕ†

i(y)

(3.25)

with

λi∈ℓ∞

such that

Kt(x, y)

exist for all

t, x, y

Denition 3.8.

Let us call the kernel in

(3.25)

frame kernel

. Set

IN⊂I

with

∥IN∥0=N

and

I⊥

N=I\IN

. Denote by

IN(x, y) = ∑︁

i∈IN

λi(t)ϕi(x)ϕ†

i(y)

⟨f, Kt(x, ·)⟩=f(x)

, then

fIN(x) = ⟨f(·), KIN(x, ·)⟩

is called a

-term kernel

approximation

. When

∥⟨f(·), Kt

I⊥

N(·, y)⟩∥ ⩽∥⟨f(·), Kt

J⊥

N(·, y)⟩∥

for any

JN⊂I

with

∥JN∥o=N

we call

fIN

best

-term kernel approximation

Remark 3.9.

In contrast to the standard text in approximation theory where

-term approximation is usually reserved for nonlinear spaces, here I do not

distinguish between linear and nonlinear spaces.

Example 3.10.

Clearly frame property

(1.2)

can be reformulated in the kernel

means

A∥f∥2⩽⟨f(x),⟨Kt(x, y), f(y)⟩⟩ ⩽B∥f∥2

(3.26)

in the special case that

λi= 1

for all

i∈I

and

ϕi=ϕ†

{ϕ†

i}i∈I

is the dual frame of

{ϕi}i∈I

and

λi= 1

, then

∥f∥2=⟨f(x),⟨Kt(x, y), f(y)⟩⟩

(3.27)

3.1. KERNEL APPROXIMATION

namely

is a reproducing kernel.

λi(t) = e−cit

with

the eigenvalues of Laplace-Beltrami operator on a

manifold, we call

ht(x, y) = ∞

∑︂

i=0

e−citϕi(x)ϕ†

i(y)

(3.28)

generalized heat kernel

Another good example of frame kernel is the multiscale kernel we have de-

ned in

(3.6)

, its reproducing property is given in Theorem

3.13

. In addition,

if we denote by

A†

and

B†

the lower and upper bound of frame

{ϕ†

i}i∈I

respec-

tively, from the denition it is obvious that

∥⟨f(·), KI⊥

N(·, y)⟩∥ = sup

∥g∥=1 ⟨⟨f(·), KI⊥

N(·, y)⟩, g(y)⟩

= sup

∥g∥=1 ∥{λi⟨f, ϕi⟩}i∈I⊥

N∥ℓ2∥{⟨ϕ†

i, g⟩}i∈I⊥

N∥ℓ2

⩽√BB†max

i|λi|∥f∥

(3.29)

and by assuming without loss of generality that

λi= 0

, there is

∥⟨f(·), KI⊥

N(·, y)⟩∥ ⩾√AA†min

i|λi|∥f∥

(3.30)

The error of the best

-term Legendre kernel approximation of image with

smooth boundary decays as

l−1/2

when

l→ ∞

, which is shown in the next

proposition by choosing the characteristic function of the spherical cap

C(ξ0,Θ)

centered at

ξ0

. The following asymptotic result about Legendre polynomials

can be derived from our Lemma 2.14 and can also be found in [18] and [71].

Lemma 3.11.

Let c be a xed positive constant,

l→ ∞

. Then

P(m)

l(cos θ) = {︃θ−m−1

2O(lm−1

c/l ≤θ≤π/2

O(l2m)

0≤θ≤c/l

(3.31)

In fact, the rst term of the asymptotic expansion of

l(cos θ)

l(cos θ)=(−l)m(2

πl sin θ)1

2cos[(l+1

2)θ+(m−1)

2π] + O(l−3

(3.32)

where

l(cos θ)=(−1)lsinmθP(m)

l(cos θ)

ϵ≤θ≤π−ϵ

ϵ > 0

l≫m

l≫1

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

Proposition 3.12.

min

IL∥∑︁

l/∈IL

PlχC(ξ,Θ)∥2=O(L−1/2)

, namely the error of

best

-term kernel approximation with respect to

{Pl}l⩾0

decays at least as

O(N−1/2)

;

Proof.

Let

f=χC(ξ,Θ)

, then

⟨f, Pl(ω·)⟩=b(l, Θ)Pl(ω·ξ)

(3.33)

where

b(l, Θ) = 2π∫︁Θ

0Pl(cos θ) sin θdθ

By integrating

(4.44)

we obtain

∫︂Θ

Pl(cos θ) sin θdθ = (l+ 1)−1(Pl−1(cos Θ)) −cos ΘPl(cos Θ))

=O(l−3/2)

(3.34)

(3.31)

, if

ξ·ω= 1

Pl(ω·ξ) = O(l−1/2)

l→ ∞

; If

ξ·ω= 1

Pl(ω·ξ) = O(1)

. Therefore

∥⟨χ(ξ,Θ), Zl(ω·)⟩∥2

2= (∫︂|ω·ξ−1|⩾1

+∫︂|ω·ξ−1|⩽1

)|b(l, Θ)Zl(ω·ξ)|2dΩ(ω)

=O(l−2) + O(l−1·l−1)

=O(l−2)

due to the observation that surface area of the region

{ω:|ω·ξ−1|⩽1

is of

size

O(l−1)

∞

∑︂

L∥Plf∥2

2=∞

∑︂

L∥⟨χ(ξ,Θ), Zl(ω·)⟩∥2

2=O(L−1)

(3.35)

Thus the error of best

-term kernel approximation is

∥f−fL∥2=O(L−1/2)

(3.36)

As I promised, now let us turn our attention back to the kernel

KΦ,λ

dened

(3.6)

. Endow the space

with norms

∥f∗Φj∗Φj∥2

Vj=∑︁

lf

ˆ(l)

, we can

claim the following result.

3.1. KERNEL APPROXIMATION

Theorem 3.13.

Given

satisfying the conditions in

(2.5)

and a positive se-

quence

λ={λj,l}j,l

such that

∑︁

j,l

λj,l <∞

and

µj=γ(2−jl)4/λj,l

independent

, then

KΦ,λ(x, y)

forms a reproducing kernel of the Hilbert space

HΦ,λ ={g∈L2:g=∑︂

gj∈⋃︂

and

∑︂

µj∥gj∥2

Vj<∞}

(3.37)

with norm

∥g∥2

Φ,λ = inf{∑︂

µj∥gj∥2

Vj:g=∑︂

gj}

(3.38)

Proof.

Recall that

ˆ︁

Φj(l) = γ(2−jl)

and let us dene

A∗

{Φj}j⩾0(

) = ∑︂

l,j⩾0

cl,j ˆ︁

Φj(l)2Pl(x·ξ0)

(3.39)

or abbreviated as

A∗

without confusion, where

∈ℓ2

µ={{cl,j}l,j :∑︂

l,j

µj|cl,j|2<∞}

(3.40)

Denote by

NA∗={

∈ℓ2

µ:A∗(

)=0}

the null space of

A∗

. With the inner

product

⟨︄∑︂

l,j

cl,j ˆ︁

Φj(l)2Pl(x·ξ0),∑︂

l,j

c′

l,j ˆ︁

Φj(l)2Pl(x·ξ0)⟩︄A∗

=⟨︂PN⊥

A∗(

), PN⊥

A∗(

′)⟩︂ℓ2

(3.41)

the range of

A∗

becomes a Hilbert space, where

PN⊥

A∗

is the projection onto the

sequence subspace

N⊥

A∗

. Indeed, it is obviously bilinear and symmetric. It is

well-dened, for

A∗(

) = A∗(

)

implying that

PN⊥

A∗(

) = PN⊥

A∗(

−

) =

PN⊥

A∗(

)

. Furthermore, it is clear that

⟨A∗(

),A∗(

)⟩⩾0

, and equality holds

when and only when

PN⊥

A∗(

) =

. The completeness of Ran

A∗

follows from

that of

ℓ2

, due to the fact that

A∗

is an isometric isomorphism.

Fix

on the sphere.

µjγ(2−jl)2Pl(x·ξ0)}j,l

forms a sequence in

ℓ2

under

our assumption, hence

KΦ,λ(x, ·)∈

Ran

A∗

. Notice that

⟨A∗(

), KΦ,λ(x, ·)⟩A∗

=⟨︄∑︂

l,j⩾0

cl,j ˆ︁

Φj(l)2Pl(y·ξ0),∑︂

l,j⩾0

µjˆ︁

Φ4

j(l)Pl(x·ξ0)Pl(y·ξ0)⟩︄A∗

=∑︂

l,j

cl,j ˆ︁

Φ2

j(l)Pl(x·ξ0)

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

Thus

KΦ,λ

is a reproducing kernel for the Hilbert space

(

Ran

A∗,⟨·,·⟩A∗)

. Fur-

thermore, since for any

′∈ℓ2

such that

A∗(

′) = A∗(PN⊥

A∗(

))

there is

∥PN⊥

A∗(

)∥2

ℓ2

µ=∥PN⊥

A∗(

′)∥2

ℓ2

⩽∥

′∥2

ℓ2

, it follows that

∥A∗(

)∥2

A∗=∥PN⊥

A∗(

)∥2

ℓ2

= min{∑︂

l,j

µj|c′

l,j|2:

′∈ℓ2

µ,A∗(

′) = A∗(

)}

=∥A∗(

)∥2

hence the claim of the theorem holds.

3.2 Construction of spherical frames

The goal of this section is to establish frame properties for two types of represen-

tation systems on the sphere that we have met in Chapter 2. Recall that the rst

type considers dilation in the frequency domain. For this type, starting with a

general basis, for instance the spherical harmonics, our next theorem allows us

to transform them into a new frame suited to the multiresolution structure and

at the same time equips them with good properties such as local support and

fast decay. Interestingly, as far as I know when I am writing this part, it has

never been discussed by any before. For that purpose, we rstly need a lemma,

whose proof can be found in [81] for instance, hence omitted here.

Lemma 3.14.

{ϕk}k⩾0

is a frame for Hilbert space

with bounds

and

, and

is an bounded operator on

with closed range, then

{Tϕk}k⩾0

a frame for the range of

with bounds

A∥T

¯∥−2

and

B∥T∥2

, where

is the

pseudo-inverse of

Let

Ξ⊂N

and

HΞ=⨁︁l∈ΞHl

be a subspace of

L2(S2)

and denote by

Vj= Φj∗Φj∗HΞ

and

Wj= Ψj∗Ψ

˜j∗HΞ

the spaces of multiresolution for the

subspace

HΞ

Theorem 3.15.

Let

be piecewise dierentiable and admissible in the sense of

(2.5)

with

inf

l∈Ξ|γ(l)|>0

. Given a frame

{bk}

for

HΞ

, then

{ϕk,j =1

(√2c)jΦj∗

Φj∗bk}

is a frame for

. If additionally for some constant

τ > 0

t0⩾0

there

−γ′(t)⩾τγ(t)

for all

t > t0

(3.42)

then

{ψk,j =1

(√2c)jΨj∗Ψ

˜j∗bk}

is a frame for

Wj∩HΞ[t0,j]

, where

Ξ[t0, j]

any subset of

{l:l⩾t02j+1}

3.2. CONSTRUCTION OF SPHERICAL FRAMES

Proof.

Suppose

{bk}

is a frame with lower bounds

and upper bounds

Dene a new function class

G2,c ={f∈L2(Sd−1) : ∥Plf∥⩽c∥Pl′f∥

for any

l⩾2l′}

(3.43)

where

c > 0

. Notice that

{ϕk,0}

forms a frame for

. For any

f∈ HΞ

and

g= Φ ∗Φ∗f∈V0

there is

∑︂

k|⟨g, ϕk,0⟩|2⩽B∥∑︂

γ4(l)Plf∥2⩽sup

γ2(l)∥g∥2B

and similarly

∑︁

k|⟨g, ϕk,0⟩|2⩾inf

l∈Ξγ2(l)∥g∥2A > 0

Introduce the operator

Djg=1

(√2c)j∑︂

l∈Ξ

γ(2−jl)2Plf

(3.44)

V0∩G2,c

It is clearly a map onto

Vj,c = Φj∗Φj∗HΞ∩G2,c

, which consists solely of

functions like

Φj∗Φj∗f

. Besides, by separating the sum into even terms and

odd terms, under our assumption that

f∈ G2,c

we can get an estimation

∥Dj+1g∥2= (2c2)−j−1⎡

⎣∑︂

odd

γ4(2−j−1l)∥Plf∥2+∑︂

even

γ4(2−j−1l)∥Plf∥2⎤

⎦

⩽(2c2)−j−1∑︂

l⩾0

γ4(2−j(l+1

2))∥P2l+1f∥2

+ (2c2)−j−1∑︂

l⩾0

γ4(2−jl)∥P2lf∥2

⩽∥Djg∥2

for

j⩾0

, hence

(√2c)−jI⩽Dj⩽I

(3.45)

and

I⩽Dj=D−j⩽(√2c)jI

(3.46)

Meanwhile

Dj−√2cDj+1

is an operator onto

Wj,c

, and there is the estima-

tion

(︂Dj−√2cDj+1)︂g

2= (2c2)−j∑︂

l(︁γ(2−jl)2−γ(2−j−1l)2)︁2∥Plf∥2

⩽2c2∥Dj+1g∥2

(3.47)

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

and for

g∈Wj,c ∩HΞ[t0,j]

(︂Dj−√2cDj+1)︂g

−2

=(︂Dj−√2cDj+1)︂g

= (2c2)−j∑︂

l⩾t02j+1 ∫︂2−jl

2−j−1l−γ′(t)dt

2(︁γ(2−jl) + γ(2−j−1l))︁2∥Plf∥2

⩾(2c2)−j∑︂

l⩾t02j+1

γ2(2−jl)∫︂2−jl

2−j−1l

tdt

·4γ2(2−jl)∥Plf∥2

= 4τ2ln22∥Djg∥2

(3.48)

due to

(3.42)

Those together with Lemma 3.14 imply that

{ϕl,j}l⩾0

form a frame for

Vj,c

, while

{(Dj−Dj+1)ϕl,0}l⩾0

form a frame for the space

Wj,c ∩HΞ[t0,j]=

Φj∗Φj∗HΞ∩G2,c ∩HΞ[t0,j]

Taking any

h∈L2(Sd−1)

and

c > 1

, suppose

is the smallest degree such

that

Pl1h= 0

. Let

hl0(x)=0

and

ul0(x) =

min{l1−1,0}

∑︂

l=0

c−l√︄4π

(2l+ 1)Zl(x·ξ0)

(3.49)

Dene inductively, for the smallest degree

¯k+1 ⩾l

¯k

such that

Plk+1 h= 0

, the

functions

hlk+1 =hlk+ulk+∥hlk−hlk−1−ulk−1∥

clk+1−lk

Plk+1 h

∥Plk+1h∥

(3.50)

and

ulk+1 =

lk+1−1

∑︂

l=lk+1

c−l+lk√︄4π

(2l+ 1)Zl(x·ξ0)∥hlk+1 −hlk−ulk∥

(3.51)

Then

h∈

span

{hlk+1 −(hlk+ulk) : k⩾0} ⊂ G2,c

, namely

G2,c

is dense in

L2(Sd−1)

. Consequently

Vj,c

is dense in

and

Wj,c

is dense in

, hence

{ϕk,j}

and

{ψk,j}

are frames of

and

Wj∩ HΞ[t0,j]

with the same bounds

respectively.

3.2. CONSTRUCTION OF SPHERICAL FRAMES

Corollary 3.16.

Under the same assumptions of theorem

3.15

with

(3.42)

being

replaced by

γ(2−1t)−γ(t)⩾τγ(t)

for all

t>t0

(3.52)

{ψk,j}

form a frame for

Wj∩HΞ[t0,j−1]

Proof.

This claim follows from the same line proof of the above theorem except

that here we have for

g∈Wj∩HΞ[t0,j−1]

that

(︂Dj−√2cDj+1)︂g

= (2c2)−j∑︂

l⩾t02jγ(2−j−1l)−γ(2−jl)2(︁γ(2−jl) + γ(2−j−1l))︁2∥Plf∥2

⩾(2c2)−jτ2∑︂

l⩾t02j

γ2(2−jl)·4γ2(2−jl)∥Plf∥2

= 4τ2∥Djg∥2

(3.53)

Example 3.17.

The Shannon type wavelets is a good example for our theory

here. Let

ˆ︁

Φj(l) = {︃1l⩽2j−1

0l⩾2j

(3.54)

with corresponding wavelets

ˆ︁

Ψj(l) = ˆ︁

˜j(l) = {︃1 2j⩽l⩽2j+1 −1

else (3.55)

It is clear that

Wj⊥Vj

and

Vj⨁︁Wj=Vj+1

in this situation. Thus

Wi⊥Wj

for any

i=j

. By choosing

t0= 1

HΞ[1,j−1] ={l:l⩾2j}

, we see that for

γ(t) = {︃1t∈[0,1)

0t⩾1

inequality

γ(t

2)−γ(t)⩾γ(t)

holds for any

t > 1

, hence by Corollary

3.16

⋃︁

j=0{ψk,j :k⩾0}

form a wavelet frame for

⨁︁J

j=0 Wj∩HΞ[1,j−1] =⨁︁J

j=0 Wj

The second type spherical wavelets we have encountered is constructed through

stereographic projection. In proposition

2.18

it has been proved that a plane

wavelet under certain conditions gives an admissible wavelet on the sphere.

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

Since stereographic projection

is conformal, hence preserves angle, a plane

wavelet frame

{ψi}i∈I

of unit norm with lower and upper frame bounds

respectively immediately implies that for any

f∈L2(S2)

there is

AC∥f∥2

L2(S2)⩽∑︂

i∈I⟨π−1πf, π−1ψi⟩L2(S2)⩽BC∥f∥2

L2(S2)

(3.56)

where

C=∥πf∥2

L2(R2)/∥f∥2

L2(S2)

, namely

{π−1ψi}i∈I

is automatically a frame

on the sphere. It seems that nothing needs to be done further, due to all kinds of

plane wavelets with good frame properties having been well studied. However,

a closer look tells a dierent story. Since the natural lattice on the plane does

not generate a good grid on the sphere through projection, it is necessary to

establish principles or conditions, under which we can obtain spherical frames

directly on the sphere. This is the next theorems in this section about.

I would like to mention that main dierences between our second type frame

construction and those kernel based frames in [63] that apply to more general

Lie groups, include but not limited to: the latter is solely designed for isotropic

case while ours includes anisotropic case; and the latter starts from contin-

uous wavelets without a result like Theorem 3.19 but rather directly utilizes

the Calderón reproducing formula with respect to the scaling parameter, hence

immediately having reproducing property, while in the system we introduced

above, Calderón's formula does not apply. However, in this work we are able

to achieve tight frames for continuous and discrete

-wavelets/shearlets on the

sphere in a dierent way.

To meet our purpose let us rstly extend the traditional spherical harmonics,

as a natural generalization of the solutions of

(4.41)

Denition 3.18.

Let the

spherical harmonics of fractional degree

and order

to be

Yβ

λ(θ, φ) = cλ,βeiβφPβ

λ(cos θ)

(3.57)

where

λ, β ∈R

cλ,β =eiβπ√︂(2λ+1)Γ(λ−β+1)

4πΓ(λ+β+1)

and

Pβ

are associated Legendre

functions which solve the dierential equation

(1 −z2)u′′ −2zu′+[︃λ(λ+ 1) −β

1−z2]︃u= 0

(3.58)

and such that

∫︁1

−1|Pβ

λ(t)|2dt =2Γ(λ+β+1)

(2λ+1)Γ(λ−β+1)

For

|z−1|<2

in the complex domain, there is the following expression(see

for instance [11])

Pβ

λ(z) = dλ,β 2F1(λ+β+ 1,−λ+β, β + 1; 1−z

2)(z2−1)β

(3.59)

3.2. CONSTRUCTION OF SPHERICAL FRAMES

where

2F1(a, b, c;z)

is a solution of the hypergeometric equation

z(1 −z)d2u

dz2+ [c−(a+b+ 1)z]du

dz −abu = 0

(3.60)

and

dλ,β =(−1)β

2βΓ(β+ 1)

Γ(λ+β+ 1)

Γ(λ+ 1) (−λ+β−1)(−λ+β−2) ···(−λ)

(3.61)

when

λ > 0

−λ+β

and

−λ−β

are negative integers; by identity

Γ(x)Γ(1−x) =

sin(πx)

for non-integer values

, there is

dλ,β =(−1)β

2βΓ(β+ 1)

Γ(λ+β+ 1)Γ(−λ+β)

Γ(λ+ 1)Γ(−λ)

=(−1)β

2βΓ(β+ 1)

Γ(λ+β+ 1)

Γ(λ−β+ 1)

(3.62)

when

β∈Z+

−λ+β

and

−λ

are not integers.

Theorem 3.19.

Let

h∈ B ∩C∞(S2)

be admissible such that there exists

ν∈

(0,π

such that

h(θ′, φ′) = 0

for

|φ′|< ν

and for

|φ′−π|< ν

. Assume that,

on subset

Λ=∅

of integer pairs

(λ0, m0)

such that

⟨e−im0φ′

h, Y m0

λ0⟩=bλ0,m0

for some

bλ0,m0>0

. Then for any

f∈⨁︁

m⩽M

and

α=1

, there exists

0< A ⩽B < ∞

such that

A∥f∥2⩽∫︂∞

a3∫︂SO(3) |⟨f, ψσ,a⟩|2dσ ⩽B∥f∥2

(3.63)

where

ψ(θ′, φ′) = h(θ′, φ′)√︃θ′

sin θ′(1 + J2) cos2θ′

(3.64)

In particular we can take

A=∑︂

(λ0,m0)∈Λ

γλ0,m0(︄m0

∏︂

s=1

λ0

λ0+s+ 1)︄2ln(λ0+ 1/2)

ln(λ0+ 1/2−ϵλ0,m0)

(3.65)

with

ϵλ0,m0

solely depending on

bλ0,m0

and

γλ0,m0=b2

(2λ0+2)2m0+2

(λ0+m0−1)!

(λ0−m0+1)!

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

Proof.

Suppose that

⟨e−im0φ′

h, Y m0

λ0⟩=b

for some integer

λ0∈Λ

and

b > 0

Take any large

l∈N

there exists

such that

λ0+1

2=√a0(l+1

. Let

λ(a)+ 1

√a(l+1

, then there exists

δ∈(0, a0/2)

such that for

a∈(a0−δ, a0+δ)

there

|λ−λ0|< ϵ

for some xed

ϵ∈(0,1/4)

and

⟨e−im0φ′

h, Y m0

λ⟩>b

For instance we can choose

δ=2ϵ(λ0+1/2)−ϵ2

(l+1/2)2

. Notice that

a−1/2(λ+1

2)−1

2+k

λ+k=a−1/2(︃λ+ 1/2

λ+k)︃+O(1)

for any

1⩽k⩽2m−1

and that

Pm0

l(cos θ) = (︁a−1/2(λ+1

2) + 1

2)︁(︁a−1/2(λ+1

2) + 3

2)︁···(︁a−1/2(λ+1

2) + m0−1

2)︁

(︁a−1/2(λ+1

2))︁m0

Γ(︁(a−1/2(λ+1

2)+1)︁

(︁a−1/2(λ+1

2)︁−m0)! (︄θsin θ′

θ′sin θ)︄1

2(︄θ′

sin θ′)︄1

Jm0(︃(λ+1

2)θ′)︃

+O(am0/2+1)

=q(a)(︄sin θ′

θ′)︄1

Pm0

λ(cos θ′) + O(am0/2+1)

(3.66)

where

q(a) = a−m0/2m0−1

∏︁

s=0

λ+1/2

λ+s+1 +O(a−m0−1

From the previous analyzing we know that for

admissible

ˆ︃

(ψa)l0

van-

ishes, hence we can without loss of generality assume that

is positive. Since

ψ(θ′, φ′) = 0

for

|φ′|< ν

and for

|φ′−π|< ν

, along with

α=1

, there is

Il,m0=1

2l+ 1 ∫︂∞

0ˆ︃

(ψa)l,m0

2da

⩾4|cl,m0|2

(2l+ 1) ∫︂a0+δ

a0−δ|cλ,m0|−2⟨e−im0φ′

h, Y m0

λ⟩

q2(a)a−3/2da

⩾b2(l−m0)!

4π(l+m0)! ∫︂a0+δ

a0−δ|cλ,m0|−2(︄m0−1

∏︂

s=0

λ+ 1/2

λ+s+ 1)︄2

a−m0−3

2da

⩾b2(l−m0)!

(l+m0)!

(λ0+m0−1)!

(2λ0+ 1)(λ0−m0+ 1)! ∫︂a0+δ

a0−δ(︄m0−1

∏︂

s=0

λ+ 1/2

λ+s+ 1)︄2

a−m0−3

2da

3.2. CONSTRUCTION OF SPHERICAL FRAMES

Using the estimations

(2l+ 1)−2m0⩽(2l2)−m0⩽(l−m0)!

(l+m0)!

(3.67)

and

(l−m0)!

(l+m0)! ⩽l−2m0⩽2m0(︃l+1

2)︃−2m0

for

l⩾m2

(3.68)

it follows that

Il,m0⩾b2

(2λ0+ 2)2m0+2

(λ0+m0−1)!

(λ0−m0+ 1)! (︄m0

∏︂

s=1

λ0

λ0+s+ 1)︄2

ln a0+δ

a0−δ

Furthermore, by the choice of

, we have

ln (︃a0+δ

a0−δ)︃⩾2 [ln(λ0+ 1/2) −ln(λ0+ 1/2−ϵ)]

independent of

. Thus we have arrived at

(3.65)

when we consider all those

integer pairs

(λ0, m0)

For an arbitrary pair

(λ0, m0)∈Λ

, due to our analyzing in Theorem 2.15,

the estimation

(3.68)

and the observation

|cλ,m0|−2⩽4π(λ0+m0+ 1)!

(2λ0+ 1/2)(λ0−m0−1)!

(3.69)

we have for

l⩾m2

that

Il,m0[0, α0] = 1

2l+ 1 ∫︂a0

0ˆ︃

(ψa)l,m0

2da

⩽8|cl,m0|2

(2l+ 1) ∫︂a0

0|cλ,m0|−2⟨e−im0φ′

h, Y m0

λ⟩

q2(a)a−3/2da

⩽22m0+2(l+1

2)−2m0−1(λ0+m0+ 1)!

(2λ0+1

2)(λ0−m0−1)! (︄m0−1

∏︂

s=0

s+ 1)︄2

∫︂a0

(λ+ 1/2)2m0⟨e−im0φ′

h, Y m0

λ⟩

a−m0−3/2da

(3.70)

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

Let

λ+1

2=λ

dλ

¯=λ

2ada

, hence

∫︂a0

(l+1

2)−2m0−1(λ+ 1/2)2m0⟨e−im0φ′

h, Y m0

λ⟩

a−m0−3/2da

⩽1

2∥h∥2

1∫︂λ0+1

0∥Ym0

λ∥2

∞λ

¯2m0λ

¯−2m0dλ

⩽1

2∥h∥2

1sup

λ∈[0,λ0+1

2]∥Ym0

λ∥2

∞(λ0+1

Thus we claim that

Il,m0[0, α0]

is bounded. Furthermore, the assumption that

ψ∈C1(S2)

immediately leads to

∥Pλ0ψa∥2=O(λ−1

, hence the integral

on the domain

(a0,1)

is bounded by some constant. Finally the fact that

∥Plψ∥⩽∥ψ∥

suces to conclude the boundedness on

(1,∞)

. In sum, the

integral

2λ0+1 ∫︁∞

0|Pλ0ψa|2a−3da

is bounded by some

B < ∞

independent of

λ0

and it nishes the proof.

Combining the above result with Proposition 2.11 we arrive at a reproducing

formula:

Corollary 3.20.

Given

that satises the conditions in Theorem

3.19

, there

∫︂∞

a3∫︂SO(3)⟨f, ψ♯

σ,a⟩ψ♯

σ,adσ =f

for any

f∈L2(S2)

, where

Plψ♯:= Plψ/√︂Cl

for each

In practice we cannot do integration on the whole domain, but rather re-

course to a discretized summation to approximate the smooth case.

Theorem 3.21.

Under the same assumption of Theorem

3.19

, let

Υδ,ϵ

be the

set of sequences

{aj}j⩾0

such that

aj→0

|aj−aj+1|⩽δ

a3α−1

j/a2

j+1 ⩽C

and

1⩽a2

0ϵ

with

and

suciently small. Then for each

{aj}j∈Υδ,ϵ

{√sjψaj}j∈N

forms a semi-frame system for

L2(S2)

, where

sj=1

j+1 −1

Proof.

Firstly notice that

dcos θ

da =O(a2α−1)

and that

dcos θPm

l(cos θ) = −mcos θsinm−2θΓ(l+m+ 1)

2mΓ(l+ 1) Pm,m

l−m(cos θ)

+ sinmθΓ(l+m+ 2)

2m+1Γ(l+ 1)Pm+1,m+1

l−m−1(cos θ)

=O(aα(m−2))

3.2. CONSTRUCTION OF SPHERICAL FRAMES

hence

da Pm

l(cos θ) = O(aαm−1)

. Since

ψ(θ, φ)=0

for

φ<ν

, there is

dae−imφ =−me−i(m+1)φ(︄itan φ′+a1−α

±√︁tan2φ′+a2−2α)︄′

=O(a−α)

Besides, a direct calculation gives

da (1 + J2) = O(a2α−1)

. From

(2.66)

there is

(︂ˆ︂

ψa)︂′

lm =O(a(α−1)/2+αm)

, hence combined with our previous result we arrive

(︂ˆ︂

ψa)︂2

lm

′

=O(aα(2m+1))

. Thus

2l+ 1 

2∑︂

1⩽|m|⩽l

∞

∑︂

j=0 (︂ˆ︃

ψaj)︂lm

2(︄1

j+1 −1

j)︄−∑︂

|m|⩽l∫︂∞

da3(︂ˆ︂

ψa)︂lm

2

⩽1

2l+ 1 ∑︂

1⩽|m|⩽l

∞

∑︂

j=0 ∫︂aj

aj+1 ∫︂aj

a(︂ˆ︂

ψs)︂2

lm

′

dsda

a3+1

2l+ 1 ∑︂

1⩽|m|⩽l∫︂∞

a3(︂ˆ︂

ψa)︂lm

⩽∞

∑︂

j=0

(aj−aj+1)2a3α−1

j+1

+ϵ

where

is arbitrarily small for suciently large

. Furthermore, under our

assumption that

|aj−aj+1|⩽δ

and

a3α−1

j+1

⩽C

, the last line of the above

inequality is bounded by

Cδ +ϵ

, thus the conclusion holds.

The next result from [62] provides us one candidate sampling method on the

rotation group. I include its simple proof here for completeness.

Lemma 3.22.

If the quadrature formulae

4π∫︂2π

0∫︂π

l(φ, θ) sin θdθdφ =∑︂

i∈I1

wi(S2)Ym

l(φi, θi)

(3.71)

with

l≤N

and

2π∫︂2π

einϕdϕ =∑︂

j∈I2

wj(S1)einϕj

(3.72)

with

|n| ≤ N

, then all polynomials

f∈ ⊕l≤NHl

can be expressed by

∫︂SO(3)

fdσ =∑︂

i∈I1∑︂

j∈I2

wi(S2)wj(S1)f(φ1i, θi, φ2j)

(3.73)

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

where

(φ1, θ, φ2)

represents the Euler angles.

Proof.

For

n= 0

, by

(3.71)

and

(3.72)

we have

8π2∫︂2π

0∫︂π

0∫︂2π

m,n(φ¯1, θ

¯, φ¯2) sin θ

¯dφ¯1dθ

¯dφ¯2

4π∫︂2π

0∫︂π

e−imφ¯1dl

m,n(θ

¯) sin θ

¯dθ

¯dφ¯1

2π∫︂2π

e−inφ¯2dφ¯2

=∑︂

i∈I1

wi(S2)e−imφ1idl

m,n(θi)∑︂

j∈I2

wj(S1)e−inφ2j

For

n= 0

, note that

m,0(φ¯1, θ

¯, φ¯2)=(−1)m√︂4π

2l+1 Y−m

l(φ¯1, θ

¯)

and

∑︁

wj(S1) =

, hence

8π2∫︂2π

0∫︂π

0∫︂2π

m,0(φ¯1, θ

¯, φ¯2) sin θ

¯dφ¯1dθ

¯dφ¯2

=∑︂

wi(S2)(−1)m√︃4π

2l+ 1Y−m

l(φ¯1i, θ

¯i)

=∑︂

i∑︂

wi(S2)wj(S1)Dl

m,0(φ¯1i, θ

¯i, φ¯2j)

Now (3.73) is an immediate result of the expression that

f=∑︂

dl∑︂

m∑︂

n⟨f, Dl

m,n⟩SO(3)Dl

m,n

(3.74)

This lemma enables us to have a fully discretized frame, as a consequence

of Theorem 3.19. Let

ψ,Υ=1

2l+ 1 ∑︂

ak∈Υδ,ϵ

sk∥Plψak∥2

(3.75)

Corollary 3.23.

Under the assumptions of Theorem

3.19

and Theorem

3.21

{ψi,j,k =√︂wi(S2)wj(S1)skR(σi,j)ψak:i∈I1, j ∈I2,{ak}k⩾0∈Υδ,ϵ}

is a fully discrete frame for

L2(S2)

, where

σi,j = (φi, θi, ϕj)

. In particular, let

Plψ♯

i,j,k := Plψi,j,k/√︂Cl

ψ,Υ

(3.76)

3.2. CONSTRUCTION OF SPHERICAL FRAMES

then

{ψ♯

i,j,k}i,j,k

is super tight, and there is the reproducing formula:

f=∑︂

i′∈I1

wi′(S2)∑︂

i∈I1∑︂

j∈I2∑︂

ak∈Υ

f(φi′, θi′)ψ♯

i,j,k(φi′, θi′)ψ♯

i,j,k

(3.77)

for any

f∈L2(S2)

Proof.

Similar to Proposition 2.11, we can prove that

∑︂

i∈I1∑︂

j∈I2∑︂

ak∈Υδ,ϵ⟨f, ψi,j,k⟩L2(S2)ψi,j,k =∑︂

l∑︂

|m|⩽l

ψ,Υf

ˆlmYm

(3.78)

Since Theorem

3.21

implies that

ψ,Υ>0

, there is

f=∑︂

i∈I1∑︂

j∈I2∑︂

ak∈Υ⟨f, ψ♯

i,j,k⟩ψ♯

i,j,k

(3.79)

Using

(3.71)

again leads to the claimed result

(3.77)

I do not intend to give redundancy analysis of spherical frames in this dis-

sertation, and their numerical simulations are also left to my future work. How-

ever, at this point it is appropriate to mention briey their connection to solving

PDEs. Consider an operator equation

Su=g

(3.80)

where

can be a dierential operator or integral operator from

L2(S2)

into

itself.

Suppose that we have a multi-resolution structure such that

L2(S2) = ∞

⨁︁

j=0

in which each

has a nite frame. I have given such a structure for Shannon

type spherical wavelets in Example 3.17, though temporarily it has not been

done for spherical

-wavelets/shearlets. In this situation we can propose a

Galerkin scheme

⟨Su, ν⟩=⟨g, ν⟩

for

ν∈Vj

(3.81)

⟨Su, ψ⟩=⟨g, ψ⟩

for

ψ∈Wj

(3.82)

for

j= 0,··· , J

, and denote by

the solution of this nite element formulation.

In particular, under the assumption that

{ψj

i}i∈Ij

is a super tight frame for

is invertible, then there are explicite expressions for

and

, namely

uJ=

∑︂

j=0 ∑︂

i∈Ij⟨g, (S−1)∗ψj

i⟩ψj

(3.83)

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

and

u=∞

∑︂

j=0 ∑︂

i∈Ij⟨g, (S−1)∗ψj

i⟩ψj

(3.84)

although practically it is often not clear what is the inverse operator hence not

recommendable for computation.

Proposition 3.24.

Let

PWj

denote the projection onto

, we have the esti-

mation for the error

eJ=u−uJ

that

∥eJ∥L2=

∞

∑︁

j=J+1 ∥PWjS−1g∥2

∞

∑︁

j=J+1 ∥PWjS−1g∥L2

is a unitary operator, then

∥eJ∥L2=

∞

∑︁

j=J+1 ∥PSWjg∥2

∞

∑︁

j=J+1 ∥PSWjg∥L2

Proof.

This is an immediate consequence of the observation that the left-hand

side of (3.24) is equal to

sup

∥φ∥=1 ⟨eJ, φ⟩= sup

∥φ∥=1

∞

∑︂

j=J+1 ∑︂

i∈Ij⟨S−1g, ψj

i⟩⟨ψj

i, φ⟩

= sup

∥φ∥=1

∞

∑︂

j=J+1 ⟨PWjS−1g, φ⟩L2

∞

∑︁

j=J+1 ∥PWjS−1g∥2

∞

∑︁

j=J+1 ∥PWjS−1g∥L2

Similarly when

is unitary, there is

∥eJ∥L2= sup

∥φ∥=1

∞

∑︂

j=J+1 ∑︂

i∈Ij⟨g, Sψj

i⟩⟨Sψj

i,Sφ⟩

= sup

∥φ∥=1

∞

∑︂

j=J+1 ⟨PSWjg, Sφ⟩L2

3.3. A PRODUCT FORMULA ON SIMPLE SURFACES

which is further equal to the right-hand side of (3.24).

3.3 A product formula on simple surfaces

To avoid confusion with previous sections, throughout this section the symbol

is reserved for polyhedra and

denotes triangles on surfaces. For topologists,

triangulation of a topological manifold

means a simplicial complex together

with a homeomorphism from its geometric realization to the manifold. It is

a deep result in topology that every smooth surface has a triangulation and

manifolds of dimension less than four has a piecewise linear triangulation. Nev-

ertheless, I have no intention to delve into topological context or any of these

advanced questions, because the purpose in this small section is to establish a

formula for a newly introduced inner product on surfaces.

According to the denition from [70], a set of triangles in

is a

triangulation

(i) for

i=j

Ti∩Tj

is either empty or a common vertex or a common edge

(ii) the number of boundary edges incident on a boundary vertex is two

(iii)

⋃︁i

is simply connected.

Meanwhile a triangulation can be considered as embedding of the graph

and here we additionally assume that

(iv) every vertex has nite degree or

is locally nite.

A more general concept for a graph embedded into an oriented surface with-

out self-intersection is the so called

tessellation

, where the faces are not limited

to triangular ones, namely a graph that satisfy

(I) every edge is contained in two faces

(II) every two faces are either disjoint or intersect in one vertex or one edge

(III) every face is homeomorphic to a closed disc

Let

N(v)

be the set of neighboring vertices of

, and

|N(v)|=

deg

(v)

be the

degree of

. An edge

v1v2

simple

v′∈ N(v1)∩ N(v2)

, then

v′v1v2∈ T

When every edge is simple, the triangulation is called simple. Obviously for an

interior edge, it is simple if and only if

|N(v1)∩ N(v2)|

=2, where

|·|

means

cardinality.

Square summable maps on the graph form a Hilbert space

ℓ2(V)

with inner

product

(f, g)w=∑︂

v∈V

f(v)g(v)

deg

(v)

(3.85)

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

and the weighted combinatoric Laplacian reads

∆wg(v) = 1

deg(v)∑︂

v′∈N(v)(︂g(v′)−g(v))︂=−g(v)+ 1

deg(v)∑︂

v′∈N(v)

g(v′)

(3.86)

which is obviously a self-adjoint operator on the space

{f: ∆wf∈ℓ2(V)}

While at the same time due to the Green's formula

(−∆wf, g)w=1

2∑︂

v∑︂

v′

df([v, v′])dg([v, v′])δvv′

(3.87)

where

df([v, v′]) = f(v′)−f(v)

and

δvv′= 1

and

v′

are neighboring, zero

otherwise, it follows that

−∆w

is positive. Thus its spectrum

σ(−∆w) = {z∈

C:zI +∆w

has no bounded inverse

}

lies on the positive part of the real line. In

fact, Theorem

4.18

gives bounds estimation for

σ(−∆w)\{0}

, particularly the

lower bound for the rst minimal eigenvalue that is larger than zero.

By using radial projection method, once given a polyhedron with triangular

facades

T={T1,··· , TN}

, one get a natural triangulation

p(T)

on the sphere.

A norm adapted to the triangulation, as the square root of a newly introduced

inner product in

L2(∂Γ)

is dened as

⟨f, g⟩∂Γ=∑︂

T∈T

|T|∫︂T

f(x)g(x)dx

(3.88)

where we denote by

∂Γ

the surface of the polyhedron

, to distinguish this norm

from the

L2(M)

norm on the Riemannian manifold.

The pull-back of the norm

(3.88)

then gives a norm on the sphere. Fur-

thermore, a continuous function

piecewise-linear on triangle in

is uniquely

determined by the values

f(V)

, where

is the set of vertices in the triangula-

tion. Denote by the space of these functions by

, we generalize a useful result

in [28] and give a dierent but simpler proof here

Proposition 3.25.

Let

be a simple triangulation of an oriented surface with-

out boundary,

and

be two elements in

P1(T)

, and denote by

Vi={vi

k}k=1,2,3

the set of vertices of triangle

. Then

⟨f, g⟩∂Γ=1

24 ⎡

⎣2(f, g)w+∑︂

v∈V ∑︂

v′∈N(v)(︂f(v)g(v′) + g(v)f(v′))︂⎤

⎦

(3.89)

3.3. A PRODUCT FORMULA ON SIMPLE SURFACES

Proof.

Let

ϕs

and

ϕt

be two arbitrary nodal functions at vertices

and

re-

spectively. Firstly observe that, if

and

are contained in the same triangle

then

2|Ti|∫︂Ti

ϕs(x)ϕt(x)dx

=∫︂1

0∫︂1−λ2

(ϕs·ϕt)(λ1vi

1+λ2vi

2+ (1 −λ1−λ2)vi

3)dλ1dλ2

={︃1

24 s∼t

12 s=t

where

s∼t

means that they are neighboring vertices. Consequently,

Is,t =∑︂

Ti∈T

2|Ti|∫︂Ti

ϕs(x)ϕt(x)dx =∑︂

i∈N(s)∩N(t)

2|Ti|∫︂Ti

ϕs(x)ϕt(x)dx

={︃1/12 s∼t

deg

(s)/12 s=t

In the situation that

and

are not contained in any triangle, clearly there is

Is,t = 0

Since the nodal functions form a basis for

and a piecewise linear function

restricted to

is the linear combination of

{ϕv}v∈Vi

, we arrive at the result

that

∑︂

Ti∈T

2|Ti|∫︂Ti

f(x)g(x)dx

=∑︂

Ti∈T

2|Ti|∫︂Ti∑︂

v∈Vi

f(v)ϕv(x)·∑︂

v∈Vi

g(v)ϕv(x)dx

12 ∑︂

v∈V

deg

(v)f(v)g(v) + 1

24 ∑︂

v∈V ∑︂

v′∈N(v)(︂f(v)g(v′) + g(v)f(v′))︂

Note that

∥f∥2

L2(M)=∑︁

Ti∫︁Mf2(x)dx ⩽maxi|Ti|⟨f, f⟩∂Γ

. Thus when

has

normalized

L2(M)

norm and the triangulation becomes ner and ner such

that

maxi|Ti|

turns smaller and smaller,

⟨f, f⟩∂Γ

becomes unbounded. In other

words, they are not comparable on the same scale. However, Proposition 3.25

does allow us to compare

⟨f, f⟩∂Γ

with

(f, f)w

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

Corollary 3.26.

Under the assumption that each vertex in the triangulation

has degree no bigger than

for some

p⩾2

, there is

1 + 2

p⩽12⟨f, f⟩∂Γ

(f, f)w

⩽2

(3.90)

for any

f∈P1(T)⊂L2(∂Γ)

not identically zero on

Proof.

In the case that

f=g

(3.89)

reduces to

⟨f, f⟩∂Γ=1

12 ⎡

⎣(f, f)w+∑︂

v∑︂

v′

f(v)f(v′)δv,v′⎤

⎦

(3.91)

Thus on the right side

⟨f, f⟩∂Γ⩽1

12 ⎡

⎣(f, f)w+∑︂

v∑︂

v′

f(v)f(v′)δv,v′⎤

⎦

⩽1

12 ⎡

⎣(f, f)w+1

2∑︂

v∑︂

v′

(f(v)2+f(v′)2)δv,v′⎤

⎦

6(f, f)w

while the left side of

(3.90)

is obvious.

While in dierential geometry there are equivalent descriptions of one math-

ematical concept, in the discrete world dierent denitions motivated by the

various smooth properties do not usually lead to the same thing. For instance

in [8] a discrete Laplace operator is constructed through the discretization of

mean curvature, while another discrete Laplace operator can be formulated

based on a discrete analogue of Green's theorem(see [4] for instance),

∑︂

i| ∇g|Ti|2·|Ti|=−∑︂

g(vj)∆Tg(vj)w(vj)

(3.92)

where

w(vj)

is some area weight at vertices

On an oriented surface which has a consistent triangulation, the discrete

gradient of a function

can be dened at each triangle

as the solution

⟨∇g|Ti, vi

k−ci⟩=g(vi

k)−g(ci)i= 1,2,3

(3.93)

3.3. A PRODUCT FORMULA ON SIMPLE SURFACES

where

ci=1

∑︁

k=1

(vi

and the inner product is in the Euclidean sense. From this

denition it is clear that

⟨∇g|Ti, vi

j−vi

k⟩=g(vi

j)−g(vi

. Suppose

1=vi

3−vi

2=vi

1−vi

3=vi

2−vi

and

∇g|Ti=c1ei

1+c2ei

, then

G(︃c1

c2)︃=(︃⟨∇g|Ti, ei

1⟩

⟨∇g|Ti, ei

2⟩)︃=(︃g(vi

3)−g(vi

g(vi

1)−g(vi

3))︃

(3.94)

where

G=(︃⟨ei

1, ei

1⟩ ⟨ei

1, ei

2⟩

⟨ei

1, ei

2⟩ ⟨ei

2, ei

2⟩)︃

Thus, if we omit the index

without bringing confusion

(︃c1

c2)︃=1

det G·(︃⟨e2, e2⟩ −⟨e1, e2⟩

−⟨e1, e2⟩ ⟨e1, e1⟩)︃(︃g(v3)−g(v2)

g(v1)−g(v3))︃

and

(e1, e2)(︃c1

c2)︃=1

det G[−(⟨e2, e2⟩e1−⟨e1, e2⟩e2)g(v2)

+ (⟨e1, e1⟩e2−⟨e1, e2⟩e1)g(v1)

−(⟨e3, e2⟩e1−⟨e3, e1⟩e2)g(v3)]

Note that

[e1, e2] = e1eT

2−e2eT

1=eT

3eT

1−eT

1eT

3=eT

2eT

3−eT

3eT

, hence

∇g|Ti= (e1, e2)(︃c1

c2)︃=−1

4|Ti|2[e1, e2]·(g(v1)e1+g(v2)e2+g(v3)e3)

(3.95)

since

det G=|e1|2|e2|2sin2θ3= 4|Ti|2

There is

| ∇g|Ti|2

= (c1e1+c2e2)T(c1e1+c2e2)

= (c1c2)G(c1c2)T

4|Ti|2(︃g(v3)−g(v2)

g(v1)−g(v3))︃T(︃⟨e2, e2⟩ −⟨e1, e2⟩

−⟨e1, e2⟩ ⟨e1, e1⟩)︃(︃g(v3)−g(v2)

g(v1)−g(v3))︃

4|Ti|gT

iEigi

(3.96)

where

gi= (g(v1), g(v2), g(v3))T

and

3×3

matrix with elements

j,k =

⟨ej, ek⟩/|Ti|

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

Therefore,

∑︂

Ti∈T | ∇g|Ti|2·|Ti|=1

4∑︂

v∈V

g(v)∑︂

Ti∋v[︂|ev|2g(v) + ⟨ev, ev′⟩g(v′) + ⟨ev, ev′′ ⟩g(v′′ )]︂/|Ti|

which implies that we shall dene the discrete Laplace-Beltrami operator at

∆Tg|v=−1

4wv∑︂

Ti∋v

|Ti|[︂|ei

v|2g(v) + ⟨ei

v, ei

v′⟩g(v′) + ⟨ei

v, ei

v′′ ⟩g(v′′ )]︂

(3.97)

where

v′

and

v′′

are the other two vertices in the triangle

. It is clear that

when

is a constant function the above equality gives

∆Tg≡0

As a small step forward and a wonderful ending for this chapter, I give the

eigenvalues estimation for

∆T

. Without loss of generality, assume that

w(v) = 1

for all vertices. Let us also assume that the restriction

r⩽CT

(3.98)

applies to all the triangles uniformly, where

is the mesh size and

is the radius

of the largest interior circle inside a triangle.

Denition 3.27.

Let

be the subspace of

L2(M)

consisting of functions

such that on any

(︁g|Ti(vi0), g|Ti(vi1), g|Ti(vi2))︁T/∈ker Ei

Theorem 3.28.

On a triangular mesh satisfying

(3.98)

, there are

λmin(T) = inf

g∈U

−∑︁

g(vj)∆Tg(vj)

∑︁

g2(vj)⩾p¯

4κ¯(CT)

λmax(T) = sup

g∈U

−∑︁

g(vj)∆Tg(vj)

∑︁

g2(vj)⩽p

4κ(CT)

where

κ¯(CT) = √3CT−√︁3C2

T−12

κ(CT) = √3CT+√︁3C2

T−12

is the

maximum degree and

p¯

is the minimum degree of the vertices.

Proof.

Suppose the eigenvalues of

are reordered so that

0⩽λ0⩽λ1⩽λ2

based on the fact that

is positive semi-denite. Let

I={0,1,2}

be the index

3.3. A PRODUCT FORMULA ON SIMPLE SURFACES

set and denote by

θj(j∈I)

the interior angles of the triangle corresponding to

the edge

respectively. From the observation that

∏︂

j=0

λj=

det

|Ti|3⎛

⎝

∏︂

j=0 |ej|2−

∏︂

j=0 |ej|2|⟨ek, el⟩|2

k=l∈I\{j}+ 2⟨e0, e1⟩⟨e1, e2⟩⟨e2, e0⟩⎞

⎠

=|e0|2|e1|2|e2|2

|Ti|3(︁1−cos2θ0−cos2θ1−cos2θ2−2 cos θ0cos θ1cos θ2)︁

= 0

(3.99)

we see

λ0= 0

Besides, it holds that

∑︂

j=1

λj=

Ei=

∑︁

j=0 |ej|2

|Ti|2=

∑︂

j=0

sin θj

:= Si

(3.100)

and that

λ1λ2=1

|Ti|2(︁|e0|2|e1|2sin2θ2+|e1|2|e2|2sin2θ0+|e2|2|e0|2sin2θ1)︁= 12

(3.101)

Equations

(3.100)

and

(3.101)

together give us the eigenvalues

⎧

⎨

⎩

λ1=Si−√S2

i−48

λ2=Si+√S2

i−48

(3.102)

The the assumption

(3.98)

implies

sin θj

⩽CT

for any

, hence

⎧

⎨

⎩

sin θj⩽CT

cos θj

⩽2√3CT

3θj∈(0,π

3]∪[2π

3, π)

sin θj⩽4√3

3θj∈(π

3,2π

(3.103)

from which we deduce that

λ1⩽2√3⩽λ2

(3.104)

CHAPTER 3. EXTENSION TO MISCELLANEOUS RESULTS

Meanwhile it is easy to verify that

achieves minimum if and only if

θj=π

(j∈I)

. Thus it follows that

4√3⩽Si⩽2√3CT

(3.105)

and

{︃λ1⩾κ¯(CT)

λ2⩽κ(CT)

(3.106)

hold for any triangle in the triangulation.

Thus for any

g∈U\{0}

−∑︁

g(vj)∆Tg(vj)

∑︁

g2(vj)⩾

p¯∑︁

Ti∈T | ∇g|Ti|2·|Ti|

∑︁

Ti∑︁

v∈Ti

g2(v)

(3.107)

as a consequence of

(3.96)

. The inequality

(3.107)

holds because

∑︁

Ti∈T | ∇g|Ti|2·

|Ti|⩾1

4κ¯(CT)·∑︁

v∈Ti

g2(v)

for each

. Similarly we see that

−∑︁

g(vj)∆Tg(vj)

∑︁

g2(vj)⩽

p∑︁

Ti∈T | ∇g|Ti|2·|Ti|

∑︁

Ti∑︁

v∈Ti

g2(v)

(3.108)

which gives us the estimation

(3.28)

Remark 3.29.

automatically excludes piecewise constant functions, which

are eigenfunctions of

∆T

corresponding to eigenvalue 0. Furthermore,

λmin(T)

can be viewed as the minimum eigenvalue besides zero adapted to a suitable

triangulation, it is strictly positive and has lower bound depending on the value

according to

(3.28)

Chapter 4

Supporting topics

4.1 Orthogonal polynomials: a dierential

equation point of view

Sturm-Liouville theory has a long history dating back to the beginning of 19th

century, with thousands of papers and articles published under this topic. Here

we only give an elegant and short introduction to this profound theory, so as

to exhibit how various kinds of orthogonal polynomials can be derived from

dierential equations. For recent development and ongoing research in this area

we refer to monographs [3][96]; for a more detailed introduction we refer to [98].

Consider linear second-order ODE on

(interval, half line, real line etc.)

˜y)(x) := a0(x)y′′ (x) + a1(x)y′(x) + a2(x)y(x) = f(x)

(4.1)

where

a0>0

a.e. on

a0, a1∈Lloc(I)

. When necessary we can assume without

loss of generality that

a0= 1

and

(4.1)

is simplied to be

L0y:= y′′ +a1y′+a2y=f

(4.2)

otherwise equation

(4.1)

is called singular and the zero is called singular point.

Let

x0, a, b

be arbitrary points in

. For any given values

and

, the

general conditions imposed on

(4.1)

{︃α1y(a) + α2y′(a) + α3y(b) + α4y′(b) = c1

β1y(b) + β2y′(b) + β3y(a) + β4y′(a) = c2

(4.3)

CHAPTER 4. SUPPORTING TOPICS

with

∑︁

i=1 |αi|>0

and

∑︁

i=1 |βi|>0

. Especially, when

α3=α4=β3=β4= 0

and

a, b

are endpoints of

, we have the

separated boundary conditions

{︃α1y(a) + α2y′(a) = c1

β1y(b) + β2y′(b) = c2

(4.4)

when

a=b=x0

, we have the

initial conditions

y(x0) = c1, y′(x0) = c2

(4.5)

and

a0>0

a.e. on I,

a0, a1∈Lloc(I)

are in fact the minimal conditions required

to have a unique solution; the special case

y(a) = y(b), y′(a) = y′(b)

(4.6)

is called

periodic boundary condition

. Unlike the initial value problems, sepa-

rated boundary value problems

(4.4)

do not always have solutions.

Assume

a0= 1

. If the coecients

a1(x)

and

a2(x)

are both analytic

at some point

, then we obviously have an (locally) analytic solution of the

equation

(4.2)

of the form

∞

∑︁

n=0

cn(x−x0)n

The space of solutions of

(4.2)

L2(I)∩C2(I)

is closed under linear com-

bination. Integral by parts shows that

˜:L2(I)∩C2(I)↦→ L2(I)

satises

⟨L

˜f, g⟩=⟨f, (a0g)′′ −(a1g)′+a2g⟩+ [a0(f′g−fg′)+(a1−a′

0)fg]|b

=⟨f, L

˜∗g⟩+ [a0(f′g−fg′)+(a1−a′

0)fg]|b

(4.7)

Then

˜=L

˜∗≡(a0)d2

dx2+ (2a0

′−a1)d

dx + (a0

′′ −a1

′+a2)

⇔a0=a0,2a0

′−a1=a1, a0

′′ −a1

′+a2=a2

(4.8)

⇔a0, a1

and

are real and

a1=a′

(4.9)

In this case,

is called formally self-adjoint and

˜=d

dx(a0(x)d

dx) + a2(x)

(4.10)

The equation

˜y= 0

with

formally self-adjoint was rstly studied by Charles

Sturm and Joseph Liouville in the 1830's, and there all the coecients and

4.1. ORTHOGONAL POLYNOMIALS: A DIFFERENTIAL

EQUATION POINT OF VIEW

solutions are of two variables

and

. Under the additional assumption that

is strictly positive and decreasing in

is increasing function of

a2(a,r)

y(a,r)

∂y

∂x (a, r) = h(r)

where

solves

Ly = 0

and

is a given decreasing

function of

, Sturm claims that

a2(a,r)

y(a,r)

∂y

∂x (a, r) = h(r)

is decreasing in

for any

x∈I

By multiplying a positive weight function

to a general

˜=a0(x)d2

dx2+

a1(x)d

dx +a2(x)

, we can make

be formally self-adjoint. In fact, by (

4.9

)

is formally self-adjoint if

w′a0+wa′

0=wa1

, or equivalently

|w(x)|=|a0(a)w(a)|

|a0(x)|exp(∫︂x

a1(t)

a0(t)dt)

(4.11)

Denition 4.1.

(Sturm-Liouville Eigenvalue Problem)

Let

be formally self-adjoint operator satisfying

(4.10)

. Solve the eigenvalue

equation

−Ly =λw(x)y

(4.12)

w(I)

subject to the separated homogeneous boundary conditions

{︃α1y(a) + α2y′(a)=0

β1y(b) + β2y′(b)=0

(4.13)

How about the linear dependence of two solutions of equation

(4.12)

? If two

solutions

and

are linearly dependent, then the Wronskian

W(y1, y2)(x) = y1(x)y2(x)

y′

1(x)y′

2(x)

(4.14)

is identically zero. Conversely,

a0(y1y′′

2−y2y′′

1) + a′

0(y1y′

2−y2y′

1)=0

gives us

the identity

W′(x) + a′

0(x)

a0(x)W(x)=0

(4.15)

Thus

W(y1, y2)(x) = c·exp(−∫︁x

a′

0(t)

a0(t)dt)

for any

x∈I

W(y1, y2)(x)=0

for

some

x⇒c= 0 ⇒W(y1, y2)(x)≡0

. Hence we have arrived at the following

conclusion.

Lemma 4.2.

Solutions

and

(4.12)

are linearly independent if and only

W(y1, y2)(x)= 0

CHAPTER 4. SUPPORTING TOPICS

Now according to standard existence theorem of ODE, there exist unique

solutions

and

of the homogeneous equation

Ly = 0

such that

y1(a) = α2

y′

1(a) = −α1

;

y2(b) = β2

y′

2(b) = −β1

. Thus

{︃Ly1= 0

α1y1(a) + α2y′

1(a)=0

(4.16)

and

{︃Ly2= 0

β1y2(b) + β2y′

2(b)=0

(4.17)

In order to construct Green's function we need

and

to be linearly

independent(

and

are independent i

α1y2(a) + α2y′

2(a)= 0

). This is

guaranteed under the assumption that

0/∈spec(L)

, since otherwise

is a

multiple of

, and

solves both

(4.16)

and

(4.17)

, namely

solves the SL

eigenvalue problem with

λ= 0

, a contradiction. Furthermore, as we shall see

soon,

0/∈spec(L)

is a reasonable assumption.

Denition 4.3.

Green's function is dened as

G(t, s) = {︄y1(s)y2(t)

a0(s)W(y1,y2)(s)

a⩽s⩽t⩽b

y1(t)y2(s)

a0(s)W(y1,y2)(s)

a⩽t⩽s⩽b

(4.18)

using the fact that

a0(s)= 0

and

W(y1, y2)(s)= 0

Since

[a0W(y1, y2)]′=y1Ly2−y2Ly1= 0

(4.19)

the denominator of Green's function

a0W(y1, y2)

turns out to be a nonzero

constant

. Thus the following properties of Green's function are satised:

(1)

is symmetric

(2)

is continuous on

I×I

, and belongs to

except for the line

t=s

lim

δ→0+

∂G

∂t (s+δ, s)−∂G

∂t (s−δ, s) = lim

δ→0+c−1[y1(s)y′

2(s+δ)−y′

1(s−δ)y2(s)]

=c−1W(y1, y2)(s)

a0(s)

(4.20)

(3)

is in the kernel space of

, namely

LG(·, s) = c−1y1(s)Ly2(

c−1(Ly1)y2(s))

= 0

(4.21)

4.1. ORTHOGONAL POLYNOMIALS: A DIFFERENTIAL

EQUATION POINT OF VIEW

Proposition 4.4.

The operator

T:C(I)→C2(I)

Tf(t) := ∫︂I

G(t, s)f(s)ds ∈C2(I)

(4.22)

solves the equation

(4.1)

, i.e.

LTf =f

Proof.

(Tf)′(t) = ∫︂t

Gt(t, s)f(s)ds +G(t, t−)f(t−)

+∫︂b

Gt(t, s)f(s)ds −G(t, t+)f(t+)

=∫︂t

Gt(t, s)f(s)ds +∫︂b

Gt(t, s)f(s)ds

(4.23)

due to the continuity of

(Tf)′′ (t) = ∫︂t

Gtt(t, s)f(s)ds +Gt(t, t−)f(t−)

+∫︂b

Gtt(t, s)f(s)ds −Gt(t, t+)f(t+)

=∫︂t

Gtt(t, s)f(s)ds +∫︂b

Gtt(t, s)f(s)ds +f(t)

a0(t)

(4.24)

by the same argument as (

4.20

Hence

L(Tf)(t) = a0(t)(Tf)′′ (t) + a′

0(t)(Tf)′(t) + a2(t)Tf(t)

= (∫︂t

+∫︂b

)LtG(t, s)f(s)ds +f(t)

=f(t)

(4.25)

due to

(4.21)

CHAPTER 4. SUPPORTING TOPICS

g∈C2(I)

satises

(4.13)

TLg(t)=(∫︂t

+∫︂b

)[ d

ds(a0(s)dg(s)

ds ) + a2(s)g(s))]G(t, s)ds

=a0(s)dg(s)

ds G(t, s)|b

a−a0(s)g(s)Gs(t, s)|t

a−a0(s)g(s)Gs(t, s)|b

+ (∫︂t

+∫︂b

)LsG(t, s)g(s)ds

=a0(s)g(s)Gs(t, s)|t+

t−+a0(s)[g′(s)G(t, s)−g(s)Gs(t, s)]|b

=g(t)

(4.26)

where we have used

(4.20)

and the fact that both

and

satisfy

(4.13)

Here I give a lemma and leave its proof as an exercise.

Lemma 4.5.

is a self-adjoint compact operator on a Hilbert space

, then

there exists an eigenvalue

such that

|λ|=∥A∥

Remark 4.6.

dened in

(4.22)

is equicontinuous and uniformly bounded

due to the fact that

G(·, s)

is uniformly continuous on

, or equivalently

is compact as we have shown in Lemma 4.8. Thus Arzel

Ascoli Theorem

applies and the Lemma 4.5 holds for

. To prove

∥A∥= sup

∥x∥=1 |⟨Ax, x⟩|

, no

matter we deal with a Hilbert space

C(I)

, we only have to show that

2Re⟨Tu, v⟩⩽(∥u∥2+∥v∥2) sup |⟨Tu, u⟩|

and then replace

with

Tu/∥Tu∥

The following theorem describes the distribution of eigenvalues of a linear,

self-adjoint compact operator.

Theorem 4.7.

(Hilbert-Schmidt) Let

be a linear, self-adjoint, compact op-

erator on a Hilbert space

with

dim(H) = ∞

. All eigenvalues of

are real

and can be ordered so that

|λn+1|⩽|λn|,lim

n→∞λn→0

(4.27)

Furthermore, eigenvectors

{φn}n∈N

can be chosen to be an ONB of

Ran(A)

In particular, when

ker(A)=0

, each element

h∈H

has an expansion

∑︁

λn⟨h, φn⟩φn

4.1. ORTHOGONAL POLYNOMIALS: A DIFFERENTIAL

EQUATION POINT OF VIEW

Proof.

By Lemma 4.5, there exists

φ1

such that

∥φ1∥= 1

and

Aφ1=λ1φ1

with

|λ1|=∥A∥

. Let

H1=span{φ1}⊥

, then

A|H1

is another self-adjoint compact

operator with operator norm

∥A|H1∥⩽∥A∥

. It has an eigenvector

φ2

such that

A|H1φ2=λ2φ2

with

λ2=∥A|H1∥⩽λ1

. Continuing the same argument gives

us a sequence of eigenvalues such that

|λ1|⩾|λ2|⩾···

Assume eigenvalues of

are bounded away from zero, namely

|λn|⩾b > 0

Then the fact that

{bφn}n∈N⊂A(BH(0,1))

has no convergent subsequence(

φn

orthonormal) contradicts the compactness of

Finally, if

x∈span{φ1, φ2,···}⊥

, then

∥Ax∥⩽λn∥x∥

for each

n∈N

, hence

Ax = 0

. If

ker(A) = {0}

, then

H=span{φ1, φ2,···}

Lemma 4.8.

k∈L2

w(I×I)

, then integral operator

Kw:L2

w(I)→L2

w(I)

Kwf(t) := ∫︁Ik(t, s)f(s)w(s)ds

is compact.

Proof.

Let

{ϕi}

be an ONB of

w(I)

and suppose

k(x, y) = ∞

∑︁

i,j=1

kijϕi(x)ϕj(y)

Then

∥k∥2

w(I×I)=∑︁

ij |kij|2

. For any

f∈L2

w(I)

, denote by

fj=⟨f, ϕj⟩w

, we

have

⟨Kwf, Kwf⟩w=∫︂I⎛

⎝∞

∑︂

i,j=1

kijϕi(t)fj⎞

⎠⎛

⎝∞

∑︂

i′,j′=1

ki′j′ϕi′(t)fj′⎞

⎠w(t)dt

=∑︂

i∑︂

j∑︂

j′

kijkij′fjfj′

⩽√︄∑︂

i,j,j′

(kijfj′)2∑︂

i,j,j′

(kij′fj)2

=∥f∥2

w∥k∥2

(4.28)

namely

belongs to

Given any

n∈N

Knf:=

∑︁

i,j=1

kij⟨f, ϕi⟩ϕj

is a bounded operator with

nite dimensional range, hence compact.

∥Kw−Kn∥2=∞

∑︁

i,j=n+1 |kij|2→0

namely

Kn→Kw

in operator norm, thus we claim that

is compact. Indeed,

let

{fm}

be a bounded sequence, Arzelá-Ascoli argument tells us that there

exists a subsequence

{fmj}

such that, for each

n∈N

Knfmj

is convergent,

hence

{Kwfmj}

a Cauchy sequence. Thus the completeness of

w(I)

gives us a

convergent subsequence

{Kwfmj}

CHAPTER 4. SUPPORTING TOPICS

Let

Twf=∫︂I

G(t, s)f(s)ds

(4.29)

By Proposition

4.4

, we see that

wLTwf=f

(4.30)

for positive

. Note that if

yi∈L2

w(I)(i= 1,2)

, then

G∈L2

w(I×I)

, hence

is linear, self-adjoint and compact. Since the eigenvalues of

goes to zero,

as a left-inverse operator, the Sturm-Liouville operator has eigenvalues

{λn}

which is increase and goes to innity. When

is some eigenfunction of

corresponding to eigenvalue

˜n>0

, then

Lyn=1

˜nwyn

. Thus we have proved

the following result that I intend to give in this section, which allows us to

study the functions on

in terms of various kinds of orthogonal polynomials

with dierent properties.

Theorem 4.9.

The eigenfunctions of SL-problem

(4.12)

form a complete ONB

for

w(I)

, with

strictly positive.

Before I proceed to exhibit examples of orthogonal polynomials, let us have

a brief look at the zeros of the solution functions, which is a topic that attracted

several generations of mathematicians. The zeros of a non-trivial solution

(4.12)

(a, b)

are isolated from the simple observation that,

u(x0)

and

u′(x0)

cannot be zero at the same time, hence

must increase or drop in a neighbor-

hood of the zero

Assume

I′⊂I

is an subinterval on which a solution

(4.12)

does not

vanish. If

is another solution that is independent of

, then

u1(a0u′

2)−

u2(a0u′

1) = a0W(u1, u2)

is a non-zero constant

cˆ

, hence we have

(u2

)′(x) = cˆ

a0(x)u2

1(x),

for

x∈I′

a.e. (4.31)

Integrating on

(a′, x)⊂I′

yields

(x) = u2

(a′) + ∫︂x

a′

cˆ

a0u2

(4.32)

Suppose

and

are two linearly independent solutions. When

and

are successive zeros of a solution

, from

W(u1, u2)= 0

, we shall have

u′

1(x1)u2(x1)

and

u′

1(x2)u2(x2)

are either both positive or negative. Since

u′

1(x1)

and

u′

1(x2)

have opposite sign,

u2(x1)u2(x2)<0

. There must exist exactly one

point

x3∈(x1, x2)

such that

u2(x3) = 0

, for otherwise it would contradicts the

assumption that

and

are successive zeros of

. We conclude that

4.1. ORTHOGONAL POLYNOMIALS: A DIFFERENTIAL

EQUATION POINT OF VIEW

Proposition 4.10.

The zeroes of two linearly independent solutions of

(4.12)

must intertwine with each other.

Now we arrive at some famous polynomials coming as the eigenfunctions of

the Sturm-Liouville eigenvalue problems.

Example 4.11.

(Laguerre Polynomials)

Laguerre polynomials

Ln,α(x) = exx−α

dxnxn+αe−x

(4.33)

are non-singular solutions of

xu′′ + (1 + α−x)u′+nu = 0,0<x<∞

(4.34)

with

α > −1

and

non-negative integer; especially when

α= 0

it is equivalent

(xe−xu′)′+ne−xu= 0

(4.35)

Obviously in this example

a0(x)

vanishes at

x= 0

. Those polynomials are also

eigenfunctions of the Sturm-Liouville operator in (4.35) with respect to eigen-

values

λn=n(n+1)

, with the orthogonality property that

⟨Ln,α, Lm,α⟩e−xxα=

δn,m

e−xxα(0,∞)

. Its series expression is

Ln,α(x) =

∑︂

k=0

Γ(n+α+ 1)

Γ(k+α+ 1)

(−x)k

k!(n−k)!

(4.36)

Besides, they satisfy the recurrence relation

nLn,α(x) = (2n+α−1−x)Ln−1,α(x)−(n+α−1)Ln−2,α(x)

(4.37)

Example 4.12.

(Legendre Polynomials)

When we solve the Laplace equation in

∂

∂r(r2∂u

∂r ) + 1

sin2φ

∂2u

∂θ2+1

sin θ

∂

∂φ(sin φ∂u

∂φ)=0

(4.38)

by assuming that the solution

is independent of

and using the method

of separation of variables, we obtain the two ODEs with respect to

and

respectively, namely

r2v′′ + 2rv′−λv = 0

(4.39)

CHAPTER 4. SUPPORTING TOPICS

and

sin φ(sin φw′)′+λw = 0

(4.40)

A change of variable

u= cos φ

gives the Legendre's equation

(1 −x2)u′′ −2xu′+λnu= 0,−1<x<1

(4.41)

which has singular points

x= 1

, where

λn=n(n+ 1)

n∈N

and the corre-

sponding eigenfunctions

are Legendre polynomials, having the orthogonality

⟨Pn, Pm⟩=2

2n+1 δm,n

L2(−1,1)

if we assume

Pn(1) = 1

From

(1.19)

it can be derived that

(1 −x2)d

dxPl(x) = l(Pl−1(x)−xPl(x))

(4.42)

and that

(1 −x2)d

dxPl(x)=(l+ 1)(xPl(x)−Pl+1(x))

(4.43)

Taking derivative on both sides of the above equations and using

(4.41)

give

lPl(x) = xd

dxPl(x)−d

dxPl−1(x)

(4.44)

and

(l+ 1)Pl(x) = −xd

dxPl(x) + d

dxPl+1(x)

(4.45)

Example 4.13.

(Hermite Polynomials)

Probably the most well known polynomials in Fourier analysis is the Hermite

polynomials, due to the fact that they constitute eigenfunctions of the Fourier

transform on

, namely

ˆn= (−i)nHn

. They solve the Hermite's equation:

u′′ −2xu′+ 2nu = 0,−1<x<1

(4.46)

(e−x2u′)′+ 2ne−x2u′= 0,−1<x<1

(4.47)

Therefore, they are at the same time eigenfunctions

Hn= (−1)nex2dn

dxne−x2

the SL operator in

(4.47)

with respect to eigenvalues

λn= 2n

. By induction,

Hn(x) = (2x)n+ (−1)np(x)

where the polynomial degree of

p(x)

is less than

In fact, it has the series expression

Hn(x) =

⌞n/2⌟

∑︂

k=0

(−1)kn!

k!(n−2k)!(2x)n−2k

(4.48)

4.1. ORTHOGONAL POLYNOMIALS: A DIFFERENTIAL

EQUATION POINT OF VIEW

Setting

H−1≡0

, the recurrence relation reads

Hn(x) = 2xHn−1(x)−2nHn−2(x)

(4.49)

Hermite polynomials have the orthogonality property in

e−x2(R)

⟨Hn, Hm⟩e−x2=∫︂∞

−∞

dxnHm(x)e−x2dx = 2nn!√πδm,n

(4.50)

where we used the fact that

dxnHm(x) = 0

for polynomial degree

m<n

Besides, it is easy to verify that Hermite polynomials' generating function is

e2xt−t2=∞

∑︁

n=0

n!Hn(x)tn

Example 4.14.

(Bessel Functions)

Bessel's equation takes the form

x2u′′ +xu′+ (x2−ν2)u= 0

(4.51)

where

is a nonnegative parameter. It obviously has singular point at

x= 0

Here the SL eigenvalue problem generates non-polynomial eigenfunctions

Jν

called the Bessel functions.

In fact, by assuming

u=xs∞

∑︁

k=0

ckxk

and collecting the coecients of the

powers

xs, xs+1,···

, we get

s=ν

and

ck=−1

k(k+2ν)ck−2

, hence

c2m=

(−1)m

2ν+2mm!Γ(ν+m+1) x2m

and

c2m+1 = 0

. Let

Jν(x) := (x

2)ν∞

∑︂

m=0

(−1)m

m!Γ(ν+m+ 1)(x

2)2m

(4.52)

then

lim

x→0+Jν(x) = {︃1ν= 0

0ν > 0

(4.53)

tells us that

Jν

is well dened for

ν⩾0

. For

t=−ν

, we can similarly dene

J−ν(x) := (x

2)−ν∞

∑︂

m=0

(−1)m

m!Γ(ν+m+ 1)(x

2)2m

(4.54)

but it is not necessarily bounded at

x= 0

. In fact, when

ν=n

J−ν= (−1)nJν

;

but for

ν∈R+\N0

lim

x→0+|J−ν|=∞

. Thus we have arrived at the following

conclusion.

CHAPTER 4. SUPPORTING TOPICS

Lemma 4.15.

Bessel functions

Jν

and

J−ν

are linearly independent i

ν∈

R+\N0

If we take

a= 0

and

b < ∞

, the scaled Bessel Functions form an orthogonal

and complete basis in

x(0, b)

, namely

⟨Jn(√︁λjx), Jn(√︁λkx)⟩x=δj,k

(4.55)

where the eigenvalue

λk= (xnk

b)2

and

xnk

is the k-th zero of

, i.e.

0< xn1<

xn2<··· < xnk <···

, coming as the eigenvalue of the scaled Bessel's equation

xu′′ +u′+ (λx −ν2

x)u= 0

(4.56)

We are not going to prove those, but rather refer to [13] for a thorough and

exquisite exposition of the properties of Bessel functions.

4.2 Spectrum of discrete Laplacian

Given a nite subgraph

, analogues of the length of boundary of a

submanifold

in the continuous setting could be the number of joint edges of

vertex set in

with its complement vertex set

|E(∂vH)|=|{xy ∈E(G) : x∈V(H), y ∈V(G)\V(H)}|

(4.57)

or probably more naturally dened as the number of edges between faces in

and its complement part, namely

|E(∂fH)|=|{xy ∈E(H)∩E(F(G)\F(H))}|

(4.58)

Similarly, the surface area of

can be measured either by

A(H) = ∑︁

v∈V(H)

deg

(v)

or by the number of faces

|F(H)|

. These lead to two dierent isoperimetric

constants measuring the ratio of length and area, that are closely connected to

the curvature, namely

α(G) = inf

0<|H|⩽1

2|V(G)|{|E(∂vH)|/A(H)}

(4.59a)

α∗(G) = inf

0<|H|⩽1

2|V(G)|{|E(∂fH)|/|F(H)|}

(4.59b)

where

α(G)

is called Cheeger constant.

4.2. SPECTRUM OF DISCRETE LAPLACIAN

Let

C0(G)

be the space of real valued functions of vertices with inner product

(g1, g2) = ∑︂

v∈V(G)

g1(v)g2(v)

and

{R}

be the set of real constant functions on

. The set of 1-forms

C1(G)

dened as functions on oriented edges satisfying

ω([x, y]) = −ω([y, x])

endowed

with inner product

(ω1, ω2) = ∑︂

ω1(e)ω2(e)

In particular, there is

dg([x, y]) = ∇xyg=g(y)−g(x)

for any

g∈C0(G)

. Sim-

ilarly, we can dene

C0(G, H)⊂C0(G)

consisting of those functions vanishing

on subcomplex

. It follows that

(dg1, dg2) = ∑︂

[x,y]

(g1(y)−g1(x)) (g2(y)−g2(x))

=∑︂

x∑︂

y∈N(x)

(g1(x)−g1(y)) g2(x)

where

[x, y]

run over all the possible edges in

E(G)

with an arbitrarily chosen

direction on each of them. Thus the Laplacian on

can be dened as

∆Gg(x) = −d∗dg(x) = ∑︂

y∈N(x)

(g(y)−g(x)) = ∑︂

y∈N(x)

g(y)−

deg

(x)·g(x)

(4.60)

The restricted Laplacian

∆H

on subcomplex

can be dened correspondingly

∆G

restricted on subspace

C0(G, H)

is an innite connected graph, it was proved in [75] under the geometric

assumptions that

deg

(v)

is uniformly bounded above by some constant

there exists positive constant

such that

γV (H)⩽V(∂H)

(4.61)

for any nite subcomplex

H < G

, the minimal positive eigenvalue of

−∆H

(and

−∆G

) has lower bound

γ2

(respectively).

Theorem 4.16.

H\∂H

is connected, then the minimal eigenvalue of

−∆H

σ(−∆H)\{0}

satises

λmin = inf

g∈C0(H,∂H)\{R}

(−∆Hg, g)

(g, g)⩾γ2

(4.62)

CHAPTER 4. SUPPORTING TOPICS

and the inmum can be achieved by a positive, subharmonic and nonconstant

function

. As

asymptotically approach

, if follows that the minimal positive

eigenvalue of

−∆G

satises

(4.62)

with a positive, subharmonic and nonconstant

eigenfunction.

Proof.

The idea of the proof is based on estimation of the quantity

∑︂

ed(g2)(e)=∑︂

[x,y]g2(x)−g2(y)

where

g∈C0(H, ∂H)

while

[x, y]

run over the set of edges. On the one hand,

there is

∑︂

ed(g2)(e)⩽⎛

⎝∑︂

[x,y]|g(x) + g(y)|2⎞

⎠

1/2

·⎛

⎝∑︂

[x,y]|g(x)−g(y)|2⎞

⎠

1/2

⩽√2⎛

⎝∑︂

[x,y]

(g2(x) + g2(y))⎞

⎠

1/2

·(dg, dg)1/2

⩽√︁2p(g, g)1/2(dg, dg)1/2

(4.63)

On the other hand, note that for any

g∈C0(H, ∂H)\{R}

(g(x)−g(y))2

(g, g)⩾(|g(x)|−|g(y)|)2

(|g|,|g|)⩾inf

g /∈{R}

(−∆Hg, g)

(g, g)

hence if

is an eigenfunction corresponding to the smallest positive eigenvalue

λmin

, so is

|g|

. In fact, by the connectness assumption,

g=|g|

, for otherwise

has negative value, then there exists

and its neighboring point

such that

(g(x)−g(y))2>(|g(x)|−|g(y)|)2

, which leads to the absurdity that

λmin >

λmin

. There cannot be a point

x∈H\∂H

such that

g(x) = 0

, for otherwise

−λming(x) = ∆Hg(x) = ∑︁

y∈N(x)

(g(y)−g(x)) ⩾0

, hence

is identically zero on

, a contradiction as well. Thus we see the positivity of the eigenfunction

Now suppose the values set of

is ordered as

0 = β0< β1<··· < βN

4.2. SPECTRUM OF DISCRETE LAPLACIAN

We have

∑︂

g(x)=βi

∑︂

k=1 ∑︂

{y∈N(x):g(y)=βi−k}(︁g2(x)−g2(y))︁

=∑︂

g(x)=βi

∑︂

k=1 ∑︂

{y∈N(x):g(y)=βi−k}

∑︂

t=i−k+1 (︁β2

t−β2

t−1)︁

=∑︂

g(x)=βi

∑︂

t=1 ∑︂

k⩾i−t+1 ∑︂

{y∈N(x):g(y)=βi−k}(︁β2

t−β2

t−1)︁

=∑︂

g(x)=βi

∑︂

t=1 (︁β2

t−β2

t−1)︁∑︂

k⩾i−t+1 |{y∈ N(x) : g(y) = βi−k}|

Thus

∑︂

ed(g2)(e)=

∑︂

i=1 ∑︂

g(x)=βi

∑︂

t=1 (︁β2

t−β2

t−1)︁∑︂

k⩾i−t+1 |{y∈ N(x) : g(y) = βi−k}|

∑︂

t=1 (︁β2

t−β2

t−1)︁∑︂

g(x)⩾βt

|{y∈ N(x) : g(y)⩽βt−1}|

(4.64)

∆Hg(x)<0

indicates that there is at least a neighboring point

such that

g(y)< g(x)

. By

(4.61)

there is

∑︂

ed(g2)(e)⩾

∑︂

t=1 (︁β2

t−β2

t−1)︁|∂{x:g(x)⩾βt}|

⩾γ

∑︂

t=1 (︁β2

t−β2

t−1)︁|{x:g(x)⩾βt}|

=γ[︄β2

N|{g(x) = βN}|+

N−1

∑︂

t=1

β2

t(|{g(x)⩾βt}|−|{g(x)⩾βt+1}|)]︄

⩾γ(g, g)

(4.65)

Combining

(4.63)

and

(4.65)

gives the desired result

(4.62)

. Besides, if

λmin

is obtained by function

, then

−∆Hg=λming > 0

, namely

is subharmonic

and nonconstant.

being positive and subharmonic also gives that Harnack

CHAPTER 4. SUPPORTING TOPICS

inequality

deg

(y)g(x)⩽g(y)⩽

deg

(y)g(x)

(4.66)

which follows immediately from denition.

Choose an arbitrary point in

, and we can form a nite subgraph

including vertices connected to

by at most

edges from

E(G)

. The

second part of the assertion comes as the limit of the smallest eigenvalue

λn

and associated eigenfunctions of

∆Hn

. In fact, let

gn∈C0(Hn, ∂Hn)\{R}

such

that

gn(x0)=1

, where

∂Hn

consists of points in

that have at least one

neighboring point not in

. Notice that

C0(Hn, ∂Hn)⊂C0(Hn+1, ∂Hn+1)

hence

λn= min

g∈C0(Hn,∂Hn)\{R}

(−∆Hg,g)

(g,g)⩾λn+1 ⩾γ2

and there exists

λ⩾γ2

the limit of

λn

Denote by

the eigenfunction corresponding to

λn

. Then by

(4.66)

, if

vertex

x∈G

is of

-edge distance to

, there is

deg

(y)min{n,s}gn(x)⩽gn(y)⩽

deg

(y)min{n,s}gn(x)

(4.67)

and if

n < s

gn(x) = 0

. Therefore

{gn(x)}n⩾0

is a bounded sequence. A

diagonal argument gives a subsequence such that

lim

k→∞gnk(x)

exists for all

vertices in

, and we dene through pointwise value a function

. Clearly

h(x0) = lim

n→∞gn(x0) = 1

, and

−∆Gh=λh ⩾0

. In fact, for any

(4.67)

gives that

h(x)

is strictly positive, namely

is positive and nonconstant.

Remark 4.17.

In the original paper [75], it was somehow miscalculated that

∑︁

ed(g2)(e)=

∑︁

t=1 (︁β2

t−β2

t−1)︁|∂{x:g(x) = βt}|

, although the nal conclusion

remains correct. In fact, we see in

(4.65)

that this holds only with an inequality,

and the precise expression for

∑︁

ed(g2)(e)

is given in

(4.64)

, which is exactly

∑︁

t=1

(β2

t−β2

t−1)|E(∂vLt)|

with

Lt:= {x:g(x)⩾βt}

. Probably noticed this im-

precision, in a later work[76] the same author changed the setting for Laplacian

and used a dierent assumption

(4.59)

in replacement of

(4.61)

. Thus the same

100

4.2. SPECTRUM OF DISCRETE LAPLACIAN

argument as in

(4.65)

applies and gives the estimation

∑︂

ed(g2)(e)⩾α

∑︂

t=1

A(Lt)

=α(β2

t−β2

t−1)∑︂

x∈Lt

deg

(x)

=α

N−1

∑︂

t=1

β2

t∑︂

g(x)=βt

deg

(x) + αβ2

deg

(x)

=α∑︂

x∈H

deg

(x)g2(x)

=α(g, g)w

(4.68)

Meanwhile

(4.63)

gives

∑︁

ed(g2)(e)⩽√2(g, g)1/2

w(dg, dg)1/2

. Together it gives

the minimal positive eigenvalue estimation

λ∗

min ⩾α2

for weighted Laplacian

∆w

. The positivity of

was given in same work under the

assumption that deg

(x)⩾7

for all vertices

x∈V

, for that there is

α(G)⩾1

The other situations are veried as the main result of

[

]

. In fact, it is proved

there that

κ(v)<0

for every

v∈V(G)

implies

α∗(G)>0

; while

χ(f)<0

for

every face in

F(G)

implies

α(G)>0

. Here

χ(f)=1−|f|

2+∑︂

v∈f

deg

(v)

(4.69)

is the

Euler-characteristic

with

|f|

the number of vertices contained in

, and

κ(v)=1−

deg

(v)

2+∑︂

{f:v∈f}

|f|

(4.70)

is the

combinatorical curvature

at a vertex

. The discrete Gauss-Bonnet theo-

rem says

∑︂

κ(v)=2−

genus (4.71)

where the genus as usual is dened as the maximal numbers of nonintersecting

simple closed curves that can be drawn on a surface without separating it. For

a proof of this fundamental result I refer the readers to [50].

101

CHAPTER 4. SUPPORTING TOPICS

Theorem 4.18.

On a weighted nite graph without loops, there is

α2(G)

2⩽λ∗

min ⩽min{2α(G),V(G)−2

|V(G)|−1}

and

max{2−2α(G),|V(G)|

|V(G)|−1}⩽λ∗

max ⩽2−α2(G)

Proof.

The lower bound for

λ∗

min

has been given in the previous remark. In the

upper bound aspect, given any nonempty

W⊂V(G)

, by letting

g♯

W(x) = {︄1x∈W

−A(W)

A(Wc)x∈Wc

(4.72)

we get

(−∆wg♯

W, g♯

W)w

(g♯

W, g♯

W)w

=∑︁

x∈∂W

(1 + A(W)/A(Wc))2A(x)

A(W) + ∑︁

x∈Wc

A(x)A2(W)/A2(Wc)

=A(∂W)A(V)

A(Wc)A(W)

⩽2α

(4.73)

Let

δv

be orthonormal basis on

V(G)

such that

δv(v′)=1

for

v′=v

, and

zero otherwise. The trace of the discrete Laplacian is

∑︂

i=0

λ∗

i=∑︂

v∈V

(∆wδv, δv)

(δv, δv)

2∑︂

v∈V

∑︁

u′,v′(︂δv(u′)−δv(v′))︂2δv′,u′

∑︁

v′

δ2

v(v′)A(v′)

=|V|

(4.74)

Therefore

λ∗

N=λ∗

max ⩾|V|

(4.75)

102

4.2. SPECTRUM OF DISCRETE LAPLACIAN

where

N+ 1 = |V|

. Meanwhile, let

be the eigenfunction of the largest

eigenvalue

λ∗

, we have

λ∗

1+λ∗

N⩽(∆w|gN|,|gN|)w

(|gN|,|gN|)w

+(∆wgN, gN)w

(gN, gN)w

=∑︁

v,v′∈V(G)[︃(︂|gN(v)|−|gN(v′)|)︂2+(︂gN(v)−gN(v′))︂2]︃Avv′

2∑︁

v∈V(G)

N(v)A(v)

⩽∑︁

v,v′∈V(G)(︂g2

N(v) + g2

N(v′))︂Avv′

∑︁

v∈V(G)

N(v)A(v)

= 2

(4.76)

hence

λ∗

1⩽2−|V|

N=V−2

V−1

while

λ∗

N⩽2−α2

103

CHAPTER 4. SUPPORTING TOPICS

4.3 Other representations on the sphere

Rotation group is closely related to quaternions, which provides us with an

alternative and convenient way to represent points and compute on the sphere.

Let

denote the set of quaternions consisting of elements like

q=q0+q1i+q2j+q3k=q0+Vec(q)

(4.77)

which can be represented as

q=∥q∥(cos θ+nˆ sin θ)

, analogous to complex

numbers, where

∥q∥=√qq

and

nˆ = (n1, n2, n3)

is the unit vector in the

direction of the vector part of the quaternion

, i.e.

nˆ∥q−q0∥=q−q0

Therefore a unit quaternion means a rotation about the axis

nˆ

by the angle

2θ

is a division algebra, and the inverse of a nonzero element

p−1/∥p∥2

We denote the set of all unit quaternion by

BH,1

. Whenever two quaternions

are conjugate with each other in the sense that there exists

r∈H

such that

p=rqr−1

, they have exactly the same norm and the same real part. In fact,

we have the relation

Re(pq) = p0q0−Vec(p)·Vec(q)

Vec(pq) = p0Vec(q) + q0Vec(p) + Vec(p)×Vecq

(4.78)

Two quaternions are called

orthogonal

pq∈VecH

. It is obvious that

are

orthogonal if and only if there exists a pure quaternion

such that

p=vq

Quaternion multiplication preserves Euclidean norm, namely

∥pq∥=∥p∥∥q∥

If we identify

with

, it is easy to check that

R× {0}

is the center of

Therefore its complement is an invariant subspace under conjugation too. In

fact, if we take a unit quaternion

and a pure quaternion

, then

J(cos θ+nˆ sin θ)v:= (cos θ+nˆ sin θ)v(cos θ−nˆ sin θ)

= (nˆ×v) sin 2θ+ (1 −cos 2θ)[nˆ(nˆ·v)−v] + vcos 2θ

∈VecH

It is clear that

maps

BH,1

onto

SO(3)

. In particular, when

θ= 0

θ=π

J(cos θ+nˆ sin θ) = I

One can identify

with

if a quaternion is rewritten in the form

q0+q1i+(q2+q3i)j=z+wj

, and the multiplication rule here is

(z1+w1j)(z2+

w2j)=(z1z2−w1w¯2)+(z1w2+w1z¯2)j

. It is easy to check that

z+wj ∈BH,1→Uz,w := (︃z w

−w¯z¯)︃∈SU(2)

(4.79)

104

4.3. OTHER REPRESENTATIONS ON THE SPHERE

is an isomorphism. In other word

SU(2)

is one-to-one parameterized by unit

quaternions. The exponential map from the Lie algebra

su(2)

has the explicit

form

exp(A) = cos θI+sin θ

θA

with

A=θ(︃in1n2+in3

−n2+in3−in1)︃

In conclusion we have the identication

SO(3) = SU(2)/±I

. Another way

to look at the identication is through

˜:g∈SU(2) →[M→gMg†]∈SO(3)

(4.80)

where

is a complex Hermitian matrix satisfying

TrM = 0

. Since

U(1) = {g∈

SU(2) : Adσ3g=g}

and it is connected,

˜(g)σ3

gives a map from

SU(2)

into

, where

σ3

is the Pauli matrix

(︃1 0

0−1)︃

, hence there is an identication

between

SU(2)/U(1)

and

Six denitions of bi-invariant inner-product induced metrics on the rotation

groups are compared in [56] and proved to be functional equivalent in the sense

that there exist positive continuous strictly increasing functions

such that

hi◦Φ1= Φi

with

i= 2,··· ,6

while some of them are boundedly equivalent.

These metrics in general can be classied into three types, one measures by

using the quaternion dierence, for instance

Φ1(p,q) = arccos(|p·q|)

another measures the derivation from the identity matrix in the Euclidean space,

for instance

Φ2(σ1, σ2) = ∥I−σ1σT

2∥F

with

∥ · ∥F

the Frobenius norm; the other measures in the Riemannian way,

namely by the distance in Lie algebra

Φ3(σ1, σ2) = ∥log(σ1σT

2)∥

with

∥·∥

being either the Frobenius norm or

∥S∥2

H=1

(STS)

for

S∈so(3)

Some characterization of mean rotation with respect to

Φ1

and

Φ2

is given in

[39]. In particular there is

arg min

σ∈SO(3)

∑︂

i=1

Φ2(σ, σi)2= arg min

σ∈SO(3)

Φ2(1

∑︂

i=1

σi, σ)

105

CHAPTER 4. SUPPORTING TOPICS

namely the projection of

∑︁

i=1

σi

onto the rotation group is the rotation mean

with respect to

Φ2

; and for

σ1,··· , σN

belonging to a same one-parameter

subgroup of

SO(3)

such that

Φ3(σi, σj)<√2π

for any

i, j

with respect to

Frobenius norm, there is an explicit expression

arg min

σ∈SO(3)

∑︂

i=1

Φ3(σ, σi)2=σ1(σT

1σ2(σT

2σ3(···σN−1(σT

N−1σN)1

2)2

3···)N−2

N−1)N−1

meanwhile

∑︂

i=1

log(σT

iσ)=0

is a necessary but not sucient condition such that

∑︁

i=1

Φ3(σi, σ)2

achieves local

minima.

106

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