WASHINGTON UNIVERSITY
Department of Mathematics
Dissertation Examination Committee:
Guido L. Weiss, Chair
Edward N. Wilson, Chair
Albert Baernstein, II
Brody D. Johnson
John E. McCarthy
Salvatore P. Sutera
Existence and Accuracy Results
for
Composite Dilation Wavelets
by
Jeffrey David Blanchard
A dissertation presented to the
Graduate School of Arts and Sciences
of Washington University in
partial fulfillment of the
requirements for the degree
of Doctor of Philosophy
August 2007
Saint Louis, Missouri
Acknowledgments
I am fully indebted to my two advisors, Guido Weiss and Ed Wilson. Our frequent
meetings in Guido’s office have been truly rewarding mathematically and personally.
They have been excellent teachers, compassionate critics, timely superiors, and generous friends. Their mentorship has only been surpassed by their obvious concern for
me as a person.
I am grateful to all the members and visitors of our wavelet seminar. Specifically,
the numerous presentations by Brody Johnson, Demetrio Labate, and Darren Speegle
have been a significant portion of my education. This final year, I have been fortunate
to have Hrvoje Šikić visiting the department. He has been a great mentor and teacher
and always available for friendly and professional conversations.
I wish to thank the entire faculty and staff of the department of mathematics at
Washington University. Professors Al Baernstein and Nik Weaver have stood at the
front of many of my courses and spent hours with me outside of class. Professor John
McCarthy helped me get to Washington University and spent the summer before my
first year reminding me how to do math.
I was fortunate to learn mathematics from many great people. At Strake Jesuit
College Preparatory, Father J.B. Leininger piqued my interest in mathematics in his
calculus course. At Benedictine College, Sister Jo Ann Fellin, and the late Richard
Farrell were dedicated teachers who challenged me in many ways. They are both
caring educators who instilled a desire to learn as much mathematics as possible.
ii
Graduate school would have been a far less rewarding experience had I not been
able to share it with wonderful friends. I was very lucky to have Kim Randle as
my officemate and true friend throughout this journey. Kim has made our five years
at Washington University a thoroughly enjoyable time. My good friend Ben Braun
has been there to discuss mathematics, the job search, the usefulness of algebraic
topology, and the great questions of life. I thank Brian Maurizi for our time in the
qualifying courses and for his random appearances to discuss mathematics. I thank
Mike Duetsch for some mathematics, but mostly for our time on the golf course.
I would like to thank Alexi Savov for his considerable work on minimally supported
frequency composite dilation wavelets. His excellent senior honors thesis laid the
foundations for chapter 2 of this dissertation. Chapter 3 is joint work with Ilya
Krishtal. While a Chauvenet lecturer at Washington University, Ilya was a fantastic
teacher and friend. I thoroughly enjoy working on mathematics with him.
Finally, I would like to thank my parents, Charles and Kathryn Blanchard. They
provided me many opportunities to learn and instilled a passion for life-long learning
and the pursuit of truth.
I was supported for three years as a Department of Homeland Security Fellow.
This fellowship allowed me to focus on my graduate studies while providing for my
family. The following is a statement at their request:
Portions of this research were performed while on appointment as a U.S. Department of Homeland Security (DHS) Fellow under the DHS Scholarship and Fellowship
Program, a program administered by the Oak Ridge Institute for Science and Education (ORISE) for DHS through an interagency agreement with the U.S Department
of Energy (DOE). ORISE is managed by Oak Ridge Associated Universities under
DOE contract number DE-AC05-06OR23100. All opinions expressed in this paper
are the author’s and do not necessarily reflect the policies and views of DHS, DOE,
or ORISE.
iii
Dedication
I dedicate this dissertation to my loving wife and best friend, Amy, and our three
fabulous children, Mary, Bridget, and Beaux. They have sacrificed considerably to
afford me the time I needed. Amy has spent countless nights explaining to the kids
that I would not be home before bedtime, but I would be home some day. An amazing
mother and wife, her support has been unbelievable. She has truly done everything
for the past five years. On top of that, Amy has heard more about wavelets than
any wife should have to endure. Mary, Bridget, and Beaux have grown up with
their father constantly “playing math” and have never complained. I am absolutely
blessed, absolutely grateful, and absolutely in love with them all.
iv
Contents
Acknowledgments
ii
Dedication
iv
List of Figures
vii
1 Introduction
1.1
Composite Dilation Wavelets
1.1.1
1
. . . . . . . . . . . . . . . . . . . . . .
2
Affine Systems, Wavelets, and Composite Dilation
Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.2
Composite Dilation Multiresolution Analysis . . . . . . . . . .
5
1.1.3
Notation and Preliminaries . . . . . . . . . . . . . . . . . . . .
7
1.2
Group Actions on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3
Accuracy of Refinable Functions . . . . . . . . . . . . . . . . . . . . .
17
2 Existence of Minimally Supported Frequency Composite Dilation
Wavelets
25
2.1
Admissibility Conditions . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.2
Finite Groups and Lattices in MSF Composite Dilation Wavelets
. .
33
2.3
Existence of MSF Composite Dilation
Wavelets in L2 (Rn ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Singly Generated, MSF, Composite Dilation Wavelets
55
2.4
v
. . . . . . . .
2.4.1
Singly Generated MRA, MSF, Composite Dilation
Wavelets for L2 (Rn ) . . . . . . . . . . . . . . . . . . . . . . .
2.4.2
Singly Generated Non-MRA, MSF, Composite
Dilation Wavelets for L2 (Rn ) . . . . . . . . . . . . . . . . . . .
2.4.3
60
Singly generated, MRA, MSF, Composite Dilation
Wavelets for L2 (R2 )
2.5
56
. . . . . . . . . . . . . . . . . . . . . . .
69
MSF, Composite Dilation, Parseval Frame
Wavelets
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Accuracy of Compactly Supported Composite Dilation Wavelets
77
94
3.1
Matrix Relations from Composite Dilations . . . . . . . . . . . . . . .
95
3.2
Accuracy of Composite Dilation Systems in 1 Dimension . . . . . . . 101
3.3
A Compactly Supported, Composite Dilation Wavelet with Accuracy 2 109
3.3.1
The Accuracy Equations . . . . . . . . . . . . . . . . . . . . . 110
3.3.2
The Smith-Barnwell Equations . . . . . . . . . . . . . . . . . 114
3.3.3
The Composite Dilation Wavelet . . . . . . . . . . . . . . . . 117
4 Conclusions
133
Bibliography
136
vi
List of Figures
1.1
F is a fundamental region for B = hr1 , r2 i, the reflections through
y = x and y = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
13
The triangle R is a B-tiling set for the square S, where B = hr1 , r2 i,
the reflections through y = x and y = 0. . . . . . . . . . . . . . . . .
14
2.1
R is a Γ∗ -tiling set for R̂2 and a B-tiling set for S. . . . . . . . . . . .
27
2.2
R1 , R2 , R3 are Γ∗ -tiling sets for R̂2 and R1 ∪ R2 ∪ R3 is a B-tiling set
2.3
2.4
2.5
2.6
2.7
2.8
2.9
for Sa\S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S
P = m
j=0 Aj = A0 ∪ A1 ∪ A2 . . . . . . . . . . . . . . . . . . . . . .
i
hS
2
(P
+
jγ
)
∩ [F \(F + γ2 )].
R = A0 ∪ (A1 + γ2 ) ∪ (A2 + 2γ2 ) =
2
j=0
S
S = b∈B Rb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S
Ω = j∈Z (P + jγ2 ) and Ω = Ω−1 ∪ Ω1 . . . . . . . . . . . . . . . . .
.
28
.
37
.
40
.
40
.
44
R = R−1 ∪ R1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S
S
R (2In ) = v∈V w∈W Rvw . . . . . . . . . . . . . . . . . . . . . . . . .
S
S
Theorem 2.11: S = b∈B Rb and Sa\S = B∈b R1 b. . . . . . . . . . .
46
50
59
2.10 R∞ = limk→∞ Rk for γ = γ1 + γ2 . . . . . . . . . . . . . . . . . . . . .
68
2.11 R∞ = limk→∞ Rk for γ = γ2 .
. . . . . . . . . . . . . . . . . . . . . .
68
. . . . . . . . . . . . . . . . . . . . . . . . . .
71
2.13 R is a B-tiling set for S. R1 is a B-tiling set for Sa\S. . . . . . . . .
2.14 Raρ −π
= R ∪ (R + γ1 ). . . . . . . . . . . . . . . . . . . . . . . . .
4
73
2.12 Rab2 = R ∪ (R + γ1 ).
vii
75
2.15 R is B-tiling set for S and R1 is a B-tiling set for Sa\S. . . . . . . .
76
2.16 PF-B-admissible: R = (P − α) ∩ F . . . . . . . . . . . . . . . . . . . .
79
2.17 PF-B-admissible: R is B-tiling set for S. . . . . . . . . . . . . . . . .
79
2.18 PF-(B, Γ)-admissible: R is B-tiling set for S, R1 is a B-tiling set for
Sa\S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
2.19 A singly generated, MSF, MRA, composite dilation Parseval frame
wavelet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
2.20 A singly generated, MSF, MRA, composite dilation Parseval frame
wavelet from a sphere. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.21 For n = 5, P̃ is a B̃-tiling set for P and P ⊂ P a.
88
. . . . . . . . . . .
89
2.22 S ⊂ Sa for a double pyramid with pentagon base. . . . . . . . . . . .
92
3.1
The tent function after 0 iterations in (3.26). . . . . . . . . . . . . . . 122
3.2
The tent function after 3 iterations in (3.26). . . . . . . . . . . . . . . 123
3.3
The tent function after 6 iterations in (3.26). . . . . . . . . . . . . . . 123
3.4
The scaling function ϕ(x). . . . . . . . . . . . . . . . . . . . . . . . . 125
3.5
The wavelet ψ(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
viii
Chapter 1
Introduction
This dissertation answers some existence questions for composite dilation wavelets
generating orthonormal bases for L2 (Rn ). In this first chapter we cover some necessary background information from affine systems to finite reflection groups to accuracy of multidimensional functions. In chapter two, we begin by proving some sufficient conditions for the existence of minimally supported frequency composite dilation
wavelets. We then prove the existence of minimally supported frequency composite
dilation wavelets in every dimension. We show that they exist for arbitrary lattices
and every group with a very minor restriction to their fundamental domain. In chapter two, we also prove the existence of such wavelets with a single generator. We
conclude chapter two by answering similar questions for composite dilation wavelets
generating Parseval frames for L2 (Rn ). Chapter three examines composite dilation
wavelets in the time domain. The main portion of chapter three involves finding a
compactly supported composite dilation wavelet with greater accuracy than the Haar
wavelet. Along the way, we look at the accuracy of composite dilation systems and
establish the necessary conditions for higher accuracy composite dilation wavelets.
1
1.1
Composite Dilation Wavelets
In both pure and applied mathematics, there is significant interest in the study of
representations of multidimensional functions. The considerable literature on wavelets
has come from this pursuit and led to noteworthy advances. Wavelets, however, are
rigid in that they must maintain their orientation throughout the basis. Composite
dilation wavelets are, in a sense, a generalization of wavelets that provides the freedom
to manipulate the orientation of the functions. This has significant value in both
applications and pure mathematics.
Here we examine affine systems in a general context, see how wavelets fall in
this category, and discuss the newer subset of affine systems called composite dilation systems. We examine the multiresolution analysis associated with each type of
system and discuss some parallels. We briefly examine some properties of the dilation and translation operators that generate the affine systems while simultaneously
establishing notation to be used throughout the discussion.
1.1.1
Affine Systems, Wavelets, and Composite Dilation
Wavelets
Affine systems are formed by applying two kinds of unitary operators to a set of
functions. These operators are dilations and translations.
Definition 1.1. For a ∈ GLn (R) and f ∈ L2 (Rn ), the dilation of f by a is the
operator Da : L2 (Rn ) −→ L2 (Rn ) such that
1
Da f (x) = |det(a)|− 2 f (a−1 x).
Definition 1.2. For k ∈ Γ, Γ a full rank lattice (Γ = cZn for c ∈ GLn (R)), and
2
f ∈ L2 (Rn ), the translation of f by k is the operator Tk : L2 (Rn ) −→ L2 (Rn ) such
that
Tk f (x) = f (x − k).
Then, for a countable subset of invertible matrices C ⊂ GLn (R), Γ a full rank
lattice, and ψ 1 , . . . , ψ L ∈ L2 (Rn ), the affine system produced by C, Γ, and Ψ =
(ψ 1 , . . . , ψ L ) is the set
ACΓ (Ψ) = Dc Tk ψ l : c ∈ C, k ∈ Γ, 1 ≤ l ≤ L .
(1.1)
When C = {2j : j ∈ Z}, Γ = Z, and ψ ∈ L2 (R), then (1.1) can be written as the
familiar system
o
n j
{ψj,k (x)}j,k∈Z = 2− 2 ψ(2−j x − k) : j, k ∈ Z .
(1.2)
In this case, we recall that ψ is a wavelet when the collection of functions (1.2) is
an orthonormal basis for L2 (Rn ). In a more general situation, ψ is a Parseval frame
wavelet when (1.2) is a Parseval frame for L2 (Rn ).
In [9] and [10], Guo, Labate, Lim, Weiss, and Wilson introduce affine systems
with composite dilations. These are affine systems obtained from (1.1) when the
countable set C is the product of two, not necessarily commuting, sets of invertible
1
matrices: C = AB, A, B ⊂ GLn (R). Obviously, Da Db f (x) = |det(a)|− 2 Db f (a−1 x) =
1
1
1
|det(b)|− 2 |det(a)|− 2 f (b−1 a−1 x) = |det(ab)|− 2 f ([ab]−1 x) = Dab f (x). Therefore, for
affine systems with composite dilations we have
AABΓ (Ψ) = Da Db Tk ψ l : a ∈ A, b ∈ B, k ∈ Γ, 1 ≤ l ≤ L .
(1.3)
These systems are more general than the traditional wavelet systems. When
3
A = {2j : j ∈ Z} and B is the trivial group, then C = A = {2j : j ∈ Z} and with
Γ = Zn we obtain (1.2). Therefore, traditional wavelets are a special class of affine
systems with composite dilations. In this spirit, we say that Ψ is a composite dilation
wavelet if the system AABΓ (Ψ) is an orthonormal basis for L2 (Rn ).
In this paper, we study the situation where A is a set obtained by taking integer
powers of some invertible, expanding matrix a ∈ GLn (R); A = {aj : j ∈ Z}. Furthermore, we study the situation where B is a finite group of invertible matrices;
B ⊂ GLn (R) and |B| < ∞ where |B| is the order of the group B. When B is finite, it
is necessarily the case that for all b ∈ B, |det(b)| = 1. Therefore, for f ∈ L2 (Rn ) and
B a finite subgroup of GLn (R), Db f (x) = f (b−1 x). Therefore, we make the following
definition:
Definition 1.3. Ψ = (ψ 1 , . . . , ψ L ) ⊂ L2 (Rn ) is an ABΓ Composite Dilation Wavelet
if the set
j
Da Db Tk ψ l : j ∈ Z, b ∈ B, k ∈ Γ, l = 1, . . . , L
o
n
− 2j
l −1 −j
= |det(a)| ψ (b a x − k) : j ∈ Z, b ∈ B, k ∈ Γ, l = 1, . . . , L
is an orthonormal basis for L2 (Rn ).
Since the groups A and B and the lattice Γ are almost certainly going to be obvious
from the context, we usually suppress the ABΓ and simply refer to Ψ as a composite
dilation wavelet. In section 2.5, we will also discuss composite dilation Parseval
frame wavelets. As in the traditional wavelet case, this is simply a relaxation of the
orthonormal basis to a Parseval frame in definition 1.3. That is, we call Ψ a composite
dilation Parseval frame wavelet if Daj Db Tk ψ l : j ∈ Z, b ∈ B, k ∈ Γ, l = 1, . . . , L is a
Parseval frame for L2 (Rn ).
Chapter 2 proves sufficient conditions and existence for Minimally Supported Frequency composite dilation wavelets. First, we recall from [11] that a Minimally Sup ported Frequency (MSF) wavelet is a wavelet, ψ, such that ψ̂ is the characteristic
4
function of a measurable set R ⊂ R̂. The motivation behind the name is that the set
R supporting the MSF wavelet must satisfy m (R) = 1, where m is Lebesgue measure
and the lattice is Z. Thus, the support of ψ̂ has the minimal measure required such
that {Tk ψ : k ∈ Z} is an orthonormal system.
Definition 1.4. Ψ = ψ 1 , . . . , ψ L ⊂ L2 (Rn ) is a Minimally Supported Frequency
1
Composite Dilation Wavelet if ψ̂ l = |det(c)| 2 χRl for all l = 1, . . . , L and the set
j
Da Db Tk ψ l : j ∈ Z, b ∈ B, k ∈ Γ, l = 1, . . . , L
is an orthonormal basis for L2 (Rn ).
In this case, with Γ = cZn , each set Rl must satisfy m(Rl ) = |det(c)|−1 =
|det(c−1 )|. We call these MSF composite dilation wavelets since the support of each
wavelet ψ̂ l has the minimal measure such that Db Tk ψ l : k ∈ Γ, l = 1, . . . , L is an
orthonormal system.
1.1.2
Composite Dilation Multiresolution Analysis
One classical method for constructing orthonormal wavelets is the well known multiresolution analysis (MRA). In traditional wavelets, for example in [11], an MRA is
defined as a sequence, {Vj }j∈Z , of closed subspaces of L2 (Rn ) satisfying
Vj ⊂ Vj+1
for all j ∈ Z;
(1.4)
f ∈ Vj if and only if f (2·) ∈ Vj+1
\
Vj = {0} ;
for all j ∈ Z;
(1.5)
(1.6)
j∈Z
[
Vj = L2 (Rn );
(1.7)
j∈Z
∃ ϕ ∈ V0 such that {Tk ϕ : k ∈ Z} is an orthonormal basis for V0 .
5
(1.8)
In [10] an analogue to the classical MRA is introduced for affine systems with
composite dilations. In this setting they develop a method for constructing composite
dilation wavelets when you are given the sets of matrices A and B and the full rank
lattice Γ. The authors discuss the generality for which a composite MRA can be
constructed. Here we will use the case where A = {aj : j ∈ Z} for an expanding
matrix a ∈ GLn (R) and B ⊂ GLn (R) is a finite group. We establish the following
definition:
Definition 1.5. Let a ∈ GLn (R) be an expanding matrix, A = {aj : j ∈ Z}, B ⊂
GLn (R) a finite group, and Γ = cZn , c ∈ GLn (R), be a full rank lattice. An ABΓ
multiresolution analysis (ABΓ-MRA) is a sequence, {Vj }j∈Z , of closed subspaces of
L2 (Rn ) satisfying
(M1) Vj ⊂ Vj+1
for all j ∈ Z;
(M2) f ∈ Vj if and only if f (a·) ∈ Vj+1
\
(M3)
Vj = {0} ;
for all j ∈ Z;
j∈Z
(M4)
[
Vj = L2 (Rn );
j∈Z
(M5) ∃ ϕ ∈ V0 such that {Db Tk ϕ : b ∈ B, k ∈ Γ} is an orthonormal basis for V0 .
Suppose the sequence is defined by Vj = Da−j V0 . Then, if (M1) is true, we automatically satisfy (M2). This is the usual way to construct an ABΓ-MRA. Find a
function in L2 (Rn ) such that {Db Tk ϕ : b ∈ B, k ∈ Γ} is orthonormal and create the
shift invariant space determined by the semi-direct product of B and Γ. Let the closure of this space be V0 . Then define the sequence of closed subspaces by Vj = Da−j V0 .
Then we are only left with determining if we have satisfied (M3) and (M4).
This follows the arguments of the standard MRA (1.4)-(1.8) almost exactly. In
6
both cases V0 is generally defined to be the shift invariant space determined by applying a group action to a specific generating function, ϕ, usually called the scaling
function. The difference is that the group in the classical MRA (1.8) is the integer
lattice and the group in (M5) is B ⋊ Γ, the semi-direct product.
Again, A, B, and Γ are usually clear from the context. As with the definition
of a composite dilation wavelet, we generally refer to an ABΓ-MRA as a composite
dilation MRA. As this paper discusses composite dilation systems, after this section
we will generally refer to a composite dilation MRA as an MRA. A composite dilation
wavelet arising from an ABΓ-MRA will obviously be an ABΓ composite dilation
wavelet. Such a wavelet will be referred to as an ABΓ MRA composite dilation
wavelet. Most of the time, we will let the reader infer A, B, and Γ from context and
simply call them MRA composite dilation wavelets.
In the traditional MRA literature, much has been studied regarding different
conditions satisfied by the basis of the set determined by the scaling function in
(1.8). Most notably, the orthonormal basis is often relaxed to be a Parseval frame.
The same can done for a composite dilation MRA. We may replace (M5) with
(M5′ ) ∃ ϕ ∈ V0 such that {Db Tk ϕ : b ∈ B, k ∈ Γ} is a Parseval frame for V0 .
We will identify this case by referring to such composite dilation MRAs as Parseval
frame MRAs and wavelets arising from Parseval frame MRAs as MRA, composite
dilation Parseval frame wavelets.
1.1.3
Notation and Preliminaries
Throughout this document, we deal with functions and their properties in both time
and frequency. We will therefore transform the functions between the time and frequency domains via the Fourier transform. We adopt the notation that the time
7
domain is represented by Rn while the frequency domain is denoted R̂n . Elements of
the time domain, denoted by letters of the Roman alphabet, will be column vectors,
x = (x1 , . . . , xn )t ∈ Rn , and elements of the frequency domain, or Fourier domain,
will be row vectors, ξ = (ξ1 , . . . , ξn ) ∈ R̂n denoted by letters of the Greek alphabet.
Definition 1.6. The space of square integral functions on a set Ω is
L2 (Ω) =
(
f :Ω→C:
Z
Ω
)
12
<∞
|f (x)|2 dx
Although there are generalizations to other spaces, we focus on L2 (Rn ).
We will use the Fourier transform that is a unitary operator from L2 (Rn ) to L2 (R̂n )
sending f → fˆ; F : L2 (Rn ) → L2 (R̂n ). That is,
ˆ =
F[f ](ξ) = f(ξ)
Z
f (x)e−2πiξx dx.
(1.9)
Rn
This is defined for functions in L1 (Rn ) and has a natural extension to all of L2 (Rn ).
Of course, there is an inverse F−1 = G : L2 (R̂n ) → L2 (Rn ) sending g to ǧ.
G[g](x) = ǧ(x) =
Z
g(ξ)e2πiξx dξ.
(1.10)
R̂n
The bases generated by composite dilation wavelets are defined in terms of the
operators Daj , Db , and Tk applied to the wavelet generating functions ψ l for 1 ≤ l ≤
L. Because of this notation, we examine how the Fourier transforms acts on these
operators.
F[Da f ](ξ) = [Da f ]ˆ(ξ) =
=
Z
ZR
Da f (x)e−2πiξx dx
n
Rn
1
|det(a)|− 2 f (a−1 x)e−2πiξx dx
8
(1.11)
=
Z
Rn
=
Z
Rn
1
|det(a)|− 2 f (y)e−2πiξ(ay) |det(a)| dy
1
|det(a)| 2 f (y)e−2πi(ξa)y
1
ˆ
= |det(a)| 2 f(ξa)
(1.12)
(1.13)
(1.14)
In (1.11) we see that the n × n matrix a is acting on x on the left as defined by the
dilation operator in 1.1. Again in (1.12), a acts on y on the left. However, in (1.13)
we can interpret this as a acting on ξ on the right. The notation that Rn consists of
column vectors and R̂n consists of row vectors allows to interpret (1.12) and (1.13)
in this manner. From this point of view, (1.14) motivates a definition similar to
definition 1.1 that will be defined on the Fourier domain. In this case, we want an
operator acting on a function in L2 (R̂n ) that multiplies the variable by the dilation
matrix on the right. This operator will not involve the inverse of the dilation matrix.
Definition 1.7. For a ∈ GLn (R) and g ∈ L2 (R̂n ), the Fourier domain dilation of g
by a is the operator D̂a : L2 (R̂n ) −→ L2 (R̂n ) such that
1
D̂a g(ξ) = |det(a)| 2 g(ξa).
Therefore, using (1.14) and definition 1.7, we have
1
F[Da f ](ξ) = [Da f ]ˆ(ξ) = |det(a)| 2 fˆ(ξa) = D̂a fˆ(ξ).
If a, b ∈ GLn (R) and |det(b)| = 1, then
F[Daj Db f ](ξ) = Daj Db f ˆ(ξ) = [Daj b f ]ˆ(ξ)
= D̂aj b fˆ(ξ)
j
= |det(a)| 2 fˆ(ξaj b)
9
(1.15)
j
= |det(a)| 2 D̂b fˆ(ξaj )
= [D̂aj D̂b fˆ](ξ)
Another useful unitary operator in the Fourier domain is modulation, a multiplication by an exponentional function.
Definition 1.8. For k ∈ Γ and g ∈ L2 (R̂n ), the modulation of g by k is the operator
Mk : L2 (R̂n ) → L2 (R̂n ) such that
Mk g(ξ) = e−2πiξk g(ξ).
It is easy to verify that the Fourier transform of a translation of f by k is modulation of fˆ by k.
F[Tk f ](ξ) = [Tk f ]ˆ(ξ) =
=
Z
ZR
ZR
Tk f (x)e−2πiξx dx
n
n
f (x − k)e−2πiξx dx
f (y)e−2πiξ(y+k)dy
Rn
Z
−2πiξk
f (y)e−2πiξy dx
=e
=
Rn
= e−2πiξk fˆ(ξ)
ˆ
= Mk f(ξ).
(1.16)
Therefore, using (1.15) and (1.16) we have
F[Da Tk f ](ξ) = [Da Tk f ]ˆ(ξ) = D̂a [Tk f ]ˆ(ξ)
1
= |det(a)| 2 [Tk f ]ˆ(ξa)
1
ˆ
= |det(a)| 2 Mk f(ξa)
10
= D̂a Mk fˆ(ξ).
(1.17)
This provides us with useful properties of the dilation and translation operators
under the Fourier transform. It is also important for us to establish the following
well known propositions. They will allow us to take advantage of the dilation and
translation notation.
Proposition 1.1. For any matrix c ∈ GLn (R), the inverse dilation operator is Dc−1 =
Dc−1 .
Proof. This is obvious since
1
1
1
Dc−1 [Dc f ](x) = |det(c)| 2 [Dc f ](cx) = |det(c)|− 2 |det(c)| 2 f (c−1 cx) = f (x).
So Dc−1 = Dc−1 .
2
∗
Proposition 1.2. For any matrix c ∈ GLn (R), (Dcj ) = Dc−j .
Proof. This is a straightforward calculation involving the inner product and a change
of variables. Let f, g ∈ L2 (Rn ).
f, Dcj g =
Z
ZR
n
f (x) Dcj g (x)dx
j
f (x) |det(c)|− 2 g(c−j x)dx
Rn
Z
− 2j
= |det(c)|
f (cj y)g(y) |det(c)|j dy
Rn
Z
j
=
|det(c)| 2 f (cj y)g(y)dy
n
ZR
−j =
Dc f (y)g(y)dy
=
Rn
= Dc−j f, g
∗
Hence, (Dcj ) = Dc−j .
2
11
Proposition 1.3. For k ∈ Γ = cZn , Tk−1 = Tk∗ = T−k .
Proof. These are also straightforward observations. First of all
T−k Tk f (x) = Tk f (x + k) = f (x + k − k) = f (x)
so Tk−1 = T−k .
Now let f, g ∈ L2 (Rn ).
hf, Tk gi =
=
=
Z
f (x)Tk g(x)dx
ZR
ZR
n
n
f (x)g(x − k)dx
f (y + k)g(y)dy
Rn
=
Z
Rn
T−k f (y)g(y)dy
= hT−k f, gi .
Hence, Tk∗ = T−k . Thus, Tk−1 = Tk∗ = T−k .
1.2
2
Group Actions on Rn
This section describes how a group acts on Rn and establishes the notation and
terminology we use when discussing the groups. We are mostly interested in how
the group B will affect Rn . We primarily concern ourselves with the case when
B ⊂ GLn (R) is a finite group of invertible matrices. First of all, we establish some
nomenclature that is not necessarily standard.
Regarding notation, writing A ∩ B = ∅ is meant as essentially disjoint. That is,
this equation does not preclude the sets A and B from sharing a set of measure zero.
This is often referred to as being disjoint in the sense of measure, i.e. they have an
12
intersection of measure 0. In the same spirit, when we refer to two sets as ”disjoint”
we still imply that they may share a set of measure zero.
We make the following definitions to assist in distinguishing between how we are
interpreting a group action.
Definition 1.9. Let B be a finite subgroup of GLn (R). The fundamental region for
B is a set, F ⊂ R̂n , such that
S
(i) b∈B F b = R̂n and
(ii) for b1 6= b2 ∈ B, F b1 ∩ F b2 = ∅.
Figure 1.1 shows a fundamental region for the group generated by the reflections
through the hyperplanes y = x and y = 0. We let r1 represent the reflection through
the line y = x and r2 represent the reflection through y = 0. We will use the standard
notation for generating a group, namely B = hr1 , r2 i.
R̂2
y=x
F r2 r1
F r1
F r1 r2 r1
F
y=0
F r2
F r1 r2 r1 r2
F r2 r1 r2
F r1 r2
Figure 1.1: F is a fundamental region for B = hr1 , r2 i, the reflections through y = x
and y = 0.
Definition 1.10. Let G be a group acting on a set Ω. Then R is a G-tiling set for
Ω if
13
(i)
S
g∈G
Rg = Ω and
(ii) for g1 6= g2 ∈ G, Rg1 ∩ Rg2 = ∅.
Figure 1.2 shows a B-tiling set for the two dimensional example where B = hr1 , r2 i,
the group generated by reflections through the line y = x and y = 0. The triangle,
R, is a B-tiling set for the square, S.
R̂2
y=x
S
R
y=0
Figure 1.2: The triangle R is a B-tiling set for the square S, where B = hr1 , r2 i, the
reflections through y = x and y = 0.
These two definitions have very minor differences. The first definition refers specifically to the subset of R̂n that, when acted upon by B, will generate all of R̂n without
any measurable overlap. Obviously, if F is the fundamental region for B, then F is
a B-tiling set for R̂n . The first definition is also restricted to finite subgroups of the
invertible matrices. This is to avoid calling a lattice tiling set a fundamental region.
Many authors use this terminology, but here we want to distinguish the set F related
to the group used in the composite dilations. For example, the unit interval [0, 1]
is a Z-tiling set for R. The restriction to finite subgroups of the invertible matrices
prevents the minor confusion of calling [0, 1] a fundamental region for Z.
Much of chapter 2 deals with groups with a certain property for their fundamental
14
domains. The chapter is devoted to proving the existence of minimally supported
frequency wavelets for a group B and a lattice Γ. Whenever a group, B, has the
property that its fundamental region, F , is bounded by hyperplanes through the
origin, we are able to prove the existence of minimally supported composite dilation
wavelets for every lattice Γ (see Theorem 2.10). A hyperplane is a codimension 1
linear subspace. We simply want every hyperplane bounding the fundamental region
to pass through the origin. When this occurs, F is an n-dimensional cone-like region
with the origin as its vertex.
A large, well-known, classified family of finite groups with fundamental regions
bounded by hyperplanes is the family of Coxeter groups [7]. Coxeter groups, also
known as reflection groups, are generated by reflections through the hyperplanes
bounding their fundamental region. In [7], the hyperplanes bounding the fundamental
region are referred to as mirrors. Let us examine our notation and definitions 1.9 and
1.10 with an example.
Example 1.1. Let B be the symmetries of the square acting on R̂2 . Then B is a
Coxeter group generated by two reflections, r1 , the reflection through y = x and,
r2 , the reflection through the line y = 0. These reflections are determined by the
lines y = 0 and y = x. Theses lines are hyperplanes though the origin in R̂2 . The
reflections are:


 0 1 
r1 = 

1 0


 −1 0 
r2 = 

0 1
Let F = {(x, y) : x ≥ y} ∩ {(x, y) : y ≥ 0}. Then F is bounded by the hyperplanes
H1 = {(x, y) : y = x} and H2 = {(x, y) : y = 0}. If we let B = hr1 , r2 i act on F , we
15
see that F is a fundamental region for B.
Now let S be the unit square centered at the origin, S = (x, y) : − 21 ≤ x, y ≤ 12 .
Let R be the triangle with vertices (0, 0), ( 12 , 0), and ( 12 , 21 ). Then R is also bound by
H1 and H2 . When B acts on R, we see that R is a B-tiling set for S.
Coxeter groups have been classified and exist in every dimension. A Coxeter group
acting on Rn will be generated by at most n reflections. This, combined with the significant literature available on Coxeter groups, suggests them as excellent candidates
for use when constructing composite dilation wavelets. Another reasonably attractive option, especially in lower dimensions, is a rotation group. Rotation groups
are subgroups of Coxeter groups. These may be the best choices for finite groups
with composite dilation wavelets. As noted in chapter 2, Coxeter groups and rotation groups have fundamental regions bounded by hyperplanes through the origin.
For more information on Coxeter groups, the reader should see Grove and Benson’s
Finite Reflection Groups [7] or go directly to the source in [3] and [4].
Composite dilation wavelets require the identification of an associated full rank
lattice.
Definition 1.11. Γ is a Full Rank Lattice if Γ = cZn where c is an invertible matrix,
c ∈ GLn (R).
Definition 1.12. Let Γ = cZn be a full rank lattice. A Lattice Basis is a set {γi }ni=1
such that every element γ ∈ Γ can be written as an integer combination of the
elements:
γ=
n
X
i=1
mi γi where mi ∈ Z.
If ci is the ith column of the matrix c, then the set {ci }ni=1 is a basis for Γ. When
we move between the time and frequency domains, we must deal with the appropriate
corresponding lattices.
16
Definition 1.13. Suppose Γ = cZn is a full rank lattice in Rn . The dual lattice to Γ
is the lattice Γ∗ = Ẑn c−1 , a full rank lattice in R̂n . The rows of the matrix c−1 form
a basis for the dual lattice Γ∗ .
The notion of a dual lattice is useful when defining bases on subspaces of L2 (Rn ).
For minimally supported frequency wavelets, we start with the characteristic function
of a set, say R. When R is a Γ∗ -tiling set for R̂n , we can generate an orthonormal
basis for L2 (R) by using exponential functions associated with the lattice Γ. The
duality of the lattices makes this system orthonormal. This will be used extensively
in chapter 2.
Finally, we define a group acting on a lattice (or a subset of a lattice) by B(Γ) =
S
b∈B
Γb. We will frequently want to choose a group and a lattice with some symmetry.
For example, we will often choose Γ so that B(Γ) = Γ or choose an expanding matrix
a so that a(Γ) ⊂ Γ.
1.3
Accuracy of Refinable Functions
In chapter 3 we discuss composite dilation wavelets with some special, advantageous
properties. We look for composite dilation wavelets with compact support in the
time domain. In [13], examples are presented of composite dilation wavelets which
are characteristic functions of sets in the time domain. It would be very useful to have
composite dilation wavelets with greater smoothness than the characteristic functions.
Smooth, compactly supported wavelets provide localization in both the time and
frequency domains. In classical wavelet literature, Daubechies compactly supported
wavelets [5] have arbitrary smoothness. There has also been reasonably significant
advances in finding arbitrarily smooth, non separable wavelets in two dimensions, for
example in [1] and [12].
In [1] Belogay and Wang use the notion of accuracy to measure the smoothness of
17
a function. For affine systems with multiple generating functions, Cabrelli, Heil, and
Molter have described the necessary conditions to obtain arbitrary levels of accuracy
[2]. We adopt this approach in our hunt for composite dilation wavelets with accuracy
greater than the known Haar-type wavelets discussed in [13]. In order to fully utilize
this approach, we must understand much of the notation, ideas, and results from [2].
We continue to use full rank lattices, Γ = cZn .
In this section and in chapter 3, the notation of dimension changes from ndimensional to d-dimensional. Also, here and in chapter 3, the meaning of A has
changed. Above and in chapter 2, A was a group generated by an expanding matrix
a. Here we use A to denote the expanding matrix, not the group. This allows us
to utilize the generalized matrix notation and operator notation in [2]. These minor
inconveniences are worthwhile to use the notation from [2].
Definition 1.14. A dilation matrix associated with Γ is a d × d matrix A such that
(i) A(Γ) ⊂ Γ; and
(ii) A is expanding, i.e. all eigenvalues satisfy |λk (A)| > 1.
Suppose we have a set of functions f1 , . . . , fr ∈ L2 (Rn ). Suppose they generate
an affine system
j
AAΓ = DA
Tk fi : j ∈ Z, k ∈ Γ, i = 1, . . . , r .
In order to have an affine system that may lead to an MRA, we want to ensure
that we can write the dilations of our generating functions as linear combinations of
the translates of the generating functions. Moreover, we want to find functions with
compact support. In [6], lemma 5 states that a function satisfying the refinement
equation for a finite subset of the lattice must have compact support. Therefore, we
further limit our search to finite subsets of the associated lattice, Λ ⊂ Γ, Λ finite.
Definition 1.15. The functions f1 , . . . , fr ∈ L2 (Rd ) are refinable with respect to A
18
and Γ if there exists Λ ⊂ Γ (Λ finite) and a vector-valued function F = (f1 , . . . , fr )t :
Rd → Rr satisfying the refinement equation
F (x) =
X
k∈Λ
ck F (Ax − k)
(1.18)
for some r × r matrices ck .
We will refer to both F and {f1 , . . . , fr } as refinable. Refinable functions are
exactly those functions whose affine systems will generate an MRA.
It is valuable to have certain levels of smoothness associated with our refinable
functions. One method of measuring the smoothness of an affine system is to determine the accuracy of the system. We define accuracy in terms of the shift invariant
space generated by F .
Definition 1.16. The shift invariant space generated by F and Γ is the set of all
linear combinations of the lattice translates of the functions f1 , . . . , fr :
S(F ) =
(
X
k∈Γ
βk Tk F : βk ∈ R1×r
)
=
(
r
XX
k∈Γ i=1
)
βk,iTk fi : βk,i ∈ R .
(1.19)
Since we are searching for compactly supported functions F (or f1 , . . . , fr ), the
series in (1.19) are well-defined for all choices of bk (or bk,i ). We interpret equality
between two functions f and g to mean f = g almost everywhere. The fundamental
property of the functions F that is studied in [2] and used to measure smoothness in
[1] is the property of accuracy. Here we define the accuracy of F , which obviously
can also be interpreted as the accuracy of the collection {f1 , . . . , fr }.
Definition 1.17. The function F : Rd → Rr has accuracy p if p is the largest integer
such that all multivariate polynomials q(x) = q(x1 , . . . , xd ) with deg(q) < p lie in the
shift invariant space S(F ).
19
Composite dilation wavelets and the associated composite dilation scaling functions are elements of L2 (Rd ) while no multivariate polynomial lies in L2 (Rd ). To
generate an MRA with a scaling function Φ, we measure the accuracy of Φ in S(Φ)
and take V0 = S(Φ)∩L2 (Rd ). When Φ is a refinable function, we produce the MRA as
−j
usual by defining Vj = DA
V0 . An MRA composite dilation wavelet, Ψ, has accuracy
P
p whenever the scaling function Φ has accuracy p since Ψ(x) = k∈Γ βk DA Tk Φ(x)
for some collection {βk } ⊂ R1×r .
It is important to understand the relationship between accuracy and smoothness.
Belogay and Wang use accuracy to measure smoothness and describe an asymptotic
relationship between accuracy and smoothness [1]. Daubechies employed vanishing
moments to measure smoothness when describing her compactly supported wavelets
[5]. Since the reader is likely more familiar with the Daubechies wavelets, it is helpful
to interpret accuracy in terms of vanishing moments.
Suppose Φ is a compactly supported, refinable function as above. Suppose d = 1
so Φ : R → Rr . Now suppose Φ has accuracy p. Then the monomial xs lies in the
shift invariant space S(Φ) for all s < p. Since the restriction of a monomial to a
compact set is square integrable, then for every M ∈ N, xs χ[−M,M ] ∈ S(Φ) ∩ L2 (Rd ).
Therefore, there exists a sequence {bk : k ∈ Γ} such that
xs χ[−M,M ] (x) =
X
bk Tk Φ(x)
k∈Γ
for all s < p.
Suppose Φ was a scaling function for an MRA, S(Φ) ∩ L2 (Rd ) = V0 . Since Φ
−j
was refinable, Vj = DA
V0 will establish an MRA. If Ψ is the associated wavelet,
S(Ψ) ∩ L2 (Rd ) = W0 ⊂ V1 .
Now Ψ : R → Rr is a compactly supported, refinable function such that V0 ⊥W0
20
(S(Φ)⊥S(Ψ)). Then, for every M ∈ N,
Z
M
p
x Ψ(x)dx =
−M
Z
M
X
bk Tk Φ(x)Ψ(x)dx
−M k∈Γ
=
X
k∈Γ
bk
Z
M
Tk Φ(x)Ψ(x)dx
−M
=0
since S(Φ)⊥S(Ψ). That is, Ψ has p − 1 vanishing moments on every compact subset
of R.
Therefore, an MRA wavelet system with accuracy p has p − 1 vanishing moments
on every compact subset of R. Higher dimensional analogues are true as well. The
interpretation of MRA, composite dilation wavelets used for chapter 3 sees Ψ =
(Db1 ψ, . . . , Dbr ψ)t where B is a group of order r. This allows us to use the above
the notation and interpretation of smoothness via accuracy for composite dilation
wavelets.
To obtain the necessary conditions for the existence of such refinable functions,
Cabrelli, Heil and Molter use a generalized matrix notation and carry this notation
over to operators. We will use theorems 3.4 and 3.6 from [2]. In order to do so,
we will establish some of their notation in the following discussion. For a complete
understanding of the notation and for proofs of the theorems see [2].
We will use the following combined version of Theorems 3.4 and 3.6 from [2].
Theorem (Cabrelli, Heil, Molter, [2]): Assume Φ(x) : Rd → Cr is a refinable, integrable, compactly supported function with independent Zd translates. Then Φ has accuracy p if and only if there exists a collection
of row vectors v[s] ∈ C1×r : 0 ≤ s < p such that
(i) v[0] Φ̂(0) 6= 0, and
(i) v[s] =
P
k∈Γi
Ps
t=0
Q[s,t] A[t] v[t] ck
21
where A, the expanding matrix, and ck : k ∈ Γ , the collection of r × r
matrices, are from the refinement equation.
In order to decipher this theorem, we must understand the notation. We are
already aware of the meaning of a refinable, integrable, compactly supported function.
We are also aware of the meaning of accuracy. Therefore, we must define independent
translates, the collection of row vectors v[s] ∈ C1×r : 0 ≤ s < p , Q[s,t] , and A[t] .
Definition 1.18. F : Rd → Rr has independent Γ-translates if for every choice of
row vectors bk ∈ R1×r ,
X
bk Tk F (x) =
k∈Γ
X
k∈Γ
bk F (x − k) = 0 ⇔ bk = 0 for every k.
Cabrelli, Heil, and Molter introduce a generalized matrix notation where the entries of a matrix are not limited to scalars, but may themselves be matrices. First we
introduce a multi-index, α = (α1 , . . . , αd ) with αi a nonnegative integer. The degree
of α is |α| = α1 + · · · + αd . The number of multi-indices of degree s is ds = s+d−1
.
d−1
Write β ≤ α if βi ≤ αi for all i = 1, . . . , d. For x = (x1 , . . . , xd ) ∈ Rd we have
xα = (xα1 1 , . . . , xαd d ).
Suppose we have a collection of 1 × r row vectors
vα = (vα,1 , . . . , vα,r ) ∈ C1×r : 0 ≤ |α| < p .
(1.20)
We group the row vectors by degree: v[s] = [vα ]|α|=s . This is the generalized matrix
notation where v[s] is a ds × 1 matrix whose entries are 1 × r row vectors. Then the
collection (1.20) can be written as
v[s] ∈ C1×r : 0 ≤ s < p .
22
(1.21)
The operators Q[s,t] and A[t] are necessary to define translation and dilation on
multidimensional monomials. Since xα = (xα1 1 , . . . , xαd d ), we have a vector valued
function X[s] : Rd → Rds defined by
X[s] (x) = [xα ]|α|=s .
(1.22)
Q[s, t] is the matrix of polynomials which arise from applying a translation operator
to X[s] .
Definition 1.19. Q[s, t] is the matrix of polynomials satisfying the equation
X[s] (x − k) =
s
X
Q[s,t] (k)X[s] (x).
t=0
Similarly, A[s] is the ds × ds matrix associated with applying a dilation by A to
the vector valued function X[s] .
Definition 1.20. A[s] is the ds × ds matrix satisfying the equation
X[s] (Ax) = A[s] X[s] (x).
In [2], Q[s,t] and A[s] are developed explicitly in section 2.4. Obviously, when
dealing with one dimensional functions (i.e. d = 1) the notation becomes much
simpler. Since there is only one index with degree s, then v[s] = vs and A[s] = As .
Also, Q[s,t] becomes the coefficients of (x − k)s . Thus, for a translation by k,
s X
s
(−k)s−t .
Q[s,t] (k) =
t
t=0
(1.23)
In chapter 3 we use the ideas from [2] to determine the necessary conditions for
23
a composite dilation system to have accuracy p. Interpreting a composite dilation
system as a set of r functions where r is the order of the composite dilation group B,
r = |B|, we may apply the results from [2]. Moreover, the nature of the composite
system allows us to reduce the above the situation to equations on the entries of the
coefficient matrices, ck , defined above.
24
Chapter 2
Existence of Minimally Supported
Frequency Composite Dilation
Wavelets
This chapter provides sufficient conditions for the existence of minimally supported
frequency composite dilation wavelets. In section 2.1, we establish two admissibility
conditions that are sufficient to produce minimally supported frequency composite
dilation wavelets. Section 2.2 provides a large family of groups that will admit an
arbitrary lattice. Section 2.3 proves that, in every dimension, 2(In ) satisfies the
admissibility condition for expanding matrices. As a result, this section ends by
proving that MSF, MRA composite dilation wavelets exist in every dimension. It is
advantageous to minimize the number of generating functions in the wavelet systems.
Families of singly-generated MSF composite dilation wavelets are developed in section
2.4. The chapter concludes with section 2.5 which discusses similar results when the
composite dilation wavelets generate Parseval frames rather than orthonormal bases.
25
2.1
Admissibility Conditions
This section develops sufficient conditions on composite dilation groups, full rank lattices, and expanding matrices that admit minimally supported frequency composite
dilation wavelets. We begin by defining two admissibility conditions. The first condition describes when a full rank lattice and the group used for composite dilations will
provide the appropriate sets to support the Fourier transfrom of the scaling function.
The second admissibility condition determines when an expanding matrix will provide
appropriate sets to support the Fourier transform of the wavelet generating functions.
These two admissibility conditions are then shown to be sufficient for the existence
of MSF, MRA composite dilation wavelets. We begin with a few of definitions.
Definition 2.1. A set S is starlike with respect to x if, for every y ∈ S, the convex
combination of y and x lies in S: {tx + (1 − t)y : t ∈ [0, 1]} ⊂ S.
Definition 2.2. A Starlike Neighborhood of x is a set S such that S is starlike with
respect to x and S contains an open neighborhood of x.
Definition 2.3. A full rank lattice Γ = cZn (c ∈ GLn (R)) is B-Admissible if there
exist a group B and a measurable set R ⊂ R̂n such that:
(i) R is a Γ∗ -tiling set for R̂n and
(ii) there exists a starlike neighborhood of 0, S, such that R is a B-tiling set for
S.
To better understand definitions 2.2 and 2.3, we look at a simple example in
two dimensions. Let B be the group of symmetries of the square generated
by
 the

 1 0  2
reflection through y = x and y = 0. Let Γ be the full rank lattice, Γ = 
Z .
−1 1
Then figure 2.1 shows a set R that satisfies (i) and (ii) from definition 2.3. The set S
in figure 2.1 is a starlike neighborhood of the origin.
26
R̂2
y=x
S
R
y=0
Figure 2.1: R is a Γ∗ -tiling set for R̂2 and a B-tiling set for S.
Definition 2.4. Let Γ be B-admissible and let S be a starlike neighborhood of 0
satisfying (ii) from definition 2.3. A matrix a ∈ GLn (R) is (B, Γ)-admissible if a is
expanding with S ⊂ Sa and there exist disjoint sets R1 , . . . , RL ⊂ Sa\S such that:
(i) for all l = 1 . . . L, Rl is a Γ∗ -tiling set for R̂n and
S
(ii) Ll=1 Rl is a B-tiling set for Sa\S.
Continuing our two
example, we have B, the group of symmetries of

 dimensional
 1 0  2
the square, and Γ = 
 Z . Then a = 2(I2 ) is (B, Γ)-admissible as shown in
−1 1
figure 2.2. We see that R1 , R2 , and R3 are Γ∗ -tiling sets for R̂2 and that R1 ∪ R2 ∪ R3
is a B-tiling set for Sa\S. From this figure we see how the MSF, MRA wavelets are
defined. The scaling function ϕ is defined by ϕ̂ = χR , while the wavelets, ψ l , are
defined by ψ̂ l = χRl for l = 1, 2, 3.
It only makes sense to define these two admissibility conditions if they are sufficient
for constructing composite dilation MSF wavelets. Indeed they are as the following
theorem establishes.
27
Sa
R̂2
y=x
R1
R
R2
R3
y=0
Figure 2.2: R1 , R2 , R3 are Γ∗ -tiling sets for R̂2 and R1 ∪ R2 ∪ R3 is a B-tiling set for
Sa\S.
Theorem 2.1. Let B be a group and Γ = cZn a B-admissible lattice. Let R ⊂ R̂n
be a set satisfying (i) and (ii) from definition 2.3. Let a ∈ GLn (R) be a (B, Γ)admissible matrix with sets R1 , . . . , RL satisfying (i) and (ii) from definition 2.4. Let
1
ψ 1 , . . . , ψ L ∈ L2 (Rn ) be functions such that ψ̂ l = |det(c)| 2 χRl for 1 ≤ l ≤ L. Then
Ψ = Daj Db Tk ψ l : j ∈ Z, b ∈ B, k ∈ Γ, 1 ≤ l ≤ L
is an orthonormal ABΓ-MRA composite dilation wavelet.
1
Proof. R is a Γ∗ -tiling set for R̂n . Then kχR k2 = |det(c)|− 2 . Also, since R is a
Γ∗ -tiling set for R̂n , the set
n
o
n
o
1
−2πiξk
2
{ek (ξ) : k ∈ Γ} := |det(c)| Mk χR (ξ) : k ∈ Γ := |det(c)| e
χR (ξ) : k ∈ Γ
1
2
(2.1)
1
2
is an orthonormal basis for L2 (R). Scaling the functions Mk χR by |det(c)| provides
the necessary normalization.
28
o
n
Now D̂b−1 : L2 (R) → L2 (Rb) is a unitary operator. Thus, D̂b−1 ek (ξ) : k ∈ Γ is
S
S
an orthonormal basis for L2 (Rb). Let S = b∈B Rb. Then L2 (S) = L2 ( b∈B Rb) =
L
2
b∈b L (Rb). Therefore,
o
o n
Mn
D̂b−1 ek (ξ) : k ∈ Γ = D̂b−1 ek (ξ) : b ∈ B, k ∈ Γ
(2.2)
b∈B
is an orthonormal basis for L2 (S). We now take the inverse Fourier transform of this
basis to obtain a basis for Lˇ2 (S).
Z
D̂b−1 ek ˇ(x) =
D̂b−1 ek (ξ)e2πiξx dξ
n
ZR
=
ek (ξb−1 )e2πiξx dξ
n
ZR
=
ek (ω)e2πiωbx dω
n
R
Z
1
2
e−2πiωk χR (ω)e2πiωbx dω
= |det(c)|
n
ZR
1
χR (ω)e2πiω(bx−k) dω
= |det(c)| 2
Rn
1
= |det(c)| 2 χ̌R (bx − k)
1
= |det(c)| 2 Db−1 Tk χ̌R (x)
So we have an orthonormal basis for Ľ2 (S):
o
n
1
|det(c)| 2 Db Tk χ̌R (x) : b ∈ B, k ∈ Γ .
(2.3)
1
Let V0 = Lˇ2 (S). Define ϕ(x) = |det(c)| 2 χ̌R (x). Then ϕ(x) satisfies (M5).
Since a is (B, Γ)-admissible, then S ⊂ Sa. Therefore, for all j ∈ Z, we define
29
Vj = Da−j V0 = Lˇ2 (Saj ). From this, we see that V0 ⊂ V1 :
L2 (Sa) = L2 ((Sa\S) ∪ S) = L2 (Sa\S) ⊕ L2 (S)
Then Ľ2 (S) ⊂ Lˇ2 (Sa), which is V0 ⊂ V1 . Therefore, for all j ∈ Z,
Vj = Da−j V0 ⊂ Da−j V1 = Da−j Da−1 V0 = Da−(j+1) V0 = Vj+1.
Thus, (M1) is satisfied.
(M2) is satisfied by the way we defined the sets Vj : if f ∈ Vj = Da−j V0 , then
−(j+1)
Da f ∈ Da V j = Da
V0 = Vj+1 .
Since Saj ⊂ Saj+1 for all j ∈ Z,
N
[
lim
N →∞
SN
j=−N
Saj = SaN . Then, since a is expanding,
Saj = lim SaN = Rˆn .
N →∞
j=−N
So we have
[
L2 (Saj )
= lim
N →∞
j∈Z
N
[
L2 (Saj ) = L2 ( lim SaN ) = L2 (R̂n ).
N →∞
j=−N
Therefore, (M3) is satisfied by
[
j∈Z
Vj =
[
Ľ2 (Saj ) = Lˇ2 (R̂n ) = L2 (Rn ).
j∈Z
In a similar fashion, we establish (M4). Since Sa−(j+1) ⊂ Sa−j then
Sa−N . Therefore
\
j∈Z
(2.4)
Saj = lim
N →∞
N
\
Saj = lim Sa−N = 0.
j=−N
30
N →∞
TN
j=−N
Saj =
Hence,
\
Vj =
j∈Z
\
Lˇ2 (Saj )
j∈Z
= Ľ2 (
\
Saj )
j∈Z
= Ľ2 (0) = 0
.
This establishes an MRA. In order to have an orthonormal wavelet system, we
must find the orthogonal complement to V0 in V1 . By the standard MRA wavelet
construction, if we find an orthonormal basis for W0 = V1 ∩V0⊥ , then we have a wavelet
system. Since V0 = Lˇ2 (S) and V1 = Ľ2 (Sa), then V1 ∩ V0⊥ = Lˇ2 (Sa) ∩ Lˇ2 (S)⊥ =
Lˇ2 (Sa\S). So define W0 = Ľ2 (Sa\S).
Using arguments similar to those leading to (2.1) and (2.2), we can find an orthonormal basis for L2 (Rl b) for each b ∈ B:
o
n
1
−1
|det(c)| 2 e−2πiξb k χRl (ξb−1 ) : k ∈ Γ .
Therefore, for each b ∈ B we have an orthonormal basis for Lˇ2 (Rl b) for all 1 ≤ l ≤ L:
o
n
1
|det(c)| 2 Db Tk χ̌Rl (x) : k ∈ Γ .
1
For each 1 ≤ l ≤ L, let ψl ∈ L2 (Rn ) be a function such that ψ̂(ξ) = |det(c)| 2 χRl (ξ).
1
Then ψ l (x) = |det(c)| 2 χ̌Rl .
S
By assumption, Ll=1 Rl is a B-tiling set for Sa\S. Thus
W0 = Lˇ2 (Sa\S) = Ľ2
L
[[
b∈B l=1
!
Rl b
31
=
L
MM
Ľ2 (Rl b)
b∈B l=1
=
M
b∈B
o
n
1
span |det(c)| 2 Db Tk χ̌Rl (x) : k ∈ Γ, 1 ≤ l ≤ L
o
n
1
= span |det(c)| 2 Db Tk χ̌Rl (x) : b ∈ B, k ∈ Γ, 1 ≤ l ≤ L
= span Db Tk ψ l (x) : b ∈ B, k ∈ Γ, 1 ≤ l ≤ L .
Defining Wj = Da−j W0 provides a sequence of spaces such that Vj ⊕ Wj = Vj+1
and Vj ∩ Wj = ∅. Then, as in the standard MRA argument,
Vj+1 = Vj ⊕ Wj = Vj−1 ⊕ Wj−1 ⊕ Wj = · · · =
j
M
Wl .
(2.5)
l=−∞
Therefore, using (2.4) and (2.5), we have
M
Wj = L2 (Rn ).
(2.6)
j∈Z
Finally
M
Wj =
M
Da−j W0
=
M
span Da−j Db Tk ψ l (x) : b ∈ B, k ∈ Γ, 1 ≤ l ≤ L
j∈Z
j∈Z
j∈Z
= span Da−j Db Tk ψ l (x) : j ∈ Z, b ∈ B, k ∈ Γ, 1 ≤ l ≤ L
= span Daj Db Tk ψ l (x) : j ∈ Z, b ∈ B, k ∈ Γ, 1 ≤ l ≤ L .
Then by (2.7), the theorem is proved.
32
2
(2.7)
2.2
Finite Groups and Lattices in MSF Composite
Dilation Wavelets
It is often beneficial to choose a lattice related to the composite dilation group, B, or
the expanding matrix, a. The goal of this section is to show that every full rank lattice
is B-admissible when B is a finite group with a fundamental region that is bounded
by hyperplanes through 0. This provides a considerable freedom to the composite
dilation wavelet system we choose. So we prove the following theorem:
Theorem 2.2. If B is a finite group whose fundamental region is bounded by hyperplanes through 0, then every lattice Γ = cZn is B-admissible.
First we need a series of a lemmas.
Lemma 2.3. Let G be any group, R be any set such that R = A ∪ B, A ∩ B = ∅
and R is a G-tiling set for a set V . If R′ = A ∪ Bgo for some go ∈ G, then R′ is a
G-tiling set for V .
Proof.
[
R′ g =
g∈G
[
g∈G
=
(A ∪ Bgo )g =
[
g∈G
Ag ∪
[
g ′ ∈G
[
g∈G
Bg
′
[
Ag ∪
Bgo g
g∈G
=
[
g∈G
(A ∪ B)g =
[
Rg = V.
g∈G
Since R is a G-tiling set for V , R∩Rg = ∅ for all g ∈ G. Hence R∩R(go ggo−1) = ∅.
Then
Rgo ∩ Rgo g = (R ∩ R(go ggo−1))go = ∅go = ∅.
Since A, B ⊂ R, then for all g ∈ G,
A ∩ Ag ⊂ R ∩ Rg = ∅
33
and
Bgo ∩ Bgo g ⊂ Rgo ∩ Rgo g = ∅.
Then for all g ∈ G
R′ ∩ Rg′ = (A ∪ Bgo ) ∩ (A ∪ Bgo )g
= (A ∩ Ag) ∪ (Bgo ∩ Bgo g)
= ∅.
Therefore, R′ is a G-tiling set for V .
2
Notice also that if R = A ∪ B is a G-tiling set for V , then A ∩ Bg = ∅ for all
g ∈ G. A, B ⊂ R then A ∩ Bg ⊂ R ∩ Rg = ∅.
Lemma 2.4. Let F be a region bounded by hyperplanes through 0. Let v be a vector
contained in the interior of F . Let K be any compact set. Then K can be translated
into F by adding v to K finitely many times.
Proof. For all x ∈ K, define Lx = {x + tv | t ∈ R}. Since v is in the interior of F and
no two hyperplanes bounding F are parallel, then Lx ∩ F 6= ∅ for all x ∈ K. Also, for
t
v ∈ F since
all x ∈ K define tx = inf {|x−f | : f ∈ Lx ∩F }. Then for all t ≥ tx , x+ |v|
t
v − x| = t ≥ tx and F contains its boundary hyperplanes. Since K is compact,
|x + |v|
l m
s
. Then l < ∞. For all x ∈ K, x + lv ∈ F since
s = supx∈K tx < ∞. Then let l = |v|
l≥
s
|v|
≥
tx
.
|v|
Therefore K + lv ⊂ F . Hence we can translate K into F by adding v
to K l times where l < ∞.
2
The preceding lemma may appear to be obvious. The proof verifies our intuition
but also provides us a method of calculating l and shows that l is the minimum
integer required to move K into F . This idea is needed more than once in the proof
of Theorem 2.2 and could be useful in any possible implementation.
34
Lemma 2.5. Suppose F is a fundamental region for a group B ⊆ GLn (R) and R is
S
a set such that R ⊂ F and Bǫ (0) ∩ F ⊂ R for some ǫ > 0. Let S = b∈B Rb. Then
(i) There exists α > 0 such that Bα (0) ⊂ S
(ii) if R is starlike with respect to 0, then S is starlike with respect to 0.
Proof. Since F is a fundamental region for B, then Rn =
S
b∈B
F b. For each b ∈ B,
define Tb : F → F b by Tb (x) = xb. Since b ∈ GLn (R), then Tb is an isomorphism.
(i) Since Tb is an isomorphism for each b ∈ B, Tb (Bǫ (0) ∩ F ) is open in F b. Also,
since 0 ∈ Bǫ (0) ∩ F and Tb (0) = 0, then 0 ∈ Tb (Bǫ (0) ∩ F ). Take ǫb > 0 such that
(Bǫb (0) ∩ F b) ⊂ Tb (Bǫ (0) ∩ F ). Define α = min{ǫb | b ∈ B}. If B is finite, α > 0. For
all b ∈ B, Bα (0) ∩ F b ⊂ Tb (Bǫ (0) ∩ F ). Then
[
b∈B
[
Tb (Bǫ (0) ∩ F )
Bα (0) ∩ F b ⊂
b∈B
and, therefore
Bα (0) ∩
Since
S
b∈B
[
b∈B
Fb ⊂
[
b∈B
(Bǫ (0) ∩ F )b.
F b = Rn and Bǫ (0) ∩ F ⊂ R, then Bα (0) ⊂
S
b∈B
Rb = S.
(ii) Suppose R is starlike with respect to 0. Since 0 ∈ R, then 0 ∈ S. Let y ∈ S.
Then there exist x ∈ R and bo ∈ B such that y = xbo = Tbo (x). Then for all t ∈ [0, 1],
ty = tTbo (x) = Tbo (tx), since Tbo is an isomorphism. Sine R is starlike with respect to
S
0, then tx ∈ R. Thus ty = Tbo (tx) ∈ Tbo (R) ⊂ b∈B Rb. So, for all y ∈ S, ty ∈ S for
all t ∈ [0, 1]. Therefore, S is starlike with respect to 0.
2
Lemma 2.6. Suppose F is a region bounded by hyperplanes through 0. Then F \(F +
v) is starlike with respect to 0 for any vector v in F .
Proof. By the definition of F , F is starlike with respect to 0. Also, we observe that
35
F C is starlike with respect to 0. Let x ∈ F \(F + v). Since x ∈ F and F is starlike
with respect to 0, then tx ∈ F for all t ∈ [0, 1]. Since x ∈
/ (F + v), then x − v ∈ F C .
Since F C is starlike with respect to 0 then for all t ∈ [0, 1], t(x − v) = tx − tv ∈ F C .
So for any t, tx ∈ F and tx − tv ∈ F C , then tx ∈ F \(F + tv) ⊂ F \(F + v). So for all
t ∈ [0, 1], tx ∈ F \(F + v). Therefore, F \(F + v) is starlike with respect to 0.
2
Now, armed with the preceding lemmas, we can establish Theorem 2.2.
Proof of Theorem 2.2. Let Γ = cZn and {γi }ni=1 = {êi c−1 }ni=1 be a basis for Γ∗ . Let
P
P = { ni=1 ti γi : ti ∈ [− 21 , 12 ]; i = 1, . . . , n}. Let F be a fundamental region for B,
then F is bounded by hyperplanes through 0.
Suppose γn is in the interior of F . Since P is compact and γn is in the interior of F ,
Lemma 2.4 tells us that there exists a smallest integer m such that P +mγn ⊂ F . Since
P is a Γ∗ -tiling set for Rˆn , for any 0 ≤ k < l ≤ m we have (P + kγn ) ∩ (P + lγn ) = ∅.
Now Define A0 = P ∩ F and for j = 1, ..., m, iteratively define
j−1
Aj = [{ P \
[
k=0
Ak + jγn } ∩ F ] − jγn .
Then Aj1 ∩ Aj2 = ∅ for all j1 6= j2 and P =
(2.8)
Sm
j=0 Aj .
Let us pause to look at an example of the sets defined by (2.8) in R̂2 . For B =
hr1 , r2 i, the
 group generated by reflections through y = x and y = 0, and Γ =

 1 0  2
 Z , figure 2.3 shows how we divide P into the sets Aj . A0 is the set

−2 2
P ∩ F . After removing A0 from P , we translate the remaining set by γ2 and take the
intersection with F . When we translate this intersection back into P , we have A1 .
Likewise, we obtain A2 by (2.8).
Returning to the proof, we define R =
Sm
j=0 (Aj
+ jγn ). Then, by lemma 2.2, R is
a Γ∗ -tiling set for R̂n . Since Bǫ (0) ⊂ P and A0 = P ∩ F , then Bǫ (0) ∩ F ⊂ A0 ⊂ R.
Now we must show the second condition in the definition of B-admissibility is also
36
R̂2
y=x
F
γ2
P
A0
y=0
γ1
A1
A2
Figure 2.3: P =
satisfied. First we observe that
m
[
Sm
Sm
j=0 (P
j=0
Aj = A0 ∪ A1 ∪ A2 .
+ jγn ) is starlike with respect to zero.
(P + jγn ) = {p + sγn : p ∈ P, s ∈ {0, ..., m}}
j=0
={
={
n
X
i=1
n−1
X
i=1
1 1
ti γi + sγn : ti ∈ [− , ]; s ∈ {0, ..., m}}
2 2
1 1
ti γi + tn γn : ti ∈ [− , ] i = 1, ..., n − 1; tn ∈ [0, m]}
2 2
This is clearly a convex set containing 0, hence
Sm
j=0 (P
+ jγn ) is starlike with
respect to 0. By lemma 2.6, F \(F + γn ) is starlike with respect to 0. Now the
intersection of two sets that are starlike with respect to 0 is also starlike with respect
to zero. So if the following claim is true, then R is starlike with respect to 0.
Claim 1.
R=
m
[
(P + jγn ) ∩ F \(F + γn )
j=0
Proof of Claim: By lemma 2.4, with γn ⊂ F , there exists a unique m ∈ N such
that P +mγn ⊂ F and P +(m−1)γn ∩F C 6= ∅. Since Aj ⊂ P , then Aj +jγn ⊂ P +jγn
37
Sm
j=0 (Aj + jγn ) ⊂
for all j = 1, . . . , m. Thus
Fix any j ∈ {0, . . . , m}. Suppose
Sm
j=0 (P
+ jγn ).
j−1
x ∈ (Aj + jγn ) = { P \
[
k=0
Ak + jγn } ∩ F.
Sj−1
Sj−1
A
+
jγ
.
Therefore
x
−
jγ
∈
P
\
k
n
n
k=0
k=0 Ak . Hence
Then x ∈ F and x ∈ P \
x − jγn ∈
/ Aj−1 . Then (x − jγn ) + (j − 1)γn ∈ F C so x − γn ∈ F C . Therefore
x∈
/ (F + γn ). So x ∈ F and x ∈
/ (F + γn ), thus x ∈ F \(F + γn ). Thus, we have
S
(Aj + jγn ) ⊂ F \(F + γn ). Since j ∈ {0, ..., m} was arbitrary, m
j=0 (Aj + jγn ) ⊂
F \(F + γn ). Therefore
R=
m
[
(Aj + jγn ) ⊂
j=0
m
[
(P + jγn ) ∩ F \(F + γn ) .
j=0
Sm
+ jγn ) ∩ F \(F + γn ) . Then x ∈ F \(F + γn ) so
S
∗
n
x ∈ F and x − γn ∈ F C . Also, x ∈ m
j=0 (P + jγn ). Since P is a Γ -tiling set for R̂ ,
Now, suppose x ∈
j=0 (P
then P + jγn ∩ P + kγn = ∅ for any j 6= k. So there exists a unique j ′ ∈ {0, . . . , m}
such that x ∈ P + j ′ γn . So, x ∈ (P + j ′ γn ) ∩ F .
Since x−γn ∈ F C then x−lγn ∈ F C for all l ∈ {1, . . . , j ′ }. Since x ∈ P +j ′γn then
x − lγn ∈ P + (j ′ − l)γn for all l ∈ {1, . . . , j ′ }. Hence (x − lγn ) ∈ [P + (j ′ − l)γn ] ∩ F C .
Thus for all l = 1, ..., j ′ we have x − lγn ∈
/ [P + (j ′ − l)γn ] ∩ F . Now
′
Aj ′ −l + (j − l)γn =
P\
j ′−l−1
[
k=0
Ak + (j ′ − l)γn ∩ F
⊂ P + (j ′ − l)γn ∩ F.
Hence, for all l = 1, ..., j ′ we have x−lγn ∈
/ Aj ′ −l +(j ′ −l)γn . Then x ∈
/ Aj ′ −l +j ′ γn .
38
Thus x ∈
/
Sj ′ −1
k=0 (Ak
+ j ′ γn ). So we have
′
x ∈ (P + j γn )\
j ′ −1
[
k=0
′
(Ak + j γn ) = P \
j ′ −1
[
k=0
(Ak ) + j ′ γn .
We already have x ∈ F , thus
x∈
So for all x ∈
Sm
j=0 (P
P\
j ′ −1
[
k=0
(Ak ) + j ′ γn ∩ F = Aj ′ + j ′ γn .
+ jγn ) ∩ F \(F + γn ) there exists j ′ ∈ {0, ..., m} such that
x ∈ Aj ′ + j ′ γn . Therefore
m
[
m
[
(P + jγn ) ∩ F \(F + γn ) ⊂
(Aj + jγn ) = R.
j=0
j=0
Therefore claim 1 is true.
Let us examine claim 1 with our two dimensional example. For B= hr1 , r2
i, the
 1 0  2
group generated by reflections through y = x and y = 0, and Γ = 
Z ,
−2 2
figure 2.4 shows how we build R as a union of translations of the sets Aj . In this
figure, we can also see how R is the intersection of the set F \(F + γn ), with a
S
union of translates of the parallelogram, P . In figure 2.4, R = 3j=0 (Aj + jγ2 ) =
i
hS
3
j=0 (P + jγ2 ) ∩ [F \(F + γ2 )].
Having established claim 1, we have determined that R is a Γ∗ -tiling set for R̂n , R
S
is starlike with respect to 0, R ⊂ F , and Bǫ (0) ∩ F ⊂ R. Define S = b∈B Rb. Then
by lemma 2.5, S is a starlike neighborhood of zero. Therefore Γ is B-admissible.
Our two dimensional example provides insight into how we constructed the sets
R and S. Figure 2.5depicts the
 sets R and S with our running example taking
 1 0  2
B = hr1 , r2 i and Γ = 
Z .
−2 2
39
R̂2
F \(F + γ2 )
A2 + 2γ2
P + 2γ2
A1 + γ2
P + γ2
F \(F + γ2 )
A0
P
Figure 2.4: R = A0 ∪ (A1 + γ2 ) ∪ (A2 + 2γ2 ) =
hS
2
j=0 (P
R̂2
S
R
Figure 2.5: S =
40
S
b∈B
Rb.
i
+ jγ2 ) ∩ [F \(F + γ2 )].
We assumed that γn was in the interior of F . If this is not the case, there are four
other possibilities:
1. There exists b ∈ B such that γn is in the interior of F b.
2. γn is not in the interior of F b for all b ∈ B, but there exists k ∈ {1, ..., n − 1}
such that γk is in the interior of F b for some b ∈ B.
3. For all k ∈ {1, ..., n}, γk is in the boundary of F b0 for a fixed bo ∈ B.
4. For all k ∈ {1, ..., n}, γk is in the boundary of F b for some b ∈ B.
In case 1, simply take F b as the fundamental region and proceed as above. In
case 2, take γk and b ∈ B such that γk is in the interior of F b. Then, relabel the basis
and proceed as above. For the third case, the parallelepiped spanned by the basis is
R and choose F b0 as the fundamental region. The little that remains of the proof
remains the same.
The final case requires very little as well. First choose the fundamental region,
F , such that γn is in the boundary of F . Choose a new basis {γ̃i }ni=1 with γ̃i =
±γi for all i = 1, . . . , n such that each γ̃i and the fundamental region, F , lie in
the same half space determined by the hyperplane containing γn . Now define P =
P
{ ni=1 ti γ̃i : 0 ≤ ti ≤ 1}. We may now follow the same procedure in the proof since
there exists a minimal m ∈ Z such that P + mγ˜n ⊂ F . Therefore, theorem 2.2 is
proved. 2
The set of all finite Coxeter groups is a subset of the family of groups that have a
fundamental region bounded by hyperplanes through 0. This is presented in Chapter
4 of Finite Reflection Groups by Grove and Benson [7]. This provides the following:
Corollary 2.7. If B is a finite Coxeter Group, then every lattice Γ = cZn is Badmissible.
Corollary 2.8. If B is a finite Rotation Group, then every lattice Γ = cZn is B41
admissible.
2.3
Existence of MSF Composite Dilation
Wavelets in L2(Rn).
This section shows that for any dimension, we can always find a composite dilation
wavelet system generating an orthonormal basis for L2 (Rn ). In fact, we can do so
with the same freedom as the previous section. We begin the section by showing that
2(In ) is always (B, Γ)-admissible when we chose a group, B, and a full rank lattice,
Γ, according to the preceding section.
Theorem 2.9. If B is a group whose fundamental region is bounded by n distinct
hyperplanes through the origin and Γ is any full rank lattice (Γ = cZn ), then a = 2In
is (B, Γ)-admissible.
Proof. Recall from Theorem 2.2 that P = {
Pn
i=1 ti γi
: ti ∈ [− 12 , 21 ], i = 1, . . . , n}
where {γi}ni=1 = {êi c−1 }ni=1 are the basis vectors for a Γ∗ . We assume that we have
already constructed the set R according to theorem 2.2 so that {γi }ni=1 is ordered such
that γn is in the interior of F , a fundamental region of R̂n for B. P is simply the
P
parallelepiped formed by the basis vectors {γi}ni=1 and then translated by − 12 ni=1 γi .
Therefore, P is a Γ∗ -tiling set for R̂n .
We make the following definitions to be used throughout the proof:
Ω=
[
(P + jγn ) =
( n−1
X
i=1
j∈Z
1 1
ti γi + βn γn : ti ∈ [− , ], βn ∈ R
2 2




n


X
Ki = 0γi +
βj γj : βj ∈ R ∀j for 1 ≤ i ≤ n − 1.




j=1
)
(2.9)
(2.10)
j6=i
n−1
Then {Ki }i=1
is a set of hyperplanes, each formed by the span of the basis vectors
42
not including the ith vector. Clearly L :=
Tn−1
i=1
Ki = {βn γn : βn ∈ R} is the line
defined by γn . We observe that Ω contains L as its center (with respect to Γ∗ ). We
n−1
bound
see immediately from the definition of Ω that the hyperplanes {Ki ± 12 γi }i=1
Ω in the sense that Ω is the nonempty intersection of the half-planes determined by
each hyperplane Ki ± 12 γi . (In the remianing discussion we say a set L is bounded by
a collection of hyperplanes {H1 , . . . , Hs } if and only if L is the nonempty intersection
of the half-spaces defined by the collection {H1 , . . . , Hs }.)
Now we proceed by dividing the region Ω into 2n−1 disjoint chambers by cutting
Ω in half with each of the hyperplanes Ki . For each i = 1, . . . , n − 1, Ki divides Ω
into two regions





±si γi +
n−1
X
j=1
j6=i



1
1 1
tj γj + βn γn : si ∈ [0, ], tj ∈ [− , ], βn ∈ R .

2
2 2

We can further divide each of these regions into two more regions by inserting one
n−1
of the other hyperplanes from {Ki }i=1
. Continuing in this fashion, we divide Ω into
disjoint chambers.
Let V denote the set of all (n − 1) × 1 vectors with entries from {−1, 1}. Then
|V | = 2n−1 . For each v ∈ V , define
Ωv =
( n−1
X
i=1
Then Ω =
S
v∈V
1
vi si γi + βn γn : v = (v1 , . . . , vn−1 ), si ∈ [0, ], βn ∈ R
2
)
(2.11)
Ωv . So we have partitioned Ω into 2n−1 disjoint chambers of equal
measure. By inspection we see that for each v ∈ V , Ωv is the region bounded by the
n−1
n−1
hyperplanes {Ki }i=1
and Ki + vi 12 γi i=1 .
Here we pause to examine the regions P , Ω, and Ωv along with the hyperplanes
Ki and Ki ± 21 γi in the two dimensional example that we used in the proof of theorem
43


2.2. For Γ = 

1 0  2
 Z , we show in figure 2.6 the sets P , Ω, Ω−1 , and Ω1 . These
−2 2
sets are defined in part by the hyperplanes K1 and K1 ± 12 γ1 .
R̂2
Ω
Ω−1
Ω1
P
K1
K1 − 12 γ1
K1 + 21 γ1
S
Figure 2.6: Ω =
j∈Z (P
+ jγ2 ) and Ω = Ω−1 ∪ Ω1 .
Returning to our proof, we recall from theorem 2.2 that for some m ∈ N
m
[
R=
(P + jγn ) ∩ F \(F + γn )
j=0
so
R⊂
[
(P + jγn ) = Ω.
j∈Z
Moreover, we see that
[
(R + jγn ) =
j∈Z
[
j∈Z
=
(
[
j∈Z
(" m
[
)
!
(P + lγn ) ∩ F \(F + γn ) + jγn
l=0
"m
[
#
#
(P + lγn ) + jγn
l=0
!)
∩
(
)
[ F \(F + γn ) + jγn .
j∈Z
(2.12)
44
Now the first term in (2.12) is simply Ω.
[
j∈Z
"m
[
#
(P + lγn ) + jγn
l=0
!
=
m
[[
(P + (j + l)γn )
j∈Z l=1
=
[
(P + j ′ γn )
j ′ ∈Z
= Ω.
(2.13)
Also, the second term in (2.12) is R̂n .
[ [
F \(F + γn ) + jγn =
({F + jγn } \ {F + (j + 1)γn })
j∈Z
j∈Z
=
[
(F + jγn )
(2.14)
j∈Z
since each portion that is dropped is then recovered in the union. Now let x ∈ R̂n .
Then {x} is a compact set. From lemma 2.4 we have l ∈ N such that x + lγn ∈ F .
Then x ∈ F − lγn . Therefore
[
(F + jγn ) = R̂n .
j∈Z
Hence, from (2.13) and (2.14) we have
[
j∈Z
(R + jγn ) = Ω ∩ R̂n = Ω.
F is a fundamental region for the group B and therefore is bounded by hyperplanes
through the origin. We let H denote the union of the portion of these hyperplanes
that form the boundary of the fundamental region, F . Since R ⊂ F \F + γn , then R
lies in the region of Ω bounded by H and H + γn , i.e. R ⊂ Ω ∩ {F \F + γn }. Now we
n−1
can describe R as the region bounded by Ki + vi 12 γi i=1 , H, and H + γn . For all
45
v ∈ V we define
Rv = R ∩ Ωv .
Since Ω =
S
v∈V
(2.15)
Ωv then
[
R=R∩
Ωv
v∈V
!
=
[
v∈V
(R ∩ Ωv ) =
[
Rv
(2.16)
v∈V
Here we see that we have a partition of R into 2n−1 disjoint chambers {Rv }v∈V . Since
n−1
n−1
Ωv is bounded by the hyperplanes {Ki }i=1
and Ki + vi 21 γi i=1 , then Rv is bounded
n−1
n−1 by the hyperplanes {Ki }i=1
, Ki + vi 12 γi i=1 , H, and H + γn .


 1 0  2
Continuing with our two-dimensional example where Γ = 
 Z and
−2 2
B = hr1 , r2 i, figure 2.7 shows the regions R and Rv (where v ∈ {1, −1}) along with
the hyperplanes K1 and K1 ± 21 γ1 .
R̂2
R
K1 − 21 γ1
K1
R−1
R1
K1 + 21 γ1
Figure 2.7: R = R−1 ∪ R1 .
Now we will describe how the expanding matrix a = 2In acts on R by observing
that we may look at its effect on the hyperplanes bounding R. We will also determine
how 2In acts on the hyperplanes bounding each Rv . Since R is bounded by the hypern−1
n−1
planes Ki ± 12 γi i=1 , then R(2In ) is bounded by the hyperplanes Ki ± 2 21 γi i=1
46
n−1
or {Ki ± γi }i=1
since every hyperplane is invariant under 2In and the translation of
the hyperplane is simply multiplied by 2:
Ki (2In ) =





0γi +
n
X
βj γj
j=1
j6=i



(2In ) =







2(0γi) +
n
X
j=1
2βj γj : βj ∈ R
j6=i
1
1
Ki ± γi (2In ) = Ki (2In ) ± 2 γi = Ki ± γi
2
2



= Ki


Each hyperplane forming the boundary of F is invariant under 2In . Thus the union
of the hyperplanes forming the boundary of F is also invariant under 2In . Therefore,
the boundary of F , which we call H, is invariant under 2In . Hence H(2In ) = H and
(H + γn )(2In ) = H + 2γn . Now R is also bounded by H and H + γn , thus R(2In ) is
bounded by H and H + 2γn . So we see that R(2In ) is bounded by the hyperplanes
n−1
{Ki ± γi }i=1
, H, and H + 2γn .
Similarly, we examine the bounding hyperplanes of Rv for all v ∈ V . For each
v = (v1 , . . . , vn ) ∈ V the bounding hyperplanes are simply the subset defined by v
n−1
of those hyperplanes bounding R and the addition of the hyperplanes {Ki }i=1
. Of
course, Rv is also bounded by H and H + γn . Therefore, the bounding hyperplanes
n−1
n−1
, H, and H + 2γn . We see that this is also a
, {Ki + vi γi }i=1
of Rv (2In ) are {Ki }i=1
parallelepiped. So from equation (2.16) we see that R is a union of parallelepipeds.
Returning to Rv , we can further divide the parallelepiped Rv (2In ) by reinserting
n−1
the original hyperplanes bounding Rv , namely Ki + vi 21 γi i=1 . These new regions
are simply translations of Rv . Thus, Rv (2In ) is a union of translates of the parallelepiped Rv .
Let W be the set of all n × 1 vectors with entries in {0, 1},
W = {w = (w1 , . . . , wn ) : wi ∈ {0, 1}} .
47
Then |W | = 2n . These translates of Rv that form Rv (2In ) will be defined as Rvw . The
translation is not necessarily in any direction defined by the lattice basis. However,
they are related to this basis in that we must translate in directions along the bounding
n−1
hyperplanes H to the intersections with Ki + vi 12 γi i=1 . The following describes
these translations.
Fix v ∈ V . Let Hv be the portion of H that is bounding Rv . That is Hv = H ∩Ωv .
Now define hv(i) ∈ H as the unique vector originating at the origin that will translate
Hv to the portion of H in the strip bounded by Ki + vi 12 γi and Ki + vi γi . Since Ki
is only defined for i = 1, . . . , n − 1, hv(n) is not defined. For clarity, we can define
hv(n) = γn . That is hv(i) is the unique vector such that
Hv + hv(i)
1
= H ∩ Ωv + vi γi .
2
Now we define the regions Rvw :
Rvw = Rv +
n−1
X
wi hv(i) + wn γn .
(2.17)
i=1
We must examine the bounding hyperplanes of Rvw for a fixed v ∈ V and an arbitrary
n−1
n−1 w ∈ W . We know that Rv is bounded by the hyperplanes {Ki }i=1
, Ki + vi 21 γi i=1 ,
H, and H + γn . So for a fixed v ∈ V and any w ∈ W , Rvw is bounded by the
hyperplanes
n−1
Ki + wi hv(i) + wn γn i=1
n−1
1
Ki + vi γi + wi hv(i) + wn γn
2
i=1
n−1
X
H+
wi hv(i) + wn γn
H + γn +
i=1
n−1
X
i=1
48
wi hv(i) + wn γn
(2.18)
(2.19)
(2.20)
(2.21)
We know hv(i) points to the subspace Ki + vi 21 γi ∩ H, thus a translation by wi hv(i)
applied to the region bounded by {Ki } and Ki + vi 12 γi is a translation by wi vi 21 γi .
By the definition of Ki (2.10), Ki + wn γn = Ki for all i = 1, . . . , n − 1. Therefore
(2.18) and (2.19) become
1
Ki + wi vi γi
2
1
Ki + vi (wi + 1) γi
2
n−1
(2.22)
i=1
n−1
(2.23)
i=1
Since v ∈ V is fixed, the portion of H in the appropriate subspace determined by v
will be invariant under translations by wi hv(i) . Therefore (2.20) and (2.21) become
Now the set
S
w∈W
H + wn γ n
(2.24)
H + (wn + 1)γn .
(2.25)
Rvw will be bounded by the outermost bounding hyperplanes,
those formed by taking wi = 0 for all i = 1, . . . , n in equations (2.18) and (2.20), and
by taking wi = 1 for all i = 1, . . . , n in equations (2.19) and (2.21). This tells us
S
that w∈W Rvw is the non-empty intersection of the half-spaces determined by the
hyperplanes
n−1
n−1
{Ki }i=1
, {Ki + vi γi}i=1
, H, H + 2γn .
(2.26)
The region bounded these hyperplanes is precisely Rv (2In ). Therefore we have established
Rv (2In ) =
[
Rvw
(2.27)
w∈W
R(2In ) =
[
Rv (2In ) =
v∈V
[ [
v∈V w∈W
49
Rvw
(2.28)
In two dimensions V = {−1, 1} and W = {(0, 0), (1, 0), (0, 1), (1, 1)}. Continuing
with our example where B = 
hr1 , r2 i, the
 group generated by reflections through
 1 0  2
y = x and y = 0, and Γ = 
 Z , figure 2.8 shows us how R (2In ) =
−2 2
S
S
v∈V
w∈W Rvw . The vectors h−1(1) = h−1 and h1(1) = h1 are also shown.
R̂2
,1)
,(1
R −1
R
,0)
,(1
−1
)
0,1
1,(
−
R
)
0,1
1,(
R
h−1
)
1,1
R 1,(
,0)
,(0
R −1
)
0,0
R 1,(
)
1,0
R 1,(
h1
Figure 2.8: R (2In ) =
S
v∈V
S
w∈W
Rvw .
Now we have established the necessary understanding and notation of the action
of the expanding matrix a = 2In on the region R. We complete the proof by showing
S
that each of the sets in the collection
satisfies (i) and (ii) in
v∈V Rvw : w ∈ W
definition 2.4.
Rvw is a Γ∗ -tiling of R̂n .
S
Proof of claim: First we establish that v∈V Rvw has the necessary measure to be a
S
Q
Γ∗ -tiling of R̂n , i.e. that m( v∈V Rvw ) = m(P ) = ni=1 |γi |. For each v ∈ V , Rvw is
Claim 2. For a fixed w ∈ W ,
S
v∈V
a parallelepiped and therefore we can compute its measure by taking the products of
the distances between its bounding hyperplanes in the directions of the basis vectors
50
{γi }ni=1 . Since |{H + wn γn } − {H + (wn + 1)γn }| = |γn |, then for each v ∈ V
m (Rvw ) = |γn |
= |γn |
= |γn |
=
=
Then, since
m
S
[
v∈V
Rvw
v∈V
1
n−1
Y Ki + wi vi 1 γi
2
i=1
n−1
Y
i=1
n−1
Y
1
1 vi γi 2 2n−1
i=1
S
v∈V
1
− Ki + vi (wi + 1) γi 2
|γi |
m(P )
Rvw is a disjoint union, we have
!
=
X
v∈V
m (Rvw ) =
X 1
m(P )
n−1
2
v∈V
= |V |
Therefore,
1
|vi | |γi |
2
i=1
n
Y
2n−1
1
2n−1
m(P ) = 2n−1
1
2n−1
m(P ) = m(P )
Rvw has the appropriate measure to be a Γ∗ -tiling of R̂n .
We now must show that any Γ∗ translates of the elements in the union are disjoint.
We will establish this by examining the translates in the γi directions individually.
We will see that no translates in the γi direction of any element in the union can
intersect any of the other elements in the union.
Claim 3. Let u, v ∈ V such that ui 6= vi . For any li ∈ Z and a fixed w ∈ W ,
(Ruw + li γi ) ∩ Rvw = ∅.
Proof of claim: Since ui 6= vi then ui = −vi . Without loss of generality, assume
vi = 1. Then Ruw is contained in the region
1
1
1
1
Ki + ui wi γi + tui γi : t ∈ [0, 1] = Ki − vi wi γi − tvi γi : t ∈ [0, 1]
2
2
2
2
51
=
1
Ki − (wi + t) γi : t ∈ [0, 1] .
2
So Ruw + li γi is contained in the region Ki + li − (wi + t) 21 γi : t ∈ [0, 1] . Also,
Rvw is contained in the region Ki + (wi + t) 21 γi : t ∈ [0, 1] . If there exists a point
x ∈ (Ruw + li γi ) ∩ Rvw then li − (wi + t) 21 = (wi + t) 21 so li = wi + t. Since t ∈
[0, 1], wi ∈ {0, 1}, li ∈ Z, then li ∈ {0, 1, 2}.
Suppose li = 0. Then wi + t = 0 so t = −wi . Since wi ∈ {0, 1} and t ∈ [0, 1] then
t = wi = 0. In this case, x ∈ Ki is a point in the boundary of each set Ruw and Rvw .
Suppose li = 1. Then 1 − t = wi ∈ {0, 1}. If wi = 0, then t = 1. If wi = 1, then
t = 0. In either case, x ∈ Ki + 21 γi , a bounding hyperplane.
Suppose li = 2. Then wi + t = 2. Since wi , t ≤ 1, then wi = t = 1. In this case,
x ∈ Ki + γi is a point in a hyperplane bounding the sets.
Therefore, we see that if u, v ∈ V such that ui 6= vi , then for any li ∈ Z, (Ruw +
li γi ) ∩ Rvw = ∅. Therefore, claim 3 is valid.
Claim 4. Suppose u, v ∈ V such that ui = vi . Let li ∈ Z and fix w ∈ W . If u 6= v,
then (Ruw + li γi ) ∩ Rvw = ∅.
Proof of claim: Ruw + li γi is contained in the region {Ki + vi wi 12 γi + tvi 12 γi + li γi :
t ∈ [0, 1]} and Rvw is contained in the region {Ki + vi wi 21 γi + tvi 21 γi : t ∈ [0, 1]}. Let
x ∈ (Ruw + li γi ) ∩ Rvw . Then there exists tx ∈ [0, 1] such that li + vi (wi + tx ) 21 =
vi (wi + tx ) 21 . Thus, li = 0. Therefore, either u = v or uj 6= vj for some j 6= i in which
case claim 3 verifies that they are essentially disjoint. Thus, claim 4 holds.
S
From claims 2 - 4 we see that for every w ∈ W , v∈V Rvw is a Γ∗ -tiling for R̂n .
S
To establish part (ii) of definition 2.4 we define S = b∈B Rb as in the proof of
theorem 2.2. Then
S(2In ) =
[
b∈B
52
!
Rb (2In )
=
[
Rb(2In )
[
R(2In )b
b∈B
=
(2.29)
b∈B
Therefore, using equations (2.28) and (2.29) we get
S(2In )\S =
(
[
b∈B
=
[
) (
R(2In )b \
[
)
Rb
b∈B
(R(2In )\R) b
b∈B
=
=
b∈B
(
[
[ [
[

[

b∈B
Then equation (2.30) tells us that
[
w∈W
w6=(0,...,0)
S
Rvw
v∈V w∈W
v∈V w∈W
b∈B
=
[ [
) (
\
Rvw \Rv0
[
v∈V
w∈W
w6=(0,...,0)
S
!
[
v∈V
b
b


Rvw  b
v∈V
Rv0
)!
(2.30)
Rvw is a B-tiling set of S(2In )\S.
Therefore claims 2, 3, 4 and equation (2.30) verify that for each w ∈ W the regions
(
[
v∈V
)
Rvw : w ∈ W \(0, . . . , 0)
satisfy (i) and (ii) from definition 2.4.
Therefore, a = 2In is (B, Γ)-admissible.
We began the proof assuming that we had ordered the basis of Γ∗ such that γn is
in the interior of the fundamental region, F . The other possible cases overlap those
from the proof of theorem 2.2. If γn is not in the interior of the fundamental region,
then each element of the basis {γi }ni=1 lies in a hyperplane bounding the fundamental
region, F . This actually simplifies the proof as each Ki is a hyperplane bounding the
set R. We eliminate the need to divide the region R into sectors using the hyperplanes
53
Ki + vi 12 γi . This is a specific case of the preceding proof.
2
Theorems 2.2 and 2.9 were proved by subtracting the vector
Pn
1
i=1 2 γi
from the
parallelepiped formed by the basis of Γ∗ . This is not the most general setting in
which we could have proved these theorems. Let P̃ be the parallelepiped formed by
P
the basis, {γi }ni=1 , of Γ∗ . Choose a vector α = (α1 , . . . , αn ) such that P̃ − ni=1 αi γi
P
contains an open neighborhood of the origin. Define P = P̃ − ni=1 αi γi . Let Ki
remain the same from the proof of theorem 2.9. Then P and subsequently Ω are
bounded by the hyperplanes {Ki − αi γi } and {Ki + (1 − αi )γi }. Keeping the same
definition for the set V of 1 × (n − 1) row vectors with entries from {1, −1}, then we
can define these hyperplanes by
1 + vi
− αi γi .
Ki +
2
We may then proceed to define R, Ωv , Rv , hv(i) , etc. This slightly more difficult
notation does not prevent us from proving theorems 2.2 and 2.9. In our proofs, we
i
− αi = vi 21 .
simplified the notation by taking αi = 21 for all i = 1, . . . , n so that 1+v
2
This proof of 2.9 highlights three things. First of all, in general, it is not that
easy to demonstrate that a matrix is (B, Γ)-admissible when we do not specify B
and Γ. On the other hand, if we specify B and Γ, it may not be any easier. Second,
the freedom of choosing a lattice does not complicate things too much. In potential
applications, one may need to choose the lattice so that R closely resembles the
geometry of the signal in the frequency domain. Finally, since we proved that 2In is
always (B, Γ)-admissible, we have the following theorem:
Theorem 2.10. If B is any group with fundamental region bounded by n hyperplanes
through the origin and Γ = cZn is any full rank lattice, then there exists a Minimally Supported Frequency, Multiresolution Analysis, Composite Dilation Wavelet
generating an orthonormal basis for L2 (Rn ).
54
Proof. From theorem 2.2, Γ is B-admissible. By theorem 2.9, a = 2In is (B, Γ)admissible. Therefore, by theorem 2.1 there exists an MSF composite dilation wavelet
generating an orthonormal basis for L2 (Rn ).
2
Example 2.1. Let us take the example we have had running through the proofs of
theorems 2.2 and 2.9. Let B be the group of symmetries of the square, i.e. B = hr1 , r2 i
where r1 is the reflection
 reflection through the
 the line y = x andr2 is the
 through
 1 0 
 1 0  2
line y = 0. Let Γ = 
. Let a = 2(In ). Then
 Z . Then Γ∗ = Ẑ2 
1 21
−2 2
Γ is B-admissible, a is (B, Γ)-admissible. So using theorem 2.1 we produce three
wavelets to generate the space W0 , our wavelet space. From figure 2.8, we observe
that these are the wavelets ψ 1 , ψ 2 , ψ 3 such that
ψ̂ 1 =
2.4
√
2χ[R−1,(1,0) ∪R1,(1,0) ] ,
ψ̂ 2 =
√
2χ[R−1,(0,1) ∪R1,(0,1) ] ,
ψ̂ 3 =
√
2χ[R−1,(1,1) ∪R1,(1,1) ] .
Singly Generated, MSF, Composite Dilation
Wavelets
In the preceding section, theorem 2.10 tells us that we can always find an MSF
composite dilation wavelet for L2 (Rn ). In [10] and [9], it is demonstrated that in
the case of MRA composite dilation wavelets, when the expanding matrix has integer
determinant the number of wavelet generating functions is the size of this determinant
minus 1. In Definition 1.3, this corresponds to L = |det(a)| − 1.
In section 2.3 we have shown that there exists a composite dilation wavelet with
2n wavelet generators. Obviously, as n grows, this would no longer be useful in applications. Even in dimension three, we already need 7 wavelet generating functions.
However, theorem 2.10 was quite general. We could pick almost any finite group
acting on Rn and any full rank lattice. If we make certain choices, such as picking
55
the lattice according to the group, we may reduce the number of wavelet generators.
Also, we generated MRA wavelets. If we are willing to surrender the MRA structure,
we may reduce the number of generators.
This section provides MSF composite dilation wavelets with a single wavelet generating function. The first subsection shows that a singly generated, MRA, MSF,
composite dilation wavelet exists for L2 (Rn ) for all n ∈ N. This requires us to give
up the freedom to choose the group or lattice. We may keep the freedom of choosing
our group if we are willing to give up the MRA structure as seen in the second part
of this section. The final portion of this section demonstrates that we may keep both
the MRA structure and some freedom in our choice of group when we restrict our
attention to two dimensions.
2.4.1
Singly Generated MRA, MSF, Composite Dilation
Wavelets for L2(Rn )
Here we pick a specific group, a specific full rank lattice, and a specific expanding
matrix in order to generate a singly generated MRA, MSF, composite dilation wavelet.
We want to generate an orthonormal basis for L2 (Rn ). In order to to let the dimension
be arbitrary, we give up the freedom of the previous section.
Theorem 2.11. There exists a singly generated, MRA, MSF, composite dilation
wavelet for L2 (Rn ).
Proof. First, we take the lattice to be Zn . Therefore, Γ∗ = Γ = Zn has the standard
basis vectors {êi }ni=1 = {ei }ni=1 . We let Hi be the hyperplane perpendicular to êi ,
i.e. Hi = ê⊥
i . Let B be the group generated by reflections through Hi for every
i = 1, . . . , n. We already know that Zn is B-admissible by theorem 2.2. Since êi
lies in a hyperplane bounding the fundamental region, this is case 3 in the proof of
56
theorem 2.2. Therefore, we define
R=
( n
X
i=1
)
ti êi : ti ∈ [0, 1] .
Then R is clearly a Ẑn tiling set for R̂n . We let
S=
[
Rb =
( n
X
i=1
b∈B
)
si êi : si ∈ [−1, 1] .
Then S is clearly a starlike neighborhood of the origin and R is obviously a B-tiling
set for S. Therefore, with this set R, Zn is B-admissible.
Define a as the modified permutation matrix sending ξ1 to ξn and ξi to ξi−1 for
i = 2, . . . , n with the modification that it doubles ξ2 . So








a=








0 0 0 ··· 0 1 

2 0 0 ··· 0 0 



0 1 0 ··· 0 0 


0 0 1 ··· 0 0 

.. .. .. . . .. .. 
. . . 
. . .


0 0 0 ··· 1 0
Then |det(a)| = 2 and an = 2In .
Now S = {(ξ1 , . . . , ξn ) : ξi ∈ [−1, 1] for i = 1, . . . , n} so
Sa = {(ξ1 , . . . , ξn ) : ξ1 ∈ [−2, 2], ξi ∈ [−1, 1] for i = 2, . . . , n} .
Then S ⊂ Sa. Also,
Sa\S = {(ξ1 , . . . , ξn ) : ξ1 ∈ [−2, −1], ξi ∈ [−1, 1] for i = 2, . . . , n}
∪ {(ξ1 , . . . , ξn ) : ξ1 ∈ [1, 2], ξi ∈ [−1, 1] for i = 2, . . . , n} .
57
We observe that R = {(ξ1 , . . . , ξn ) : ξi ∈ [0, 1]} and that
Ra = {(ξ1 , . . . , ξn ) : ξ1 ∈ [0, 2], ξi ∈ [0, 1], i = 2, . . . , n}
= R ∪ {(ξ1 , . . . , ξn ) : ξ1 ∈ [1, 2], ξi ∈ [0, 1], i = 2, . . . , n}
Let R1 = Ra\R. Then R1 = {(ξ1 , . . . , ξn ) : ξ1 ∈ [1, 2], ξi ∈ [0, 1], i = 2, . . . , n} = R +
ê1 . So R1 is a Ẑn tiling set for R̂n .
Let bj be the reflection through the hyperplane Hj . Then we have the following
action of B on R1 :
R1 b1 = {(ξ1 , . . . , ξn ) : ξ1 ∈ [−2, −1], ξi ∈ [0, 1], i = 2, . . . , n}
(2.31)
R1 bj = {(ξ1 , . . . , ξn ) : ξ1 ∈ [1, 2]; ξi ∈ [0, 1], i 6= j, i = 2, . . . , n; ξj ∈ [−1, 0]}
(2.32)
The group B, taking combinations from (2.32), gives all possible combinations of [0, 1]
and [−1, 0] for the every i = 2, . . . , n. Hence, for every i ∈ {2, . . . , n}, ξi can range
over [−1, 1]. The reflection through H1 (2.31) gives us the rest of Sa\S.
[
b∈B
R1 b = {(ξ1 , . . . , ξn ) : ξ1 ∈ [−2, −1], ξi ∈ [−1, 1] for i = 2, . . . , n}
∪ {(ξ1 , . . . , ξn ) : ξ1 ∈ [1, 2], ξi ∈ [−1, 1] for i = 2, . . . , n}
= Sa\S.
(2.33)
By (2.33), R1 is a B-tiling set for Sa\S. Therefore, a is (B, Zn )-admissible.
From theorem 2.1, with ψ = χ̌R1 ,
Ψ = Daj Db Tk ψ : j ∈ Z, b ∈ B, k ∈ Zn
is an orthonormal, aBΓ-MRA, composite dilation wavelet for L2 (Rn ).
58
2
Here our example is slightly different than the running example in the previous
sections. Remaining in R̂2 , let B = hb1 , b2 i be the group generated by b1 , the

reflection
 0 1 
through x = 0, and b2 , the reflection through y = 0. Let Γ = Z2 and a = 
.
2 0
S
Then figure 2.11 shows the set Sa, and we can clearly see how S = b∈B Rb and
S
Sa\S = b∈B R1 b. In this case, the singly generated, MRA, MSF, composite dilation
wavelet is the function ψ defined by ψ̂ = χR1 . The associated composite dilation
scaling function for the MRA is the function ϕ defined by ϕ̂ = χR .
R̂2
Sa
R
Figure 2.9: Theorem 2.11: S =
S
b∈B
R1
Rb and Sa\S =
S
B∈b
R1 b.
This demonstrates one of the potentially very useful properties of composite dilation wavelets. Without the dilations by the group B, we could generate a wavelet
system using ã = 2(In ) and a unit square centered at the origin. In that case, we
know the wavelet space would need |det(a)|−1 = 2n −1 wavelet generators. However,
the composite dilations allow us to transfer this need for generators into the group
B. Since B is finite, B has a finite number of generators. When this number is
considerably smaller that the determinant of the expanding matrix (in this section
B has n generators while |det(ã)| = 2n ) we actually reduce the input from 2n − 1 to
n+1 to create the orthonormal basis. If we used ã, any implementation would require
the definition of each of the 2n − 1 wavelet generators. Using composite dilations, an
implementation would only require the n elements of the dilation group B and the
single wavelet generator. It is certainly plausible that implementation algorithms can
exploit the group properties to take advantage of this fact.
59
2.4.2
Singly Generated Non-MRA, MSF, Composite
Dilation Wavelets for L2(Rn )
The theorem in the previous section provides MRA wavelets for any dimension at the
cost of surrendering the ability to choose the group or lattice. In this section, we can
retain the freedom to let the group B come from a very large family if we are willing
to sacrifice the MRA structure. It is often advantageous to have the lattice be related
to the group. We choose the basis for the lattice Γ∗ such that each basis vector lies
in a unique bounding hyperplane of the fundamental region for the group B.
Theorem 2.12. For any group, B, with fundamental region bounded by n hyperplanes
through the origin, there exists a non-MRA, MSF, composite dilation wavelet for
L2 (Rn ).
Proof. Let B be a group with fundamental region, F , bounded by n hyperplanes
through the origin. Choose a lattice Γ such that the basis vectors of Γ∗ each lie in a
unique hyperplane bounding F . We let R0 be the parallelepiped formed by the basis
vectors of Γ∗ , {γi }ni=1 :
Q0 = R0 =
(
n
X
i=1
)
ti γi : ti ∈ [0, 1] .
Clearly, R0 is a Γ∗ -tiling set for R̂n . Now we take a = 2In and for i = 1, . . . , n we
P
choose mi ∈ {0, 1} to form γ = ni=1 mi γi ∈ Γ∗ \ {0}. For k ∈ N, we define
−k
Qk = R0 a
k−1
X
+
γa−j
(2.34)
j=0
Rk =
(
k
[
j=0
Qj
) (
\
k
[
j=1
(Qj − γ)
)
(2.35)
The plan is to dilate the region Rk by a−1 and determine the intersection. This
60
nonempty intersection keeps the dilations by a from being orthogonal. In each step, we
remove this intersection. However, to keep the Γ∗ -tiling property, we must somehow
reinsert the intersection we have removed. We do that by translating the intersection
by γ ∈ Γ∗ . Very rapidly, these pieces become quite small, but we must take this
process to its limit to ensure the orthogonality of the a dilations.
First of all, for all k ∈ N,
Qk a−1 =
R0 a−k +
k−1
X
γa−j
j=0
= R0 a−(k+1) +
k−1
X
!
a−1
γa−(j+1)
j=0
−(k+1)
= R0 a
+
k
X
j=1
γa−j + γ − γ
= Qk+1 − γ.
(2.36)
So, we can have an equivalent description of Rk for all k ∈ N:
Rk =
(
=
(
=
(
k
[
Qj
) (
Qj
) (
j=0
k
[
j=0
\
\
k
[
j=1
k
[
(Qj − γ)
Qj−1 a−1
j=1
)
)
) (k−1
)
k
[
[
Qj \
Qj a−1 .
j=0
j=0
Now since a = 2In , we can describe the sets Qk explicitly.
−k
Qk = R0 a
+
k−1
X
γa−j
j=0
=
(
n
X
i=1
)
ti γi : ti ∈ [0, 1]
k−1
X 1
1
+
γ
2k
2j
j=0
61
(2.37)
) 1
1
+ 2 − k−1 γ
=
ti γi : ti ∈ 0, k
2
2
i=1
)
( n
n X
X
1
1
+
2 − k−1 mi γi
=
ti γi : ti ∈ 0, k
2
2
i=1
i=1
( n
)
X
1
1
1
=
ti γi : ti ∈
2 − k−1 mi , 2 − k−1 mi + k
2
2
2
i=1
(
n
X
(2.38)
From (2.38) and the fact that mi ∈ {0, 1}, we see that when a−1 is applied to the sets
Qk , the resulting set is contained in R0 :
Qk a−1
)
1
1
1
=
ti γi : ti ∈
2 − k−1 mi , 2 − k−1 mi + k
2
2
2
2
)
( i=1
n
X
1
1
1
=
ti γi : ti ∈
1 − k mi , 1 − k mi + k+1
2
2
2
( i=1
)
n
X
⊂
ti γi : ti ∈ [0, 1] = R0
( n
X
1
(2.39)
i=1
By (2.39) and (2.36) we have established for all k ∈ N
Qk+1 ⊂ R0 + γ
Qk a−1 = Qk+1 − γ ⊂ R0 .
(2.40)
(2.41)
Using the fact that R0 is a Γ∗ -tiling set for R̂n , (2.40), and (2.41), we can establish
that for all k ≥ 1 and all j ≥ 0
Qk ∩ Qj a−1 = ∅.
(2.42)
From equation (2.38) we obtain that for k 6= l
Qk ∩ Ql = ∅.
62
(2.43)
To establish (2.43), assume that x ∈ Qk ∩ Ql for k 6= l. Without loss of generality,
we assume l < k. Since γ 6= 0 and γ ∈ Γ∗ , there exist j ∈ {1, . . . , n} such that
mj 6= 0. Since x ∈ Ql , then (2.38) tells us that there is an s ∈ [0, 1] such that
1
mj + 2sl . Since x ∈ Qk then we have
xj = 2 − 2l−1
2−
1
2k−1
mj ≤
2−
1
2l−1
s
mj + l ≤
2
1
1
2 − k−1 mj + k .
2
2
(2.44)
Solving (2.44) for s in the center we arrive at
2−
1
2k−l−1
Since s ∈ [0, 1] then
mj ≤ s ≤ 2 −
2−
1
2k−l−1
1
2k−l−1
mj +
1
2k−l
.
(2.45)
mj ≤ 1
or
mj ≤
1
2−
1
2k−l−1
=
2k−l−1
.
2k−l − 1
(2.46)
With 2k−l−1 = 12 2k−l = (1 − 21 )2k−l = 2k−l − 2k−l−1, (2.46) becomes
2k−l − 2k−l−1
.
mj ≤
2k−l − 1
(2.47)
Since l < k, then 2k−l−1 ≥ 1 and we have mj ≤ 1. We know mj 6= 0, thus mj = 1.
However, if mj = 1, then (2.45) becomes
2−
1
2k−l−1
≤s≤ 2−
1
2k−l−1
+
1
2k−l
.
(2.48)
Therefore, 1 ≤ s and s ∈ [0, 1]. Thus, s = 1 and k = l + 1. Then the intersection can
only occur on the boundary. Therefore, in the measure sense, equation (2.38) holds.
Now we want to proceed by showing that the sets Rk ∩ Rk a−1 are shrinking to a
set of measure zero. We establish the following claim:
63
Claim 5. For all k ≥ 1, Rk ∩ Rk a−1 = Qk a−1 .
Proof of claim 5: We use the alternative definition of Rk we established by (2.37).
Rk ∩ Rk a−1 =
(
k
[
Qj
j=0
!
k−1
[
\
Qj a−1
j=0
!)
\
(
k
[
Qj a−1
j=0
!
\
k−1
[
Qj a−2
j=0
!)
.
(2.49)
The left hand set of the intersection in (2.49) has already removed every Qj a−1 for
0 ≤ j ≤ k − 1. So we have
Rk ∩ Rk a−1 =
(
k
[
j=0
Qj
!
k−1
[
\
Qj a−1
j=0
!)
\
(
k−1
[
Qk a−1 \
Qj a−2
j=0
!)
.
(2.50)
Using (2.42), we see Qk a−1 ∩ Qj a−2 = ∅ for all j ≥ 0. Then (2.50) is now
Rk ∩ Rk a−1 =
(
k
[
Qj
j=0
!
\
k−1
[
Qj a−1
j=0
!)
\
Qk a−1 .
(2.51)
Again, we use (2.42) to see that for j ≥ 1, Qj ∩ Qk a−1 = ∅. Thus (2.51) becomes
Rk ∩ Rk a−1 =
(
=
(
Q0 \
R0 \
k−1
[
Qj a−1
!)
−1
!)
j=0
k−1
[
Qj a
j=0
\
Qk a−1
\
Qk a−1 .
(2.52)
But we know from (2.41) that Qk a−1 ⊂ R0 so we have reduced this intersection to
Rk ∩ Rk a−1 = Qk a−1 \
k−1
[
Qj a−1
j=0
!
.
(2.53)
Now applying a−1 to equation (2.43), we see that Qk a−1 ∩ Qj a−1 = ∅ for every
0 ≤ j ≤ k − 1. Hence, we have demonstrated the claim as (2.53) is now
Rk ∩ Rk a−1 = Qk a−1 .
64
(2.54)
We use claim 5 to show that the measure of these intersection is going to zero as
k tends to infinity. Let m denote Lebesgue measure.
Claim 6. m(Qk ) =
1
m(R0 ).
2nk
Proof of claim 6: This is a simple induction argument on the sets. From the
definition of Q0 , R0 = Q0 . Hence
m(Q0 ) = m(R0 ) =
Assume that for all j < k, m(Qj ) =
1
m(R0 ).
20
1
m(R0 ).
2nj
We use (2.36) and the induction
hypothesis to see
m(Qk ) = m(Qk − γ)
= m(Qk−1 a−1 )
1
m(Qk−1 )
2n
1
1
= n n(k−1) m(R0 )
2 2
1
= nk m(R0 )
2
=
So the claim is valid.
Therefore, claims 5 and 6 show limk→∞ m(Rk ∩ Rk a−1 ) = limk→∞ m(Qk a−1 ) = 0.
Define R∞ = limk→∞ Rk . Then R∞ ∩ R∞ a−1 = ∅. Thus dilations by a of R∞ are
orthogonal. Also, if Rk was a Γ∗ -tiling set for all k ∈ N, then R∞ is also a Γ∗ -tiling
set for R̂n . We removed the intersections to keep orthogonality, but we added a Γ∗
translate of the intersection back in to maintain the tiling property. We see this in
the following straightforward claim.
Claim 7. For all k ∈ N, Rk is a Γ∗ -tiling set for R̂n .
Proof of claim 7: We proceed by induction on the sets Rk . First of we already
know that R0 is a Γ∗ -tiling set for R̂n . Assume that for all j < k + 1, Rj is a Γ∗ -tiling
65
set for R̂n .
k+1
[
Rk+1 =
Qj
j=0
=
(
=
(
k
[
!
k
[
\
Qj
!
Qj
!
j=0
j=0
k
[
−1
Qj a
j=0
\
\
k
[
Qj a−1
j=0
k−1
[
!
!)
Qj a−1
j=0
!
∪
(
Qk+1 \
\Qk a−1
)
∪
k
[
Qj a−1
j=0
(
Qk+1 \
!)
k
[
Qj a−1
j=0
= Rk \Qk a−1 ∪ Qk+1
= Rk \Qk a−1 ∪ Qk a−1 + γ .
!)
(2.55)
Let A = Qk a−1 and B = Rk \Qk a−1 = Rk \A. We know from (2.36) and claim 5
that Rk ∩ Rk a−1 = Qk a−1 . So A ⊂ Rk . Hence Rk = A ∪ (Rk \A) = A ∪ B. By the
induction hypothesis, Rk is a Γ∗ -tiling set for R̂n and by (2.55) Rk+1 = (A + γ) ∪ B.
Then lemma 2.2 establishes that Rk+1 is a Γ∗ -tiling set for R̂n . Hence, the claim is
true.
Since Rk is a Γ∗ -tiling set for R̂n for all k, then R∞ = limk→∞ Rk is also a Γ∗ -tiling
set for R̂n . This, along with the fact that the a dilates of R∞ are orthogonal, allows
us to define a orthonormal basis for L2 (R∞ ), namely
n
o
1
2
n
1
2
−2πiξk
|det(c)| Mk χR∞ (ξ) : k ∈ Γ = |det(c)| e
We observe that F =
S
j
j∈Z R∞ a .
o
χR∞ (ξ) : k ∈ Γ .
(2.56)
Therefore
o
n
1
|det(c)| 2 D̂aj Mk χR∞ (ξ) : j ∈ Z, k ∈ Γ
(2.57)
is an orthonormal basis for the fundamental region, F . Since F is a fundamental
66
region for B, then by definition F is a B-tiling set for R̂n and
R̂n =
[
F b.
b∈B
Therefore, since a is diagonal,
R̂n =
[
R∞ aj b =
[
b∈B
b∈B
j∈Z
j∈Z
R∞ baj .
(2.58)
Using (2.57) and (2.58) we establish an orthonormal basis for L2 (R̂n ), namely
n
|det(c)|
1
2
D̂aj D̂b Mk χR∞ (ξ)
o
: j ∈ Z, b ∈ B, k ∈ Γ .
(2.59)
Hence, taking the inverse Fourier transform of (2.59) we establish an orthonormal
1
basis for L2 (Rn ). Therefore, with ψ = |det(c)| 2 χ̌R∞ ,
Ψ = Daj Db Tk ψ : j ∈ Z, b ∈ B, k ∈ Γ
is a non-MRA, composite dilation wavelet for L2 (Rn ).
(2.60)
2
Example 2.2. Let B = hr1 , r2 i be the group generated
by the
 reflections through

 1 0  2
y = x and y = 0. Choose the full rank lattice, Γ = 
 Z , so that the dual
−1 1
lattice has basis elements in the lines bounding the fundamental region. As in the
proof, let a = 2(I2 ). Figure 2.10 depicts the set R∞ when we take γ = γ1 + γ2 . Figure
2.11 depicts the set R∞ when γ = γ2 . It is apparent in each fiugre how the sets Rk
are formed. In both cases, the wavelet ψ is defined by ψ̂ = χR∞ .
67
R̂2
00
11
11
00
00
11
00
11
000
111
000
111
1111
0000
000
111
0000
1111
0000
1111
000000
111111
0000
1111
000000
111111
000000
111111
000000
111111
000000
111111
00000000000
11111111111
000000
111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
0000000000000000
1111111111111111
00000000000
11111111111
0000000000000000
1111111111111111
2
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
R∞
γ
γ1
Figure 2.10: R∞ = limk→∞ Rk for γ = γ1 + γ2 .
R̂2
00
11
00
11
00
11
000
111
00
11
000
111
1111
0000
000
111
0000
1111
0000
1111
000000
111111
0000
1111
000000
111111
000000
111111
000000
111111
000000
111111
00000000000
11111111111
000000
111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
0000000000000000
1111111111111111
00000000000
11111111111
0000000000000000
1111111111111111
2
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
R∞
γ
γ1
Figure 2.11: R∞ = limk→∞ Rk for γ = γ2 .
68
2.4.3
Singly generated, MRA, MSF, Composite Dilation
Wavelets for L2(R2 )
The two previous subsections dealt with singly generated MSF composite dilation
wavelets for L2 (Rn ). Letting the dimension be arbitrary forced us to give something
up. If we restrict ourselves to two dimensions, we can maintain both the MRA
structure and the freedom in choosing our group B.
Theorem 2.13. If B is a finite Coxeter group generated by reflections through two
lines through the origin in R2 , then there exists a singly generated, MRA, MSF composite dilation wavelet for L2 (R2 ).
Proof. Let B be a finite Coxeter group acting on R2 with fundamental region F
bounded by two lines through the origin, l1 and l2 . Then B = hb1 , b2 i where bj is the
reflection through lj for j = 1, 2. Then b1 b2 is a rotation that generates the subgroup
H = hb1 b2 i. Since B is finite, H is finite and [B : H] = 2. Also, we know that H has
a fundamental region G = F ∪ F b2 . Now a finite rotation group in R2 is generated
by a rotation of
2π
.
|H|
Since l2 must bisect this angle, we know the angle between l1
and l2 , call it ∠l1 l2 , must be
and l2 is
π
n
2π
2|H|
=
π
.
|H|
Let |H| = n ∈ N. Thus the angle between l1
for some n ∈ N.
Choose Γ such that the basis vectors for Γ∗ , {γ1 , γ2}, lie in l1 and l2 , respectively.
√
Furthermore, we choose this basis so that |γ1 | = 1, |γ2 | = 2. As in the proof of
theorem 2.2, we let
R = {t1 γ1 + t2 γ2 : t1 , t2 ∈ [0, 1]} .
(2.61)
S
Rb a starlike neighbor-
Then by the same theorem, Γ is B-admissible with S =
hood of the origin.
b∈B
We denote the counter-clockwise rotation by an angle of
√
π
n
by ρ( πn ). Let a =
2ρ( πn ). Recall that in the frequency domain, the dilations act on the right. Therefore
69
a =
√
2ρ( πn ) =
∠l1 l2 =
π
n

√ 
2
cos( πn )

sin( nπ )

. Then a is an expanding matrix. Since
− sin( πn ) cos( nπ )
√
and |γ1 | = 1, |γ2 | = 2, then γ1 a = γ2 and γ2 a = (2γ1 ) b2 . Therefore, since
γ2 b2 = γ2 , we have
Ra = {t1 γ1 a + t2 γ2 a : t1 , t2 ∈ [0, 1]}
= {t1 γ2 + 2t2 γ1 b2 : t1 , t2 ∈ [0, 1]}
= {t1 γ1 b2 + t2 γ2 : t1 , t2 ∈ [0, 1]} ∪ ({t1 γ1 b2 + t2 γ2 : t1 , t2 ∈ [0, 1]} + γ1 b2 )
= Rb2 ∪ (Rb2 + γ1 b2 )
(2.62)
Applying b2 = b−1
2 to (2.62) we have
Rab2 = R ∪ (R + γ1 ) .
(2.63)
Figure 2.12 depicts equation (2.63) for a specific example. Let B be the group
generated by b1 , the reflection through y = 0, and b2 , the reflection through y = x.
π
ThenB is also 
generated by the
 rotation
ρ( 4 ) = b1 b2 and b2 ; B = hb1 , b2 i = hb1 b2 , b2 i.
 1 0  2
 1 1  √ π
Γ=
 Z and a = 
 = 2ρ( 4 ).
−1 1
−1 1
Resuming the proof, we look at the B orbit of Ra:
[
Rab =
b∈B
[
Rab2 b
[
[R ∪ (R + γ1 )] b
b∈B
=
b∈B
=
[
!
Rb
b∈B
=S∪
"
[
b∈B
70
∪
"
[
(Rb + γ1 b)
b∈B
#
(R + γ1 ) b .
#
(2.64)
R̂2
Ra
γ2
R
R + γ1
γ1
Figure 2.12: Rab2 = R ∪ (R + γ1 ).
In R2 , all rotations commute since ρ(θ)ρ(φ) = ρ(θ + φ) = ρ(φ)ρ(θ). Also when
we fix a line in R2 , call it l, then the reflection through the line l, call it rl , has a
commuting relationship with all rotations. This relationship is
ρ(θ)rl = rl [ρ(θ)]−1 = rl ρ(−θ).
(2.65)
Therefore, we can easily see that for any angle θ
ρ(θ)rl [ρ(θ)]−1 = ρ(θ)ρ(θ)rl = ρ(2θ)rl .
(2.66)
In our situation we had fixed the lines l1 , l2 in R2 and defined b1 = rl1 , b2 = rl2 as
the reflections through these lines. We also know that B = hb1 , b2 i = hb1 b2 , b2 i. From
√
) since ∠l1 l2 = πn . Also, a = 2ρ( πn ).
the previous discussion, we have b1 b2 = ρ( 2π
n
Therefore,
−1
a(b1 b2 )a
√
π 2π π 1
√
= 2ρ
ρ
ρ −
n
n
n
2
2π
=ρ
n
71
ab2 a−1
= b1 b2
π 1
√ π √
rl2 ρ −
= 2ρ
n
n
2
2π
rl2
=ρ
n
(2.67)
= (b1 b2 )b2
= b1
(2.68)
Since b1 b2 and b2 generate B, then by (2.67) and (2.68) we have aBa−1 = B. Hence
Sa =
[
b∈B
!
Rb a =
[
Rba =
b∈B
[
Rab.
(2.69)
b∈B
Thus, combining (2.64) and (2.69) and defining R1 = R + γ1 , we observe that S ⊂ Sa
and
Sa\S =
[
R1 b.
(2.70)
b∈B
Since R was a Γ∗ -tiling set for R̂n , γ1 ∈ Γ∗ , and R1 = R + γ1 , the lemma 2.2 provides
that R1 is a Γ∗ -tiling set for R̂n . Equation (2.70) provides that R1 is a B-tiling
√
set for Sa\S. Therefore, a = 2ρ( πn ) is (B, Γ)-admissible. So by theorem 2.1, for
1
ψ = |det(c)| 2 χ̌R1 , we have
Ψ = Daj Db Tk ψ : j ∈ Z, b ∈ B, k ∈ Γ
is a singly generated, orthonormal, MRA, MSF, composite dilation wavelet for L2 (R2 ).
2
Figure 2.13 depicts the sets Sa, Sa\S, S, R, and R1 for the example that ran
through the proof. Let φ̂ = χR and ψ̂ = χR1 . B = hb1 , b2 i = hb1 b2 , b2 i is the group
generated by b1 , the reflection through y = 0, and b2 , the reflection through y = x.
72


B is also generated by the rotation ρ( π4 ) = b1 b2 and b2 . Γ = 


a=

1 1  √ π
 = 2ρ( 4 ).
−1 1

1 0  2
 Z and
−1 1
R̂2
Sa
R
R1
Sa\S
S
Figure 2.13: R is a B-tiling set for S. R1 is a B-tiling set for Sa\S.
Now we may also obtain a singly generated, MRA, MSF composite dilation wavelet
with rotation groups and an expanding matrix that is a scalar multiple of an appropriate reflection.
Theorem 2.14. If B is a finite rotation group acting on R2 , there exists a singly
generated, MSF, MRA composite dilation wavelet for L2 (R2 ).
Proof. This proof is quite similar to the preceding proof. Let B be a rotation group,
73
|B| = n ∈ N. A fundamental region for B is necessarily bounded by two lines
2π 2π
= 2π
.
Thus
B
=
ρ( n ) . Choose Γ
through the origin, l1 and l2 . Here, ∠l1 l2 = |B|
n
and R as in the previous proof. Again, Γ is B-admissible with S = ∪b∈B Rb a starlike
neighborhood of the origin.
√
2r so that a is expanding.
√
√
2π
√1 γ2 . Similarly, with |γ2 | =
2,
γ
2γ1 r. So,
With |γ1 | = 1, then γ1 ρ( 2π
)
=
)
=
ρ(
2
n
n
2
√
√
√
) and γ2 a = 2γ2 r = 2γ2 = 2γ1 ρ( 2π
). Then
γ1 a = 2γ1 r = γ2 ρ( 2π
n
n
Let r = rl2 be the reflection through l2 . Define a =
Ra = {t1 γ1 a + t2 γ2 a : t1 , t2 ∈ [0, 1]}
2π
2π
+ t2 2γ1 ρ
: t1 , t2 ∈ [0, 1]
= t1 γ2 ρ
n
n
2π
= {2t1 γ1 + t2 γ2 : t1 , t2 ∈ [0, 1]} ρ
n
= [{t1 γ1 + t2 γ2 : t1 , t2 ∈ [0, 1]} ∪ ({t1 γ1 + t2 γ2 : t1 , t2 ∈ [0, 1]}) + γ1 ] ρ
2π
.
= [R ∪ (R + γ1 )] ρ
n
2π
n
(2.71)
Figure 2.14 depicts equation (2.71) for a specific example. Let B = ρ π4 be
√
the rotation group generated
by arotation of π4 , a = 2r where r is the reflection

 1 0  2
through y = x, and Γ = 
Z .
−1 1
) = R∪R1 .
We continue the proof by defining R1 = R+γ1. Then we have Raρ(− 2π
n
Using arguments similar to those in the previous proof, we show that aBa−1 = B:
aρ
2π
n
−1
a
√
1
2π
= 2rl ρ
rl √
n
2
2π
rl rl
=ρ −
n
2π
=ρ −
.
n
74
R̂2
Ra
γ2
R + γ1
R
γ1
−π
4
Figure 2.14: Raρ
= R ∪ (R + γ1 ).
Then, with S = ∪b∈B Rb, we have
Sa =
[
Rba
[
Rab
b∈B
=
b∈B
=
[
[R ∪ (R + γ1 )] b
b∈B
=
2π
Raρ −
n
b∈B
=
[
(
[
)
Rb
b∈B
=S∪
(
[
b∈B
∪
(
b
[
)
R1 b
b∈B
)
R1 b .
Then S ⊂ Sa and R1 is a B-tiling set of Sa\S. Lemma 2.2 provides that R1 is a
Γ∗ -tiling set for R̂n . Therefore, a is (B, Γ)-admissible.
1
Hence, with ψ = |det(c)| 2 χ̌R1 ,
Ψ = Daj Db Tk ψ : j ∈ Z, b ∈ B, k ∈ Γ
75
is a singly generated, orthonormal, MRA, MSF, composite dilation wavelet for L2 (R2 ).
2
Figure 2.15 depicts the sets Sa, Sa\S, S, R, and R1 for the example that
π through the proof of theorem 2.14. Let φ̂ 
= χR andψ̂ = χR1 . B = 
ρ 4 is
√  1 1
 1 0  2
group generated by the rotation ρ( π4 ). Γ = 
 Z and a = 2 
−1 1
−1 1
R̂2
Sa
R
R1
Sa\S
S
Figure 2.15: R is B-tiling set for S and R1 is a B-tiling set for Sa\S.
76
ran
the


.
2.5
MSF, Composite Dilation, Parseval Frame
Wavelets
So far, we have focused on composite dilation wavelets generating orthonormal bases
for L2 (Rn ). Relaxing the orthonormality condition can be advantageous in many
ways. Frames, especially Parseval frames, are used extensively in applications as
they may possess a useful redundancy with very little additional computational cost.
Additionally, it is usually easier to produce a Parseval frame than to produce an
orthonormal basis.
The same is true for minimally supported frequency composite dilation wavelets.
Obviously, every orthonormal basis is a Parseval frame, so the existence of Parseval
frame composite dilation wavelets has been established in the preceding sections.
However, it becomes easier to find Parseval frame composite dilation wavelets as the
orthonormality requires the characteristic functions to be supported on lattice tiling
sets for R̂n . To find MSF, composite dilation, Parseval frame wavelets, this condition
is relaxed to the situation where the characteristic functions are supported on sets
contained in lattice tiling sets for R̂n . This significantly reduces the effort required to
find such wavelets. We begin with versions of the admissibility conditions for Parseval
frames.
Definition 2.5. A lattice Γ = cZn (c ∈ GLn (R)) is PF-B-admissible if there exist a
group B and a measurable set R ⊂ R̂n such that:
(i) R ⊂ W where W is a Γ∗ -tiling set for R̂n and
(ii) there exists a starlike neighborhood of 0, S, such that R is a B-tiling set for
S.
Definition 2.6. Let Γ be PF-B-admissible and let S be a starlike neighborhood of
0 satisfying (ii) from definition 2.5. A matrix a ∈ GLn (R) is PF-(B, Γ)-admissible if
a is expanding with S ⊂ Sa and there exist R1 , . . . , RL ⊂ Sa\S such that:
77
(i) for all l = 1 . . . L, Rl ⊂ Wl where Wl is a Γ∗ -tiling set for R̂n and
S
(ii) Ll=1 Rl is a B-tiling set for Sa\S.
Because we only need to find sets contained in lattice tiling sets, these Parseval
frame admissibility conditions are much easier to satisfy. In the previous sections we
showed that we can satisfy them since any lattice that is B-admissible is also PFB-admissible and any matrix that is (B, Γ)-admissible is also PF-(B, Γ)-admissible.
However, we will prove analogous theorems in this section to highlight how the Parseval frame requirements are easier to satisfy.
Theorem 2.15. If B is a finite group whose fundamental region is bounded by hyperplanes through 0, then every lattice Γ = cZn is PF-B-admissible.
Proof. Suppose B is a finite group whose fundamental region is bounded by hyperplanes through 0 and Γ = cZn . Let F be a fundamental domain of R̂n for the
group B; that is, F is a B-tiling set for R̂n . Let {γi }ni=1 be a basis for Γ∗ . Define
P
P = { ni=1 ti γi : ti ∈ [0, 1]}. Then P is a Γ∗ -tiling set for R̂n . If we translate P , it
is still a Γ∗ -tiling set for R̂n . Therefore, translate P such that the origin is in the
interior of P . Say 0 ∈ P − α for some α ∈ R̂n . Define R = (P − α) ∩ F . Then
R ⊂ P − α. Also, since P − α and F are both starlike with respect to 0, then R is
S
starlike with respect to 0. By lemma 2.5, S = b∈B Rb is a starlike neighborhood of
0. Therefore, Γ is PF-B-admissible.
2
Again, we look at a two dimensional example. Figures 2.16 and 2.17 depict R and
1
),
S for α = ( 14 , 16
 B = hr1 ,r2 i, the group generated by reflections through y = x and
1 0  2
Z .
−2 2
For Parseval frames, we eliminated the need to replace the portion of the par-

y = 0, and Γ = 
allelepiped formed by the lattice basis that does not lie in F . It is also no longer
78
R̂2
y=x
F
γ2
R
P −α
γ1
y=0
Figure 2.16: PF-B-admissible: R = (P − α) ∩ F .
R̂2
y=x
S
F
R
Figure 2.17: PF-B-admissible: R is B-tiling set for S.
79
necessary to concern ourselves with how the lattice relates to the fundamental domain for B as we no longer need to translate any of the set (P − α) ∩ F c .
In the orthonormal case, finding an expanding matrix a ∈ GLn (R) is rather difficult. We actually only established that twice the identity matrix is always (B, Γ)admissible. However, every expanding matrix, a, that normalizes B and satisfies
S ⊂ Sa is PF-(B, Γ)-admissible since we can simply intersect the region Sa\S with
lattice tilings.
Theorem 2.16. Let B be a finite group whose fundamental region is bounded by hyperplanes through 0 and Γ = cZn . Suppose S is a starlike neighborhood of 0 satisfying
(ii) from definition 2.5. If a ∈ GLn (R) is expanding such that S ⊂ Sa and aB = Ba,
then a is PF-(B, Γ)-admissible.
Proof. Suppose R is any set satisfying the conditions of definition 2.5. Suppose
S
S = b∈B Rb is a starlike neighborhood of the origin and a ∈ GLn (R) is expanding
such that S ⊂ Sa and aB = Ba. Define Q = Ra\S. Then
[
Qb =
b∈B
[
(Ra\S)
b∈B
=
(
[
b∈B
[
) (
Rab \
[
 b∈B

Rb̃a \S
=


b̃∈B
!
[
=
Rb a\S
)
Sb
b∈B
= Sa\S.
(2.72)
Therefore Q is a B-tiling set of Sa\S.
Let P be the parallelepiped formed by taking the convex hull of the lattice basis for
Γ∗ as in the proof of theorem 2.15. Then P is a Γ∗ -tiling set of R̂n . Since Q is compact,
80
there exist a smallest integer L and α1 , . . . , αL ∈ Γ∗ such that Q ⊂
SL
l=1 (P
+ αl ).
Define Rl = Q ∩ (P + αl ). Then Rl ⊂ (P + αl ) and (P + αl ) is a Γ∗ -tiling set for R̂n .
S
Furthermore, by (2.72), Q = Ll=1 Rl is a B-tiling set of Sa\S.
Therefore, a is PF-(B, Γ)-admissible.
2
1
),
Continuing with the example from figures 2.16 and 2.17, we let α = ( 41 , 16
B =hr1 , r2 i,
 1 0
Γ=
−2 2
the
 group generated by reflections through y = x and y = 0, and
 2
 Z . The for a = 32 (I2 ), figure 2.18 shows Sa and how R and R1 are
B-tiling sets for S and Sa\S, respectively.
R̂2
Sa
y=x
F
R
R1
Sa\S
S
a = 23 I2
Figure 2.18: PF-(B, Γ)-admissible: R is B-tiling set for S, R1 is a B-tiling set for
Sa\S.
81
Again, it is important to have admissibility conditions that actually produce
wavelets. We will establish that given a, B, Γ satisfying the Parseval frame admissibility conditions, we are able to generate an MSF, composite dilation, Parseval frame
wavelet for L2 (Rn ). First, we cover a few elementary lemmas that will be useful.
∞
Lemma 2.17. If, for each l = 1, . . . , L, xli i=1 is a Parseval frame for L2 (Rn ), then
SL 1 l ∞
2
n
l=1 L xi i=1 is a Parseval frame for L (R ).
Proof. Let f ∈ L2 (Rn ). For each l,
P∞ f, xl 2 = kf k2 . Therefore
i
2
i=1
2
L X
∞ L X
∞
X
X
f, 1 xl = 1
f, xl 2
i
i L
L l=1 i=1
l=1 i=1
L
1X
kf k22
=
L l=1
=
1
L kf k22
L
= kf k22 .
Hence,
∞
xl
L i i=1
SL 1
l=1
is a Parseval frame for L2 (Rn ).
2
For the remaining discussion in this section, we simplify notation by defining
1
1
ek (ξ) = |det(c)| 2 Mk χR (ξ) = |det(c)| 2 e−2πiξk χR (ξ) where c ∈ GLn (R) and R are
obvious from context.
Lemma 2.18. Let Γ = cZn . If W is a Γ∗ -tiling set of R̂n and R ⊂ W , then
{ek : k ∈ Γ} is a Parseval frame for L2 (R).
Proof. We simply establish the Parseval identity. Let f ∈ L2 (R̂n ). Since R ⊂ W ,
n
o
1
kf kL2 (R) = kf χR kL2 (W ) . We know that the set |det(c)| 2 Mk χW : k ∈ Γ is an or-
thonormal basis for L2 (W ) and thus satisfies the Parseval identity. Hence,
kf k2L2 (R) = kf χR k2L2 (W )
82
E2
X D
1
=
f χR , |det(c)| 2 e−2πi(·)k χW k∈Γ
E2
X D
1
=
f, |det(c)| 2 e−2πi(·)k χW χR k∈Γ
E2
X D
1
=
f, |det(c)| 2 e−2πi(·)k χR k∈Γ
=
X
k∈Γ
|hf, ek i|2
Therefore, {ek : k ∈ Γ} is a Parseval frame for L2 (R).
2
Lemma 2.19. Let Γ = cZn . If W is a Γ∗ -tiling set of R̂n , R ⊂ W , and R is a
n
o
B-tiling set for S, then D̂b ek : b ∈ B, k ∈ Γ is a Parseval frame for L2 (S).
Proof. We know that {ek : k ∈ Γ} is a Parseval frame for L2 (R). Let f ∈ L2 (S).
Then
E2 X X D
E2
X X D
−1
e
f,
D̂
D̂
f,
e
=
b k k b
b∈B k∈Γ
b∈B k∈Γ
2
X
=
D̂
f
b 2
L (R)
b∈B
2
X Z =
D̂b f (ξ) dξ
b∈B
=
XZ
b∈B
=
XZ
b∈B
=
Z
R
R
|f (ξb)|2 dξ
Rb
S
S=
|f (ξ)|2 dξ
b∈B
Rb
|f (ξ)|2 dξ
= kf k2L2 (S) .
Therefore,
n
D̂b ek : b ∈ B, k ∈ Γ
frame for L2 (S).
2
o
satisfies the Parseval identity and is a Parseval
Now we are ready to establish that with a very large family of groups B we can
83
generate MRA, MSF, composite dilation Parseval frame wavelets for L2 (Rn ). The
following theorem uses the preceding lemmas and the standard MRA arguments.
Theorem 2.20. Suppose B is a finite group with fundamental domain bounded by
hyperplanes through the origin, Γ = cZn is PF-B-admissible with R and S satisfying
definition 2.5, and a ∈ GLn (R) is PF-(B, Γ)-admissible with R1 , . . . , RL satisfying
definition 2.6. Then, for ψ l =
1
L
1
|det(c)| 2 χ̌Rl ,
j
Da Db Tk ψ l : j ∈ Z, b ∈ B, k ∈ Γ, l = 1, . . . , L
is an MRA, MSF, composite dilation Parseval frame wavelet.
Proof. An MRA, Parseval frame wavelet has a scaling function ϕ and wavelets ψ l
for l = 1, . . . , L that generate Parseval frames for the spaces V0 and W0 respectively.
Define V0 = Lˇ2 (S) and Vj = Da−j V0 = Ľ2 (Saj ). Since S ⊂ Sa, then {Vj }∞
j=−∞ is a
nested sequence with Vj ⊂ Vj+1.
1
Define ϕ(x) = |det(c)| 2 χ̌R (x) = eˇ0 (ξ). Then [Tk ϕ]ˆ(ξ) = ek (ξ). Since R is a
n
o
B-tiling set for S, lemma 2.19 tells us that D̂b [Tk ϕ]ˆ : b ∈ B, k ∈ Γ is a Parseval
frame for L2 (S). Thus,
{Db Tk ϕ : b ∈ B, k ∈ Γ}
is a Parseval frame for V0 .
Suppose L = 1 so Sa\S =
1
S
b∈B
R1 b and R1 is contained in a Γ∗ -tiling set for R̂n .
Then, with ψ = |det(c)| 2 χ̌R1 , a similar argument provides that
{Db Tk ψ : b ∈ B, k ∈ Γ}
is a Parseval from for L2 (Sa\S). Define W0 = Lˇ2 (Sa\S). Then W0 = V1 ∩ V0⊥ .
The argument demonstrating that this is an MRA follows the argument in the
proof of theorem 2.1. Since ϕ generates a Parseval frame for V0 and ψ generates a
84
Parseval frame for W0 ,
j
Da Db Tk ψ : j ∈ Z, b ∈ B, k ∈ Γ
is an MSF, MRA, composite dilation Parseval frame wavelet.
If L > 1, then with ψ l =
1
L
1
|det(c)| 2 χRl , lemma 2.17 provides that
j
Da Db Tk ψ l : j ∈ Z, b ∈ B, k ∈ Γ, l = 1, . . . , L
is an MRA, MSF, composite dilation Parseval frame wavelet for L2 (Rn ).
2
The relaxation of the lattice tiling property creates a situation where Parseval
frame wavelets with the MRA structure are much easier to find. In the preceding
section, we discussed singly generated MSF, composite dilation wavelets. When we
sacrificed the MRA structure, we could achieve the desired single generator for every
group with the appropriate fundamental domain. If we want to keep the MRA structure and are willing to accept Parseval frames in lieu of the orthonormal basis, we may
obtain singly generated, MSF, MRA, composite dilation Parseval frame wavelets.
Theorem 2.21. For any group, B, with fundamental region bounded by n hyperplanes
through the origin, there exists a singly generated, MRA, MSF, composite dilation
Parseval frame wavelet for L2 (Rn ).
Proof. Recall the proof of theorem 2.12. Since we only need characteristic function
supported on a subset of a lattice tiling set, we can simply take the set R0 a−1 as the
support of our scaling function. Let’s see this explicitly.
Let B be a group with fundamental region, F , bounded by n hyperplanes through
the origin. Choose a lattice Γ = cZn such that the basis vectors of Γ∗ each lie in a
unique hyperplane bounding F . We let P be the parallelepiped formed by the basis
85
vectors of Γ∗ , {γi }ni=1 :
P =
( n
X
i=1
)
ti γi : ti ∈ [0, 1] .
Then P is clearly a Γ∗ -tiling set for R̂n . Now take a = 2In . Define R = P a−1 so
R=
( n
X
i=1
si γi : si ∈ 0,
1
2
)
.
Then R ⊂ P . By definition, R is a starlike with respect to the origin. Define
S
S = b∈B Rb. Then R is a B-tiling set for S and S is a starlike neighborhood of the
origin. Hence, with R and S as defined, Γ is PF-B-admissible.
The matrix a = 2In is expanding, S ⊂ Sa, and aB = Ba. By theorem 2.16, a is
PF-(B, Γ)-admissible. For clarity, we describe the sets explicitly. Since R = P a−1 ,
then Ra = P . Let Q = Ra\S = Ra\R as in the proof of theorem 2.16. Since
R ⊂ Ra ⊂ F , then Q ⊂ F . Therefore, R1 = Q ∩ F = Q = Ra\R. Then, the scaling
function is supported on R and the wavelet on R1 .
1
With ψ = |det(c)| 2 χ̌R1 , we have
j
Da Db Tk ψ : j ∈ Z, b ∈ B, k ∈ Γ
is a singly generated, MSF, MRA, composite dilation Parseval frame wavelet for
L2 (Rn ).
2
Figure 2.19 depicts a singly generated, MSF, MRA, composite dilation Parseval
frame wavelet. Let B = hr1 , r2 i be the group generated by reflections
 y=x
 through
 1 0  2
and y = 0. This is the group of symmetries of the square. Let Γ = 
 Z and
−1 1
a = 2(I2 ). Then taking P to be the parralellogram formed by the two basis vectors
for Γ∗ , we construct R by R = P a−1 as in the proof above. We let R1 = Ra\R.
Then the scaling function, ϕ, is defined by ϕ̂ = χR and the wavelet, ψ, is defined by
86
ψ̂ = χR1 .
R̂2
111111111111111
000000000000000
γ2
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
1
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
R
R
γ1
Figure 2.19: A singly generated, MSF, MRA, composite dilation Parseval frame
wavelet.
This shows us a particular way to construct such composite dilation Parseval frame
wavelets. Since we seek only subsets of the lattice tiling sets, this also significantly
loosens the restriction on the geometry of the supporting sets. In the orthonormal
case, we always ended with some union of parallelepipeds. However, here we may
take any geometry we desire as long as the B-tiling of the set supporting the scaling
function is a starlike neighborhood of the origin. For example, we may choose a ball.
Example 2.3. Let B be a group generated by reflections through n hyperplanes
through the origin. Let Γ = cZn be a lattice such that the basis elements of Γ∗ each
lie in a unique bounding hyperplane. Let t = min {|γi | : i = 1, . . . , n}. Define S to
be the n dimensional ball centered at the origin with radius 12 t. A ball centered at
the origin is invariant under B. Let R = S ∩ F where F is a fundamental domain for
P
B. Then with R and S, Γ is PF-B-admissible since R ⊂ { ni=1 ti γi : ti ∈ [0, 1]} and
S
S = b∈B Rb is obviously a starlike neighborhood of the origin.
Let a = 2In . Then R ⊂ Ra and since Ra has radius t, t ≤ |γi | for all i = 1, . . . , n,
87
Ra ⊂ {
Pn
i=1 ti γi
Sa =
[
b∈B
: ti ∈ [0, 1]}. Now, with R1 = Ra\R,
Rba =
[
b∈B
Rab =
(
[
)
R1 b
b∈B
∪
(
[
)
Rb
b∈B
=
(
[
)
R1 b
b∈B
∪ S.
Thus S ⊂ Sa, R1 is contained in a Γ∗ tiling set for R̂n , and R1 is a B-tiling set for
1
Sa\S. Therefore, a is PF-(B, Γ)-admissible. Hence, with ψ = |det(c)| 2 χ̌R1 , we have
j
Da Db Tk ψ : j ∈ Z, b ∈ B, k ∈ Γ
is an MSF, MRA, composite dilation Parseval frame wavelet for L2 (Rn ).
2
Figure 2.20 depicts
such
 a wavelet in R when B is the group of symmetries of

 1 0  2
the square, Γ = 
 Z and a = 2(I2 ). In this case the scaling function, ϕ, is
−1 1
defined by ϕ̂ = χR and the wavelet, ψ, is defined by ψ̂ = χR1 .
R̂2
γ2
R
R1
γ1
Sa\S
S
Figure 2.20: A singly generated, MSF, MRA, composite dilation Parseval frame
wavelet from a sphere.
Next we present an example due to Alexi Savov [14] which provides a singly
88
generated, MRA, MSF composite dilation Parseval frame wavelet for L2 (R3 ). This
system exploits the symmetries of a regular polygon in R2 by looking at a double
pyramid. This gives us an example in three dimensions that has a potentially useful
geometric structure.
Example 2.4. Let P be a regular n-gon in R̂2 , n > 4. We will denote R̂2 as the ξ1 , ξ2
plane. Let B̃ be the group of symmetries of P and P̃ a B̃-tiling set for P under the
S
action of B, P = b∈B P̃ b is a disjoint union.
Figure 2.21 shows P̃ and P when n = 5. When B is the group of symmetries of
a pentagon, the triangle P̃ is a B̃-tiling set for the darker pentagon P .
P̃
Figure 2.21: For n = 5, P̃ is a B̃-tiling set for P and P ⊂ P a.
Select a point h = (0, 0, h) on the ξ3 axis and generate the pyramid, Q, determined
S
by P and h. Define R as the convex hull of P̃ and h. Then Q = b̃∈B̃ Rb̃.
Let B be the group obtained by adding the reflection through the ξ1 , ξ2 plane to
B̃; with r the reflection through the ξ1 , ξ2 plane,

 1 0 0

r=
 0 1 0

0 0 −1
D



,


E
S
we may write B = B̃, r . Also, we let S = Q∪Qr. Then B fixes S and S = b∈B Rb.
89
Let

1


π
a=
 − tan( n )

0

tan( nπ )
0
1
0
0
2 cos2 ( πn )


.


(2.73)
From (2.73), we see that we can write a as the product of a rotation by − nπ in the
ξ1 , ξ2 plane and an expanding matrix. Namely,

cos( πn )
sin( nπ )

0 




π
π
a = ρα = 
)
cos(
)
0
−
sin(


n
n


0
0
1
1
cos( π
)
n
0
0
1
cos( π
)
n
0
0
0
2 cos2 ( nπ )
0



.


(2.74)
Since we chose n > 4, the matrix α is expanding. With n > 4, cos( πn ) > √12 .
√
1 Therefore, the eigenvalues of α are cos( π ) > 2 and 2 cos2 ( πn ) > 1. Since ρ is a
n
rotation, |det(ρ)| = 1. So a is the composition of a rotation in the ξ1 , ξ2 plane and an
expanding matrix, and is therefore expanding.
Now we observe that a normalizes B. We write B = aBa−1 = ραBα−1 ρ−1 . Since
α is diagonal, α will commute with every element of the group B. Thus B = ρBρ−1 .
However, ρ is a rotation by − nπ in the ξ1 , ξ2 plane and, therefore, as discussed in
section 2.4.3, ρ normalizes B̃. Since r fixes R̂2 , ρ and r commute. Thus ρ normalizes
D
E
B = B̃, r .
Finally, we show that S ⊂ Sa. First we observe the action of a−1 on P , the n-gon
base of S in the ξ1 , ξ2 plane. Let, v = (v1 , v2 , 0), w = (w1 , w2 , 0) be any two adjacent
vertices of P . Then |v| = |w|. Also ∠vw =
2π
.
n
Hence, the midpoint of the edge of P
determined by v and w is defined by a vector of length cos( nπ ) |v|. Then the inscribed
circle of P has radius cos( πn ) |v|.
For all ξ0 = (ξ1 , ξ2 , 0) ∈ P , we know |ξ| ≤ |v|. Since a−1 = α−1 ρ−1 and ρ−1 is
a rotation of P , then |ξ0 a−1 | = |ξ0 α−1 | = cos( nπ ) |ξ0 | < cos( πn ) |v|. Thus, P a−1 is
contained in the inscribed circle of P establishing P a−1 ⊂ P .
90
Figure 2.21 depicts P ⊂ P a for a pentagon base. With a pentagon base, n = 5
and a is defined by equation (2.74).
Now we observe the action of a−1 on S. Recall that Q was the pyramid determined
by P and a point (0, 0, h). We observe the
(0, 0, h)a−1 = (0, 0, h)α−1 =
1
|h| < |h| .
2 cos2 ( nπ )
Since a−1 is also the product of a rotation in the ξ1 , ξ2 plane and a diagonal matrix,
then a−1 will take the convex hull of P and (0, 0, h) to the convex hull of P a−1 and
(0, 0, h)a−1 . With P a−1 ⊂ P and |(0, 0, h)a−1 | < |h|, then Qa−1 ⊂ Q. Now
Sa−1 = [Q ∪ Qr] a−1 = Qa−1 ∪ Qra−1 .
Since r is diagonal, ra−1 = a−1 r. So we have
Sa−1 = Qa−1 ∪ Qa−1 r = Qa−1 ∪ Qa−1 r ⊂ Q ∪ Qr = S.
Therefore, S ⊂ Sa.
With our example for n = 5, figure 2.22 shows the relation S ⊂ Sa for a defined
by equation (2.74).
Now we have a set R that is a B-tiling set of S. We need to choose a lattice
Γ = cZn such that R is contained a Γ∗ -tiling set of R̂n . With such a lattice, Γ is
PF-B-admissible.
Since a is expanding, S ⊂ Sa, and a normalizes B, a is PF-(B, Γ)-admissible.
However, to obtain a singly generated composite dilation wavelet, we also need that
R1 = Ra\S is contained in a lattice tiling set of R̂n . Therefore, we simply choose Γ
so that both R and R1 are contained in lattice tiling sets.
Alternatively, we can simply scale S such that R and R1 are contained in [0, 1]3 .
91
Figure 2.22: S ⊂ Sa for a double pyramid with pentagon base.
Choose P such that the length of any vertice is cos( nπ ). Then P a has vertices of length
1. Additionally, choose h = cos−2 ( πn ). With n > 4, we can choose R, R1 ⊂ [0, 1]3 . If
we choose P so that one of the vertices lies on the positive ξ1 axis, then the largest S
such that R, R1 ⊂ [0, 1]3 is the double pyramid with vertices
π
2π
1
cos( ) cos( j) : j = 0, . . . , n − 1 ∪ (0, 0, ± 2 π
.
n
n
cos ( n )
This section highlights the ease with which we may obtain MRA, MSF, composite
dilation Parseval frame wavelets. Because the measure of the support sets for the
scaling function and wavelet are no longer important, it is always the case that we
may take a Parseval frame composite dilation wavelet and reduce the number of
92
generators to one. To do so, we may simply scale the original support of the scaling
function so that it and the support of the wavelet are contained in a lattice tiling.
With this in mind, relaxing the requirement of an orthonormal basis to that of a
Parseval frame may be extremely useful.
93
Chapter 3
Accuracy of Compactly Supported
Composite Dilation Wavelets
From Cabrelli, Heil, and Molter, [2] we have necessary conditions for the existence
of a system generated by several, multidimensional functions to have a specified level
of accuracy. Here we are concerned with the situation where the several functions
are defined by a group acting on a single function by dilation. That is, if ϕ is an
integrable function and B ∈ GLd (R) is a group such that |B| = r < ∞, then ϕb (x) =
Db ϕ(x) = ϕ(b−1 x). When we fix a specific ordering to the list of elements in B, namely
B = {I, b2 , . . . , br } we can write ϕi = ϕbi = Dbi ϕ where ϕ1 = DI ϕ = ϕ. Then, writing
the r functions as a column vector, we have Φ(x) = (ϕ1 (x), ϕ2 (x), . . . , ϕr (x))t .
We want Φ(x) to have compact support and be a refinable function, i.e.
Φ(x) =
X
k∈Λ
ck Φ(Ax − k)
(3.1)
for some Λ ⊂ Γ, Λ finite, some r × r matrices ck , and A and expanding matrix. We
also require Λ to be invariant under the action of B, namely B(Λ) = Λ. In this
j
case, we call {Φj,i,k } = {DA
Dbi Tk ϕ | i = 1, . . . , r; j, k ∈ Z} a compactly supported,
refinable Composite Dilation System.
94
First of all, we want to show that Φ : Rd → Cr is integrable. Since ϕ ∈ L1 (Rd ),
we have kϕk1 < ∞. With B ⊂ GLd (R) and |B| = r < ∞ then for all b ∈ B we know
|det(b)| = 1. Thus, for all b ∈ B,
kϕb k1 =
Z
Rd
|ϕb (x)|dx =
=
Z
Rd
Z
−1
Rd
|ϕ(b x)|dx =
Z
bRd
|ϕ(y)||det(b)|dy
|ϕ(y)|dy = kϕk1 < ∞.
Therefore, ϕb ∈ L1 (Rd ) for all b ∈ B. Hence, Φ ∈ L1 (Rd , Cr ).
3.1
Matrix Relations from Composite Dilations
This section introduces a simplification for composite dilation systems to the necessary
conditions provided in [2]. As we mentioned at the end of section 1.3, we may take
the point of view that composite dilation systems have r = |B| generating functions
related by the composite dilation group, B. This relationship from the group action
defines a set of identities for the entries of the coefficient matrices from the refinment
equation. Therefore, we are able to significantly reduce the number of free entries
in our coefficient matrices by establishing matrix relations for composite dilation
systems.
Suppose Φ(x) is a compactly supported refinable function. Then there exist Λ ⊂ Γ,
Λ finite, a set of r × r matrices ck (where ck = 0 for k ∈
/ Λ), and an expanding matrix
A such that
Φ(x) =
X
k∈Λ
ck Φ(Ax − k).
95
Then

ϕ (x)
 1

 ϕ2 (x)

 .
 .
 .

ϕr (x)


ϕ (Ax − k)
 1



 X  ϕ2 (Ax − k)


ck 
=
..


.
 k∈Λ 


ϕr (Ax − k)


P
Pr
k
j=1 c1,j ϕj (Ax
− k)
  k∈Λ
  P
Pr
k
 
  k∈Λ j=1 c2,j ϕj (Ax − k)
=
..
 
.
 
 
P
Pr
k
j=1 cr,j ϕj (Ax − k)
k∈Λ





.



This Composite Dilation System places certain relations on the entries of the r × r
matrices ck . We know from the above equation that for each i = 1, . . . , r we have
ϕi (x) =
r
XX
k∈Λ j=1
cki,j ϕj (Ax − k).
Simultaneously, we know that ϕi is defined by the action of bi providing
ϕi (x) = ϕ1 (b−1
i x) =
r
XX
k∈Λ j=1
ck1,j ϕj (Ab−1
i x − k).
So we have the following equation:
r
XX
cki,j ϕj (Ax
k∈Λ j=1
− k) =
r
XX
k∈Λ j=1
ck1,j ϕj (Ab−1
i x − k).
Suppose that A normalizes B. That is, for each b ∈ B there exists b̃ ∈ B such that
b̃ = AbA−1 . Then, writing ϕbi = ϕi , we have
r
XX
k∈Λ l=1
cki,l ϕbl (Ax
− k) =
=
=
r
XX
k∈Λ j=1
r
XX
k∈Λ j=1
r
XX
k∈Λ j=1
96
ck1,j ϕbj (Ab−1
i x − k)
ck1,j ϕbj (b̃−1
i Ax − k)
ck1,j ϕbj (b̃−1
i (Ax − b̃i k))
=
=
r
XX
k∈Λ j=1
r
XX
k∈Λ j=1
ck1,j ϕb̃i bj (Ax − b̃i k)
b̃−1 k
i
ϕb̃i bj (Ax − k).
c1,j
Since B is a group, there exists bl ∈ B such that b̃i bj = bl . In each case we have
b̃−1 k
i
cki,l = c1,j
or ck1,j = cb̃i,li k .
(3.2)
Therefore, the free entries in the r×r matrices ck are only the elements of the first row
of the matrix. So when attempting to find appropriate matrices for the refinement
equation, we have |B||Λ| = r|Λ| free variables.
Example 3.1. Let d = 1, B = {b1 , b2 } = {1, −1}, Λ = {−n, . . . , 0, . . . , n} ⊂
Z, A = 2. Here Φ(x) = (ϕ(x), D−1 ϕ(x))t = (ϕ(x), ϕ(−x))t . We see that |B| = 2 and
|Λ| = 2n + 1 so there are 4n + 2 free entries for the associated matrices. From (3.2),
we have the general relations ck1,j = cb̃i,li k . Here B consists of the two scalars 1 and -1
and A is the scalar 2 so A and B commute. Thus b̃ = b for all b ∈ B. Then:
b̃1 b1 = b1 b1 = 1 · 1 = 1 = b1
b̃1 b2 = b1 b2 = 1 · −1 = −1 = b2
b̃2 b1 = b2 b1 = −1 · 1 = −1 = b2
b̃2 b2 = b2 b2 = −1 · −1 = 1 = b1
Hence, for j = 1 we have l = i, and for j = 2 we have l = (i mod2) + 1. So
−k
k
k
k
ck1,1 = ck1,1 , ck1,1 = c−k
2,2 , c1,2 = c1,2 , c1,2 = c2,1 .
97
Now dropping the comma in the subscript, the useful relations are
k
−k
ck11 = c−k
22 , c12 = c21 .
(3.3)
Therefore, for |k| < n, the matrices for this system are:


c−k = 
c−k
11
c−k
12
ck12
ck11




0
, c = 
c011
c012
c012 c011




k
, c = 
ck11
ck12
c−k
c−k
12
11


,
Here we take the free variables to be the entries of the first row of the matrix
−k
k
associated to each lattice point in Λ. Alternatively, since ck21 = c−k
12 and c22 = c11 ,
we could look at the free variables as the entries of the matrices associated with the
nonnegative lattice points in Λ. Suppose n = 2, so Λ = {−2, −1, 0, 1, 2}; then these
matrices would be


c0 = 
c011
c012
c012 c011




1
, c = 
c111
c112
c121 c122




2
, c = 
c211
c212
c221 c222


.
Then the remaining matrices, c−1 and c−2 , are defined by the relations, namely




2
2
1
1
 c22 c21 
 c22 c21 
−2
,
c
=
c−1 = 
.


2
2
1
1
c12 c11
c12 c11

 

0 
 −1 0   1
Example 3.2. Let d = 2, B = {b1 , b2 , b3 , b4 } = {I, 
 , −I},
,
0 −1
0 1


 0 2 
t
and A = 
 . Here Φ(x) = (φ(x), Db2 φ(x), Db3 φ(x), Db4 φ(x)) . We see that
1 0
|B| = 4. We first observe that b˜1 = b1 , b˜2 = b3 , b˜3 = b2 , and b˜4 = b4 where
98
b̃j = Abj A−1 . From (3.2), we obtain the following identities for B:
b˜1 b1 = b1
b˜1 b2 = b2
b˜1 b3 = b3
b˜1 b4 = b4
b˜2 b1 = b3
b˜2 b2 = b4
b˜2 b3 = b1
b˜2 b4 = b2
b˜3 b1 = b2
b˜3 b2 = b1
b˜3 b3 = b4
b˜3 b4 = b3
b˜4 b1 = b4
b˜4 b2 = b3
b˜4 b3 = b2
b˜4 b4 = b1 .
Therefore, using (3.2) we find the coefficient identities. For example, when i = 2
and j = 1, we have ck11 = cb̃232 k = cb233 k . We have the following identities.
ck11 = cb233 k = cb322 k = cb444 k
b2 k
ck12 = cb243 k = c31
= cb434 k
b2 k
= cb424 k
ck13 = cb213 k = c34
b2 k
= cb414 k
ck14 = cb223 k = c33
This leads us to an explicit definition of the coefficient matrices, namely,

.
ck11
ck12
ck13
ck14

 bk bk bk bk
 c3 c3 c3 c3
14
11
12
 13
ck = 
 b2 k b2 k b2 k b2 k
 c11 c12 c13 c14

cb144 k cb134 k cb124 k cb114 k





.



(3.4)
Now let us pick a specific subset of the lattice Z2 . Let
0
0
−1
1
0
.
,
,
,
,
Λ=
−1
1
0
0
0
Recall that in the time domain, the points in R2 are written as column vectors. We
see that B(Λ) = Λ. Also, for Λ̃ = 00 , 10 , 01 , we have B(Λ̃) = Λ. We know we
99
have |B| |Λ| = 4 · 5 = 20 free entries for the 4 × 4 coefficient matrices.
0
1
To simplify the notation, we use α for the entries of c(0) , β for the entries of c(1) ,
0
and δ for the entries of c(0) . Since 10 = b3 10 , the entries in the first and third rows
1
of c(0) are free variables. Therefore, 8 of our choices are the matrix entries α1j and α3j
0
for j = 1, 2, 3, 4. Since b2 01 = b4 01 , the entries in the first and second rows of c(1)
are free variables. Therefore, an additional 8 choices are the matrix entries β1j and
0
β2j for j = 1, 2, 3, 4. Since c(0) is completely determined by the entries of its first row,
the remaining 4 free variables for our matrix entries are δ1j for j = 1, 2, 3, 4. Then
our twenty free entries are {α1j , α3j , β1j , β2j , δ1j : j = 1, 2, 3, 4}. Thus, the matrices
for this example are

α11 α12 α13 α14


 α13 α14 α11 α12
1

c(0) = 
 α
 31 α32 α33 α34

α33 α34 α31 α32


β11 β12 β13 β14


 β21 β22 β23 β24
0

c(1) = 
 β
 12 β11 β14 β13

β22 β21 β24 β23









δ
 11

 δ13
0

c (0 ) = 
 δ
 12

δ14
δ12 δ13 δ14
δ14 δ11 δ12
δ11 δ14 δ13
δ13 δ12 δ11









(3.5)





.



The two remaining matrices are now defined by the matrices above.

α32 α31 α34 α33


 α34 α33 α32 α31
−1

(
)
0
c
=

 α12 α11 α14 α13

α14 α13 α12 α11









100

β23 β24 β21 β22


 β13 β14 β11 β12
0

(
)
−1
c
=

 β24 β23 β22 β21

β14 β13 β12 β11





.



3.2
Accuracy of Composite Dilation Systems in 1
Dimension
In this section we identify the necessary conditions for a composite dilation system
in 1 dimension to have a given level of accuracy. The one dimensional case is a
natural place to begin the search for more accurate composite dilation systems. We
are not yet looking for orthonormal bases, but simply establishing when an affine
system with composite dilations will have arbitrary accuracy. We establish that such
a system exists for every level of accuracy.
Let us restrict our attention to Composite Dilation Systems for L2 (R). We need
a group B consisting of elements with |b| = 1 for all b ∈ B. In 1 dimension, the
only such elements are 1 and -1. Thus, B = {1, −1}. Let Γ = Z since for any
Γ = cZ with c ∈ R = GL1 (R) we have Γc−1 = Z. From 3.1, we have the relations 3.3
−k
k
ck11 = c−k
22 , c12 = c21 . From [2] Theorem (?), in order to have a system with accuracy
p, we must find row vectors v[s] ; 0 ≤ s < p such that
v[s] =
s
XX
Q[s,t] (k)A[t] v[t] ck .
k∈Γi t=o
In one dimension we have v[s] = vs and A[s] = As for all 0 ≤ s < p. We also know
P
that Q[s,t] (k) = st=0 (−1)s−t st k s−t . Therefore, in L2 (R) we are searching for row
vectors {vs ∈ C1×r |0 ≤ s < p} with v0 6= 0 and
vs =
s
XX
(−k)
k∈Γi t=0
s−t
s t k
A vt c .
t
Since B = {1, −1}, then r = |B| = 2 and Φ(x) = (ϕ(x), ϕ(−x))t . When we take the
traditional dyadic dilation, A = 2, then from [2], we combine theorems 3.4 and 3.6 of
Cabrelli, Heil, and Molter. In our case, this can be written as follows:
101
Theorem (Cabrelli, Heil, Molter, [2]): Assume Φ(x) = (ϕ(x), ϕ(−x))t :
R → C2 is a refinable, integrable, compactly supported function with
independent Z-translates. Then Φ has accuracy p if and only if there
exists a collection of row vectors {vs ∈ C1×2 |0 ≤ s < p} such that
(i) v0 Φ̂(0) 6= 0, and
(ii) vs =
P
k∈Γi
Ps
t=0 (−k)
s−t s
t
2t vt ck .
Now for some α ∈ R, we know (2α − k)s =
0 ≤ t < p,
Ps
t=0
s
t
(−k)s−t 2t αt . Define, for
vt = αt (1, (−1)t ).
Then
s XX
s
(−k)s−t 2t vt ck
vs =
t
k∈Γi t=0
s XX
s
(−k)s−t 2t αt (α−t vt )ck .
=
t
k∈Γ t=0
i
α−t vt = (1, (−1)t ) so


(α−t vt )ck = (1, (−1)t ) 
ck11
ck12
ck21 ck22



= ck11 + (−1)t ck21 , ck12 + (−1)t ck22 .
Then
vs =
s XX
s
(−k)s−t (2α)t ck11 + (−1)t ck21 , ck12 + (−1)t ck22
t
" s s X
X X
s
s
s−t
t k
(−k)s−t (2α)t (−1)t ck21 ,
(−k) (2α) c11 +
=
t
t
t=0
t=0
k∈Γ
k∈Γi t=0
i
102
#
s X
s
(−k)s−t (2α)t (−1)t ck22
(−k)s−t (2α)t ck12 +
t
t
t=0
t=0
"
s
X s
X
(k)s−t (−1)s−t (−1)t (2α)t ck21 ,
(2α − k)s ck11 +
=
t
t=0
k∈Γi
#
s X
s
(k)s−t (−1)s−t (−1)t (2α)t ck22
(2α − k)s ck12 +
t
t=0
X
=
(2α − k)s ck11 + (−1)s (2α + k)s ck21 , (2α − k)s ck12 + (−1)s (2α + k)s ck22
s X
s
k∈Γi
=
"
X
k∈Γi
(2α − k)s ck11 +
X
k∈Γi
X
(−1)s (2α + k)s ck21 ,
k∈Γi
(2α − k)s ck12 +
X
#
(−1)s (2α + k)s ck22 .
k∈Γi
Now we can use the relations 3.3 of the matrix entries induced by the action of the
−k
k
group B by substituting ck11 = c−k
22 , c12 = c21 . Since B must fix Λ ⊂ Z, Λ finite, then
Λ must have the form {−n, . . . , 0, . . . , n}. For all k ∈
/ Λ, we have ck = 0 or ckij = 0
for every i, j. Therefore, we make the substitutions and to obtain
X
(−1)s (2α + k)s ck21 =
k∈Γi
X
(−1)s (2α + k)s c−k
12 =
k∈Γi
X
k∈Γi
(−1)s (2α − k)s ck12 .
Similarly,
X
(−1)s (2α + k)s ck22 =
k∈Γi
X
k∈Γi
(−1)s (2α − k)s ck11 .
Thus we have the following necessary equation for vs :
vs =
"
X
k∈Γi
(2α − k)s ck11 +
X
(−1)s (2α + k)s ck21 ,
k∈Γi
X
k∈Γi
=
"
X
k∈Γi
(2α − k)s ck11 +
X
k∈Γi
(2α − k)s ck12 +
(−1)s (2α − k)s ck12 ,
103
X
k∈Γi
(−1)s (2α + k)s ck22
#
X
k∈Γi
(2α − k)s ck12 +
X
k∈Γi
(−1)s (2α − k)s ck11
#
X
(2α − k)s (ck11 + (−1)s ck12 ), (2α − k)s (ck12 + (−1)s ck11 )
=
k∈Γi
=
X
(2α − k)s (ck11 + (−1)s ck12 ), (−1)s (2α − k)s (ck11 + (−1)s ck12 )
k∈Γi
=
X
k∈Γi
[1, (−1)s ] (2α − k)s (ck11 + (−1)s ck12 ).
(3.6)
Since vs = αs (1, (−1)s ), we have
αs (1, (−1)s ) = (1, (−1)s )
X
k∈Γi
(2α − k)s (ck11 + (−1)s ck12 ).
Therefore, for all 0 ≤ s < p,
αs =
X
k∈Γi
(2α − k)s (ck11 + (−1)s ck12 ).
(3.7)
This provides the necessary conditions to find our collection of row vectors. We must
find α ∈ R and the entries of the first row of the matrices associated with lattice
points in Λ such that
X
t s
(2α − k)t ,
α t=0 =
k∈Γi
(−1)t (2α − k)t
s
t=0



ck11
ck12


.
(3.8)
Recall that Γi is a coset of Γ/A(Γ). In this case, with A = 2 and Γ = Z, we
have Γ1 = the odds and Γ0 = the evens. Since Λ ⊂ Γ is finite and of the form
{−n, . . . , 0, . . . , n}, and ck11 = ck12 = 0 for all k ∈
/ Λ, we replace Γi with Λi in equation
3.8.
Suppose n is odd. Then Λ1 = {−n, −n + 2, . . . , n − 2, n}. Let m =
104
n+1 . Then
2
|Λ1 | = 2m. We can write (3.8) as a product of the following matrices:






α
~ =





1
α
α2
..
.
αs












(s+1)×1
and

Podd
1
1


 a−n a−(n−2)


2
2
=
 a−n a−(n−2)

..
 ..
 .
.

as−n as−(n−2)
···
1
···
1
1
1
· · · an b−n · · · bn−2 bn
· · · a2n b2−n · · · b2n−2 b2n
.
..
..
..
..
..
. ..
.
.
.
.
· · · asn bs−n · · · bsn−2 bsn












(s+1)×4m
where ak = (2α − k) and bk = (−1)(2α − k). Let

~codd
Then we have












=












c−n
11
−n+2
c11
..
.
cn11
c−n
12
−n+2
c12
..
.
cn12


























4m×1
α
~ = Podd~codd .
105
.
(3.9)
In this case, we see that the (s + 1) × 4m matrix Podd has the form of a Vandermonde
matrix, namely:

Podd





=





1
1
···
1
β1 β2 · · · β4m
2
β12 β22 · · · β4m
..
..
..
..
.
.
.
.
s
β1s β2s · · · β4m












where the βi are defined by the polynomials (2α − k) or −(2α − k). If we can invert
the matrix Podd , then we can solve equation 3.9 for ~codd . Thus if α ∈ R is such that
2α − k 6= 0 for all k ∈ Λ1 , and s + 1 = 4m, then
−1
~codd = Podd
α
~.
(3.10)
In the case of Λ2 = Λ ∩ Γ2 , the even elements of Λ, the same analysis applies
except the matrix Peven has different dimensions. Recall that in the above analysis,
n was odd. Then Λ2 = {−n + 1, −n + 3, . . . , 0, . . . , n − 3, n − 1}. Let l = n2 . Then
there are 2l nonzero even numbers in Λ2 . Therefore, |Λ| = 2l + 1. Then Peven is the
(s + 1) × 2(2l + 1) matrix:

1


 a−(n−1)


 a2
 −(n−1)

..

.


as−(n−1)
···
1
···
1
1
···
1 ···
1
· · · a0 · · · an−1 b−(n−1) · · · b0 · · · bn−1
· · · a20 · · · a2n−1 b2−(n−1) · · · b20 · · · b2n−1
..
. ..
..
..
. ..
..
..
.
. ..
.
. ..
.
.
.
· · · as0 · · · asn−1 bs−(n−1) · · · bs0 · · · bsn−1












(s+1)×(4l+2)
where ak = (2α − k) and bk = (−1)(2α − k).
We also have a different column vector, ceven consisting of the entries of the first
106
row of the coefficient matrices corresponding to the even elements of Λ.

~ceven















=















−n+1
c11
..
.
c011
..
.
n−1
c11
−n+1
c12
..
.
c012
..
.
n−1
c12
































.
4m×1
In order to solve the equation
α
~ = Peven~ceven
(3.11)
we must still have 2α − k 6= 0 for all k ∈ Λ2 and we need Peven to be square, or
s + 1 = 4l + 2. When this is true, we have the solution
−1
~ceven = Peven
α
~.
(3.12)
If n is even, the analysis is the same but the matrices change slightly. In Podd , the
entries of the first column change from powers of (2α + n) to powers of (2α + n − 1)
since the largest odd number in Λ will be n − 1. The remaining entries change in a
similar fashion. The dimensions of the matrix Podd will remain the same: (s + 1) × 4m
with m = n+1
. The entries of Peven will change similarly with (2α − n + 1) replaced
2
by (2α − n). Again, the dimensions of the matrix Peven will remain (s + 1) × (4l + 2)
with l = n2 .
107
In order to find entries for the 2 × 2 coefficient matrices ck that will satisfy the
necessary conditions for Φ to have accuracy p, we must find any n such that
jnk
n+1
, 4
+ 2 ≥ p.
min 4
2
2
Then setting Λ = {−n, . . . , 0, . . . , n} and choosing α ∈ R such that 2α − k 6= 0 for all
k ∈ Λ, we can solve 3.9 α
~ = Podd~codd and 3.11 α
~ = Peven~ceven . Then, for all k ∈ Λ, we
use these solutions to find the coefficient matrices for the refinement equation:


k
k
 c11 c12 
ck = 
.
−k
−k
c12 c11
If Φ is a solution to the refinement equation with these coefficient matrices, it will
have accuracy p, where p is the largest integer such that
jnk
n+1
p ≤ min 4
, 4
+2 .
2
2
(3.13)
In the preceding discussion we have proved the following theorem:
Theorem 3.1. For every p ∈ N there exists α ∈ R, a collection of row vectors
{vs ∈ R1×2 | 0 ≤ s < p}, a finite subset Λ ∈ Z, and a collection of 2 × 2 matrices ck
such that
(i) v0 = (1, 1) 6= 0 and
P
P
P
(ii) vs = k∈Λi (2α − k)s ck = k∈Λi st=0 st (−k)s−t 2t vt ck .
This theorem states that the necessary conditions from [2] are satisfied for the
existence of a refinable composite dilation system of arbitrary accuracy on L2 (R). The
problem now becomes finding a sufficient condition for a solution to the refinement
equation
Φ(x) =
X
k∈Λ
ck Φ(2x − k).
108
The question of determining such sufficient conditions remains open.
3.3
A Compactly Supported, Composite Dilation
Wavelet with Accuracy 2
In the previous section we showed that, for dimension one, we can satisfy the necessary conditions for the existence of a Composite Dilation System with arbitrary
accuracy. However, that discussion made no attempt to minimize the support for
such a Composite Dilation generating function. Also, it does not give us a means to
find a solution to the refinement equation. Additionally, we are most concerned with
finding a Composite Dilation Wavelet; a function ψ ∈ L2 (R) such that {Daj Db Tk ψ| b ∈
B, |A| > 1, j, k ∈ Z} = {ψ(b−1 Aj x − k)} is an orthonormal basis for L2 (R). A solution when B is the trivial group is the well-known result of Daubechies compactly
supported wavelets [5]. In our case, the situation of composite dilation wavelets, we
take B = {1, −1}.
Using the relations (3.3) on the entries of the coefficient matrices, ckij , induced by
the action of B from example 3.1 and the results of Cabrelli, Heil, and Molter [2] we
arrived at Theorem 3.1. From this we can calculate the collection {vs ∈ R1×2 | 0 ≤ s <
p} and the 2 × 2 matrices ck for any level of accuracy. As expected, as the accuracy
of the system increases, so must the size of the support of the generating function.
That is, Λ must grow proportionately with p.
In this section we derive the necessary accuracy equations for a 1-dimensional
composite dilation system with Λ = {−n, . . . , 0, . . . , n}. To obtain a reproducing system, we must at least satisfy the Smith-Barnwell equation. We develop the necessary
and sufficient equations introduced by the Smith-Barnwell equation in terms of the
entries of the coefficient matrices. We then conclude the section by directly calculating the matrices ck and solving the refinement equation (3.1) for Λ = {−2, −1, 0, 1, 2}.
109
Seeking a composite dilation wavelet, we add the constraints induced by satisfying
the Smith-Barnwell induced equations and then solve some orthogonality and normalization equations.
3.3.1
The Accuracy Equations
First, we let A = 2, B = {1, −1}, Γ = Z, and Φ(x) = (φ(x), φ(−x))t . We start
by taking a general Λ = {−n . . . , 0, . . . , n} and setting up the necessary

 accuracy
k
k
 c11 c12 
conditions for a collection {v0 , v1 } with v0 6= 0 and the matrices ck = 
.
k
k
c21 c22
−k
k
Recall, from Example 1, ck11 = c−k
22 and c12 = c21 . Since A = 2 and Γ = Z we have
Γ0 = 2Z, the evens, and Γ1 = 2Z + 1, the odds. Letting v0 = (1, 1) we get
X
ck = (1, 1)
even
ck11
even
ck21
P
(1, 1) = (1, 1)
k
X
k
even
ck .
odd
With the notation
cE =
X
k
even

P
 k
ck =  P
k
P
k
even
ck12
k
even
ck22


 
=
cE
11
E
cE
21 c22
we see that
cE
11 =
X
k
ck11 =
X
k
even
c−k
11 =
X
k
even
ck22 = cE
22 .
even
E
Similarly we have cE
12 = c21 . Hence
cE =
X
k
even


ck = 
110
cE
11
cE
12
E
cE
12 c11


.
cE
12



Likewise, for the odds, we arrive at
cO =
X
k
odd


ck = 
cO
11

cO
12

.
O
cO
12 c11
Hence, we have the following two equations:
(1, 1) = (1, 1)
(1, 1) = (1, 1)
E
E
 c11 c12 
E
E
ck = (1, 1) 
 = (1, 1)(c11 + c12 )
E
E
c12 c11
even
X
k
X
k


odd


ck = (1, 1) 
cO
11
cO
12
O
cO
12 c11


O
O
 = (1, 1)(c11 + c12 ).
Therefore, our first set of accuracy equations for s = 0 is
E
O
O
cE
11 + c12 = 1 = c11 + c12
or
X
k
(ck11 + ck12 ) = 1 =
X
k
even
(ck11 + ck12 ).
odd
Now, for s = 1, we know
1 X
X
XX
s
(−k)s−t 2t vt ck =
−kv0 ck +
2v1 ck .
v1 =
t
k∈Λ
k∈Λ
k∈Γ t=0
i
i
Then
v1 (2
X
k∈Λi
ck
!
i
X
− I) = v0
kck
k∈Λi
!
.
With our even and odd notation and v0 = (1, 1), we have
E
v1 (2c − I) = (1, 1)
111
X
k
even
kc
k
!
(3.14)
and
X
O
v1 (2c − I) = (1, 1)
k
kc
k
odd
!
.
(3.15)
Looking at the right hand side of equation (3.14) for k even, we have
(1, 1)
X
k
kck
even
!
= (1, 1)
X
k∈2N∩Λ

P

= (1, 1)  P
kck − kc−k
k
k∈2N∩Λ k(c11
−
!
c−k
11 )
k(ck21 − c−k
21 )
k
k∈2N∩Λ k(c12
P
−
c−k
12 )



k(ck22 − c−k
22 )


P
P
k
k
k
k
k∈2N∩Λ k(c12 − c21 ) 
 k∈2N∩Λ k(c11 − c22 )
= (1, 1)  P

P
k
k
k
k
)
−
c
)
k(c
−
c
k(c
11
22
12
21
k∈2N∩Λ
k∈2N∩Λ
"
#
X
X
=
k(ck11 − ck22 + ck21 − ck12 ),
k(ck12 − ck21 + ck22 − ck11 )
k∈2N∩Λ
P
k∈2N∩Λ
k∈2N∩Λ
= (1, −1)
k∈2N∩Λ
X
k∈2N∩Λ
!
k(ck11 − ck22 + ck21 − ck12 ) .
Now to simplify the left hand side of equation (3.14) for k even, we use the relation
E
mandated by the accuracy equation for s = 0. That is, cE
11 = 1 − c12 . Taking
v1 = α(1, −1), α ∈ R, we obtain


α(1, −1) 
2cE
11
2cE
12
−1
2cE
12
2cE
11 − 1




 = α(1, −1) 
2cE
11
−1
2(1 −
cE
11 )
E
2(1 − cE
11 ) 2c11 − 1



E
E
E
= α 2cE
11 − 1 − 2 + 2c11 , 2 − 2c11 − 2c11 + 1
= α(1, −1) 4cE
11 − 3
"
!
#
X
= α(1, −1) 4
(ck11 + ck22 ) − 3 .
k∈2N∩Λ
112
So we can rewrite equation (3.14) as

 





X

 X k
k(ck11 − ck22 + ck21 − ck12 ) .
(c11 + ck22 ) − 3 = (1, −1) 
α(1, −1) 4 
k even
k≥0
k even
k≥0
Hence, we have
 


X
 X k


α 4 
(c11 + ck22 ) − 3 =
k(ck11 − ck22 + ck21 − ck12 ).
k even
k≥0
k even
k≥0
Similarly, for the odds we can express (3.15) as the equation

 

X


 X k
k(ck11 − ck22 + ck21 − ck12 ).
(c11 + ck22 ) − 3 =
α 4 
k odd
k>0
k odd
k>0
So gathering the equations for s = 0 and s = 1, we have the following necessary
conditions for accuracy 2:
X
(ck11 + ck12 + ck21 + ck22 ) = 1
X
(ck11 + ck12 + ck21 + ck22 ) = 1
k even
k≥0
k odd
k>0

 

X


 X k
k(ck11 − ck22 + ck21 − ck12 )
(c11 + ck22 ) − 3 =
α 4 
k even
k≥0
k even
k≥0

 

X


 X k
k(ck11 − ck22 + ck21 − ck12 ).
(c11 + ck22 ) − 3 =
α 4 
k odd
k>0
k odd
k>0
Here, as in the discussion at the end of Example 3.1, we are looking at the free entries
in the coefficient matrices as the entries of the first row of the matrix associated to
113
0 and the entries of the matrices associated to the positive values in Λ. If we solve
these four equations for α and the ckij , k ≥ 0 we will satisfy the necessary conditions
for the solution to the refinement equation having accuracy at least 2.
3.3.2
The Smith-Barnwell Equations
We now want to ensure the system will be a reproducing system. A necessary condition for this property is that the system must satisfy the Smith-Barnwell equation:
1
1
M(ω)M ∗ (ω) + M(ω + )M ∗ (ω + ) = I
2
2
(3.16)
where
M(ω) =
1 X j ∗ 2πijω
1 X k −2πikω
c e
and M ∗ (ω) =
(c ) e
.
2 k∈Λ
2 j∈Λ
First we observe that
M(ω)M ∗ (ω) =
1 X X k j ∗ −2πi(k−j)ω
c (c ) e
4 k∈Λ j∈Λ
and
1
1
1 X X k j ∗ −2πi(k−j)ω −πi(k−j)
M(ω + )M ∗ (ω + ) =
c (c ) e
e
2
2
4
j∈Λ
k∈Λ
=
1 XX
4
ck (cj )∗ e−2πi(k−j)ω (−1)(k−j) .
k∈Λ j∈Λ
Therefore
1
1
1 XX
M(ω)M ∗ (ω) + M(ω + )M ∗ (ω + ) =
(1 + (−1)k−j )ck (cj )∗ e−2πi(k−j)ω
2
2
4 k∈Λ j∈Λ
X
1
=
ck (cj )∗ e−2πi(k−j)ω
2
(k−j)∈2Z∩Λ
114
=
1 X X k k−2n ∗ −2πi(2n)ω
c (c
) e
.
2 n∈Z k∈Λ
Now, for a fixed k and a fixed n, we have

ck11

k−2n
ck11 c11

ck (ck−2n )∗ = 

=
ck12
ck21 ck22



+
k−2n
c11
k−2n
c21
k−2n
k−2n
c12
c22
k−2n
ck12 c12



k−2n
ck11 c21
+
k−2n
ck12 c22
k−2n
k−2n
k−2n
k−2n
+ ck22 c22
ck21 c21
+ ck22 c12
ck21 c11


.
Then, keeping n fixed and summing over k ∈ Λ, we can simplify the matrix using the
−k
k
relations (3.3) ck11 = c−k
22 and c12 = c21 :
X
ck (ck−2n )∗ =
k∈Λ

k−2n
ck11 c11

k−2n
ck11 c11
X

k∈Λ
=
X

k∈Λ
k−2n
ck21 c11
+
k−2n
ck12 c12
k−2n
ck11 c21
+
k−2n
ck22 c12
k−2n
ck21 c21
+
k−2n
ck12 c12
+
k−2n
ck12 c22
+
k−2n
ck22 c22
−k+2n
ck11 c12
+



−k+2n
ck12 c11
k−2n
k−2n
−k+2n
−k+2n
c−k
+ c−k
c−k
+ c−k
12 c11
11 c12
12 c12
11 c11





k−2n
k−2n −k
k−2n −k
k−2n
X  ck11 c11
c11
c12 + c12
c11
+ ck12 c12



k−2n
k−2n
−k−2n −k
−k−2n −k
c−k
+ c−k
c12
c12 + c11
c11
k∈Λ
12 c11
11 c12


P
P
k−2n −k
k−2n −k
k k−2n
k k−2n
c12 + c12 c11 )
k∈Λ (c11
 k∈Λ (c11 c11 + c12 c12 )

= P
.
P
−k k−2n
−k k−2n
−k−2n −k
−k−2n −k
+
c
c
)
c
(c
c
)
+
c
(c
c
11 12
11
12
11
k∈Λ 12 11
k∈Λ 12
=
(3.17)
For l = −k, we see that
X
−k−2n
−k−2n
(c−k
+ c−k
)=
11 c11
12 c12
X
l∈Λ
k∈Λ
115
(cl11 cl−2n
+ cl12 cl−2n
11
12 ).
One final simplification is
X
k−2n −k
k−2n
(c11
c12 + c−k
)=
11 c12
X
=
X
k∈Λ
k−2n −k
c11
c12 +
X
k−2n
c−k
11 c12
k−2n −k
c11
c12 +
X
j−2n −j
c11
c12
k∈Λ
k∈Λ
j∈Λ
k∈Λ
=2
X
k−2n −k
c11
c12 .
(3.18)
k∈Λ
Therefore, we define q1 and q2 as follows:
q1 (Λ, n) =
X
k−2n
k−2n
(ck11 c11
+ ck12 c12
)
k∈Λ
q2 (Λ, n) = 2
X
k−2n −k
c11
c12 .
k∈Λ
Therefore, from (3.17) and (3.18), we arrive at


 q1 (Λ, n) q2 (Λ, n) 
ck (ck−2n )∗ = 
.
q2 (Λ, n) q1 (Λ, n)
k∈Λ
X
Now (3.16) can be written


1
1
 1 0 
∗
∗
 = M(ω)M (ω) + M(ω + )M (ω + )

2
2
0 1
1 X X k k−2n ∗ −2πi(2n)ω
=
c (c
)e
2 n∈Z k∈Λ


1 X  q1 (Λ, n) q2 (Λ, n)  −2πi(2n)ω
.
=
e

2 n∈Z
q2 (Λ, n) q1 (Λ, n)
116
Therefore, the entries of the coefficient matrices must satisfy the following equations:
X
q1 (Λ, n) =
k−2n
k−2n
(ck11 c11
+ ck12 c12
) = 2δn,0
(3.19)
k∈Λ
and
q2 (Λ, n) =
X
k−2n −k
c11
c12 = 0,
k∈Λ
3.3.3
∀n ∈ Z.
(3.20)
The Composite Dilation Wavelet
In order to find a specific solution, we choose Λ = {−2, −1, 0, 1, 2} and continue to
use A = 2 and B = {1, −1}. We seek a composite dilation wavelet with accuracy
p = 2 and compact support. Recalling the property that we can solve the accuracy
n
, 4 2 + 2 , we are lead to believe
equation in general whenever p ≤ min 4 n+1
2
that such a solution exists since
2
2+1
,4
+ 2 = 4.
p = 2 < min 4
2
2
In this discussion, we will not use the technique from Section 3.2, but will compute
the entries of the coefficient matrices explicitly. We gather the accuracy equations
from Section 3.3.1 and the Smith-Barnwell equations from Section 3.3.2 for the coefficient matrices. In order to more descriptively represent the Smith-Barnwell equation,
we break equation (3.20) into two equations, one for n ≥ 0 and the other for n > 0,
therefore only considering n ∈ N. Therefore, the entries of our coefficient matrices
must satisfy the following system of equations:
X
(ck11 + ck12 + ck21 + ck22 ) = 1
X
(ck11 + ck12 + ck21 + ck22 ) = 1
k even
k≥0
k odd
k>0
117


 
X

 X k
k(ck11 − ck22 + ck21 − ck12 )
c11  − 3 =
α 4 
k even
k≥0
k even
k≥0

 

X


 X k
k(ck11 − ck22 + ck21 − ck12 ).
(c11 + ck22 ) − 3 =
α 4 
k odd
k>0
k odd
k>0
X
k−2n
k−2n
(ck11 c11
+ ck12 c12
) = 2δn,0
k∈Λ
X
k−2n −k
c11
c12 = 0,
∀n ≥ 0
X
k+2n −k
c11
c12 = 0,
∀n > 0.
k∈Λ
k∈Λ
With Λ = {−2, . . . , 2}, we can write these equations explicitly in terms of the
entries of the matrices ck .
Accuracy:
(c011 + c012 + c211 + c212 + c221 + c222 ) = 1
(c111 + c112 + c121 + c122 ) = 1
α 4(c011 + c211 + c222 ) − 3 = 2(c211 − c212 + c221 − c222 )
α 4(c111 + c122 ) − 3 = (c111 − c112 + c121 − c122 )
Smith-Barnwell:
n=0:
(c211 )2 + (c111 )2 + (c011 )2 + (c122 )2 + (c222 )2 + (c212 )2 + (c112 )2 + (c012 )2 + (c121 )2 + (c221 )2 = 2
c211 c221 + c111 c121 + c011 c012 + c122 c112 + c222 c212 = 0
n=1:
c211 c011 + c111 c122 + c011 c222 + c212 c012 + c112 c121 + c012 c221 = 0
118
c011 c221 + c122 c121 + c222 c012 = 0
c211 c012 + c111 c112 + c011 c212 = 0
n=2:
c211 c222 + c212 c221 = 0
c222 c221 = 0
c211 c212 = 0.
One can derive the first Smith-Barnwell equation for n = 0, the sum of the squares
of the free matrix entries, from the remaining accuracy and Smith-Barnwell equations.
Thus, this equation is redundant and adds no value to the system of equations. The
final two Smith-Barnwell equations for n = 2 allow us to make choices for the location
of zeros, i.e. if c211 c212 = 0 then either c211 = 0 or c212 = 0. So we make the arbitrary
choices c211 = c221 = 0. With these choices and the removal of the redundant equation,
in order for a solution of the refinement equation to have accuracy p = 2, we must
solve
Accuracy:
(c011 + c012 + c212 + c222 ) = 1
(c111 + c112 + c121 + c122 ) = 1
α 4(c011 + c222 ) − 3 = −2(c212 + c222 )
Smith-Barnwell:
α 4(c111 + c122 ) − 3 = (c111 − c112 + c121 − c122 )
c111 c121 + c011 c012 + c122 c112 + c222 c212 = 0
c111 c122 + c011 c222 + c212 c012 + c112 c121 = 0
119
c122 c121 + c222 c012 = 0
c111 c112 + c011 c212 = 0.
Including α, this is a system of 8 equations with 9 free variables. Therefore, we
can solve this system of equations as a one-parameter family of equations in terms of
α. Using MatLab, we get a large continuous family of solutions. When we restrict
our attention to finding real matrices, we must isolate the values of α that give us
real ckij . These values of α are
√
√ # "
√ #
√
3− 3 3+ 3
−5 − 19 −5 + 19
∪
.
,
,
α∈
6
6
6
6
"
Let us take α =
√
3− 3
.
6
(3.21)
Then we can explicitly solve this system of equations to
obtain the coefficient matrices
c0 =
√


0
1 3− 3


,
√
4
0
3− 3
c2 =
c1 =

1 3+ 3 1− 3 
,

4
0
0
√

√
√


1 0 1+ 3 

.
4
0
0
(3.22)
Then the remaining matrices, c−1 and c−2 , are defined by the relations, namely
c−1 =


0
0
1


,
√
√
4
1− 3 3+ 3
c−2 =


0
0 
1

.
√
4
1+ 3 0


(3.23)
 ϕ(x) 
Now we must find ϕ : R → R such that Φ(x) = 
 satisfies the refineϕ(−x)
120
ment equation Φ(x) =
P
k∈Λ
ck Φ(2x − k). So
Φ(x) = c−2 Φ(2x + 2) + c−1 Φ(2x + 1) + c0 Φ(2x) + c1 Φ(2x − 1) + c2 Φ(2x − 2).
Then





0
0   ϕ(2x + 2) 
 ϕ(x)  1 


 = 

√
4
ϕ(−2x − 2)
1+ 3 0
ϕ(−x)



0
0

  ϕ(2x + 1) 
+

√
√ 
1− 3 3+ 3
ϕ(−2x − 1)



√
0
 3− 3
  ϕ(2x) 
+

√ 
ϕ(−2x)
0
3− 3



√
√
 3 + 3 1 − 3   ϕ(2x − 1) 
+


ϕ(−2x + 1)
0
0



√
 0 1 + 3   ϕ(2x − 1) 
+
 .

ϕ(−2x + 2)
0
0
Then
√
√
1h
(3 − 3)ϕ(2x) + (3 + 3)ϕ(2x − 1)
ϕ(x) =
4
i
√
√
+(1 − 3)ϕ(−2x + 1) + (1 + 3)ϕ(−2x + 2)
(3.24)
and
√
√
1h
(1 + 3)ϕ(2x + 2) + (1 − 3)ϕ(2x + 1)
ϕ(−x) =
4
i
√
√
+(3 + 3)ϕ(−2x − 1) + (3 − 3)ϕ(−2x) .
121
(3.25)
We clearly see that equations (3.24) and (3.25) are equivalent by substituting −x
for x in (3.24). This allows us to consider the single function ϕ as a refinable function
with the given refinement equation. That is, ϕ is the function to which the iterated
function system
√
√
1h
(3 − 3)f (2x) + (3 + 3)f (2x − 1)
f (x) =
4
i
√
√
+(1 − 3)f (−2x + 1) + (1 + 3)f (−2x + 2)
(3.26)
converges. Writing ft (x) to represent t iterations of (3.26), we see that ϕ(x) =
lim
f (x).
t→∞ t
Using MatLab, we begin with the tent function f0 (x) = (1 − |x|) χ[−1,1]
and iterate the function in equation (3.26). Figures 3.1, 3.2, and 3.3 show these plots
for t = 0, 3, 6, respectively. We observe that as t → ∞, ft (x) appears to be converging
to a linear function times the characteristic function of [0, 1].
Figure 3.1: The tent function after 0 iterations in (3.26).
122
Figure 3.2: The tent function after 3 iterations in (3.26).
Figure 3.3: The tent function after 6 iterations in (3.26).
123
Let ϕ(x) = (αx + β)χ[0,1] and solve equation (3.24) for α and β:
(αx + β)χ[0,1]
√
1h
(3 − 3)(α(2x) + β)χ[0,1] (2x)
=
4
√
+ (3 + 3)(α(2x − 1) + β)χ[0,1] (2x − 1)
√
+ (1 − 3)(α(−2x + 1) + β)χ[0,1] (−2x + 1)
i
√
+(1 + 3)(α(−2x + 2) + β)χ[0,1] (−2x + 2)
√
1h
=
(3 − 3)(2αx + β)χ[0, 1 ] (x)
2
4
√
+ (3 + 3)(2αx + β − α)χ[ 1 ,1] (x)
2
√
+ (1 − 3)(−2αx + β + α)χ[0, 1 ] (x)(−2x + 1)
2
i
√
+(1 + 3)(−2αx + β + 2α)χ[ 1 ,1] (x)
2
i
√
√
1h
(3 − 3)(2αx + β) + (1 − 3)(−2αx + β + α) χ[0, 1 ] (x)
=
2
4
i
√
√
1h
(3 + 3)(2αx + β − α) + (1 + 3)(−2αx + β + 2α) χ[ 1 ,1] (x).
+
2
4
(3.27)
So we solve the following system of 2 equations with 2 unknowns:
i
√
√
1h
(3 − 3)(2αx + β) + (1 − 3)(−2αx + β + α)
4
i
√
√
1h
(αx + β) =
(3 + 3)(2αx + β − α) + (1 + 3)(−2αx + β + 2α) .
4
(αx + β) =
This leads us to the one parameter family of solutions
√
1− 3
β = √ α.
2 3
As a scaled refinable function will satisfy the same refinement equation, then we have
124
a one parameter family of functions that will satisfy our refinement equation (3.1):
ϕα (x) =
√ !
1− 3
αx + √ α χ[0,1] (x).
2 3
In order to normalize the scaling function, ϕ, we choose α =
ϕ(x) = ϕ√6 (x) =
√
(3.28)
√
6 and define
√ !
1− 3
6x + √
χ[0,1] (x).
2
(3.29)
The graph of our scaling function defined by (3.29) appears in figure 3.4.
Figure 3.4: The scaling function ϕ(x).
With Φ(x) = (ϕ(x), ϕ(−x))t we have a solution to the refinement equation (3.1)
Φ(x) =
X
k∈Λ
ck Φ(2x − k)
for Λ = {−2, . . . , 2} and the ck which are the solutions (3.22) and (3.23).
From the preceding discussion, this solution has accuracy 2 and satisfies the Smith√
Barnwell equation. Also, we chose α = 6 so that kϕk2 = 1. Now we see from the
125
following straightforward calculation that the translates of ϕ(x) and D−1 ϕ(x) are also
orthogonal. For any two integers k, l ∈ Z,
hTk ϕ(x), Tl D−1 ϕ(x)i
= hϕ(x − k), ϕ(−(x − l))i
Z
=
ϕ(x − k)ϕ(l − x)dx
R
√ !
√ !
Z √
√
1− 3
1− 3
χ[0,1] (x − k)
χ[0,1] (l − x)dx
6(x − k) + √
6(l − x) + √
=
2
2
R
√ !
√ !
Z √
√
√
√
1− 3
1− 3
6x − 6k + √
=
χ[k,k+1](x) − 6x + 6l + √
χ[l−1,l] (x)dx
2
2
R
√ !
√ !
Z k+1 √
√
√
√
1− 3
1− 3
6x − 6k + √
− 6x + 6l + √
dx
= δk,l−1
2
2
k
√ !
√ !
Z k+1 √
√
1 − (2k + 1) 3
1 + (2k + 1) 3
√
√
= δk,l−1
− 6x +
dx
6x +
2
2
k
k+1
1 − 3(2k + 1)2 3
2
x = δk,l−1 −2x + 3(2k + 1)x +
2
k
= δk,l−1 −6k 2 − 6k + 6k 2 + 6k
= 0.
Therefore, {Db Tk ϕ : b ∈ {1, −1}, k ∈ Z} is an orthonormal system with accuracy
2. Furthermore, this orthonormal system generates a shift invariant space that will
be the scaling space for our composite dilation MRA, namely V0 = span{Db Tk ϕ :
b ∈ {1, −1}, k ∈ Z}. To construct the composite dilation MRA, we define Vj =
D2−j V0 . Since ϕ satisfies the refinement equation, or the 2-scale equation, we know
that D2−1 V0 = V1 = span{Db Tk ϕ(2·) : b ∈ {1, −1}, k ∈ Z}.
At this point, we want to find the composite dilation wavelet Ψ = (ψ, D−1 ψ)t
associated with the scaling function Φ = (ϕ, D−1ϕ)t . We seek a function ψ ∈ V1 such
that span{Db Tk ψ : b ∈ {1, −1}, k ∈ Z} = W0 where V0 ⊥W0 and V0 ⊕ W0 = V1 . For
such a function to exist with support on [0, 1], there must exist α, β, γ, and δ (α and
126
β are different unknowns than from the family of solutions (3.28)) such that
ψ(x) = αϕ(2x) + βD−1 ϕ(2x − 1) + γϕ(2x − 1) + δD−1 ϕ(2x − 2)
= αϕ(2x) + βϕ(1 − 2x) + γϕ(2x − 1) + δϕ(2 − 2x).
If V0 ⊥W0 , then hTk ψ, Tl ϕi = hTk ψ, Tl D−1 ϕi = 0 for any k, l ∈ Z. Also, if {Db Tk ψ}
is to be an orthogonal system, then hTk ψ, Tl D−1 ψi = 0 for all k, l ∈ Z. Therefore, we
can solve these orthogonality conditions to obtain solutions for α, β, γ, and δ. (From
the discussion of the orthogonality of the translates of ϕ and D−1 ϕ, we see that the
δk,l−1 allow us to simply check the orthogonality conditions when k = 0 and l = 1.)
Using the facts that ϕ(2·) and ϕ(1 − 2·) are supported only on [0, 12 ] and ϕ(2 · −1)
and ϕ(2 − 2·) are supported only on [ 12 , 1], we find a system of equations for the
unknown scalars.
hψ, ϕi =
=
Z
ψ(x)ϕ(x)dx
R
Z
1
2
0
(αϕ(x)ϕ(2x) + βϕ(x)ϕ(1 − 2x)) dx
Z 1
+
(γϕ(x)ϕ(2x − 1) + δϕ(x)ϕ(2 − 2x)) dx
1
2
√
√
√ i
√
1h
=
(3 − 3)α + (1 − 3)β + (3 + 3)γ + (1 + 3)δ .
8
Similarly we obtain
hψ, T1 D−1 ϕi =
=
hψ, T1 D−1 ψi =
Z
R
ψ(x)ϕ(1 − x)dx
√
√
√
√ i
1h
(1 + 3)α + (3 + 3)β + (1 − 3)γ + (3 − 3)δ
8
Z
R
ψ(x)ψ(1 − x)dx
127
=
1
[αδ + βγ] .
2
Therefore, to find appropriate values for α, β, γ, δ we solve the system of equations
(3 −
(1 +
√
√
3)α + (1 −
3)α + (3 +
√
√
3)β + (3 +
3)β + (1 −
√
√
3)γ + (1 +
3)γ + (3 −
√
3)δ = 0
√
3)δ = 0
αδ + βγ = 0.
From these equations we get another one-parameter family:
α=α
√
β = 3α
γ = (2 +
(3.30)
√
3)α
√
δ = −(3 + 2 3)α.
Each of these solutions will guarantee orthogonality. For ease of computations, we
will choose α = 1 for the following calculations and normalize the wavelet later. We
now have a compactly supported scaling function ϕ and an associated compactly
supported orthogonal wavelet, ψ̃, where
ψ̃ = ϕ(2x) +
√
3ϕ(1 − 2x) + (2 +
√
3)ϕ(2x − 1) − (3 +
√
3)ϕ(2 − 2x).
Since ϕ(2·) and ϕ(1 − 2·) are supported on [0, 12 ] and ϕ(2 · −1) and ϕ(2 − 2·) are
supported on [ 21 , 1], then ψ̃ is supported on [0, 1]. Therefore, as expected, the scaling
function ϕ and the wavelet ψ̃ have the same support. Consequently, D−1 ϕ(x) =
ϕ(−x) and D−1 ψ̃(x) = ψ̃(−x) also have the same support, [−1, 0]. Thus, Φ =
(ϕ, D−1 ϕ)t and Ψ̃ = (ψ̃, D−1 ψ̃)t have precisely the same support. Since Φ and Ψ̃ are
128
orthogonal, as shown above, we know V0 ⊥W0 .
We now must show that V0 ⊕W0 = V1 = D2−1 V0 . It is possible to show this using the
dimension function for the shift invariant spaces. However, in the spirit of the above
calculations, we show this directly. Since V1 = span{Db Tk ϕ(2·) : b ∈ {1, −1}, k ∈
Z}, if we are able to construct ϕ(2·) from a linear combination of translations of
ϕ, D−1 ϕ, ψ̃, and D−1 ψ̃, then we clearly see that V1 = V0 ⊕ W0 . Therefore, let
ϕ(2x) = aϕ(x) + bϕ(1 − x) + cψ̃(x) + dψ̃(1 − x).
Expanding ψ̃ in terms of ϕ we obtain the equation
√ ϕ(2x) = aϕ(x) + bϕ(1 − x) + c − (3 + 2 3)d ϕ(2x)
√
√ √
√ 3c + (2 + 3)d ϕ(1 − 2x) + (2 + 3)c + 3d ϕ(2x − 1)
+
√ + d − (3 + 2 3)c ϕ(2 − 2x).
Again, since ϕ(2x) is supported on [0, 21 ], we get the following two two equations:
√ ϕ(2x) = [aϕ(x) + bϕ(1 − x)] χ[0, 1 ] (x) + c − (3 + 2 3)d ϕ(2x)
2
√
√ 3c + (2 + 3)d ϕ(1 − 2x)
+
√
√ 0 = [aϕ(x) + bϕ(1 − x)] χ[ 1 ,1] (x) + (2 + 3)c + 3d ϕ(2x − 1)
2
√ + d − (3 + 2 3)c ϕ(2 − 2x).
Substituting ϕ(x) =
√
6x +
√ 1−
√ 3 χ[0,1] (x)
2
into the equation, equating the linear
and constant coefficients, and dividing by appropriate scalars we obtain the following
four equations:
√
√ a − b + 2 (1 − 3)c − (5 + 3 3)d = 2
129
√ √
a − b + 2 (5 + 3 3)c − (1 − 3)d = 0
√ √ √ a − 2 + 3 b − 2 + 2 3 c − 10 + 6 3 d = 1
√ √ √ a − 2 + 3 b + 38 + 22 3 c − 2 − 2 3 d = 0.
This system of four linear equations with four unknowns has the explicit solution
√
√
√
√
3− 3
1+ 3
2− 3
3
a=
, a=
, c=
, d=−
.
8
8
16
16
Since ϕ(2x) is the generating function for V1 , ϕ(x) is the generating function for
V0 , and ψ̃(x) is the generating function for W0 , we have established that V1 = V0 ⊕W0
as
√
√
√
√
3− 3
3
1+ 3
2− 3
ϕ(2x) =
ϕ(x) +
ϕ(1 − x) +
ψ̃(x) −
ψ̃(1 − x).
8
8
16
16
Finally, to generate an orthonormal wavelet basis, we must normalize ψ̃. In order
to accomplish this, we utilize the one-parameter family defined by (3.30) and take
α= √
1
√ .
8+ 13
Then
ψ(x) = p
1
8+
√
13
ψ̃(x)
i
h
√
√
√
1
=p
ϕ(2x) + 3ϕ(1 − 2x) + (2 + 3)ϕ(2x − 1) − (3 + 3)ϕ(2 − 2x) .
√
8 + 13
(3.31)
The graph of our wavelet defined by (3.31) appears in figure 3.5.
Since ϕ is compactly supported and refinable, then ψ is compactly supported and
satisfies the same refinement equation (3.1). Therefore the system {Db Tk ψ : b ∈
{1, −1}, k ∈ Z} is a compactly supported, refinable system with accuracy 2. Therefore, ψ is a compactly supported, orthonormal, MRA, composite dilation wavelet on
L2 (R) with accuracy 2.
130
Figure 3.5: The wavelet ψ(x).
A final comment about the results of this process is that this was all done directly
using the accuracy equations and Smith-Barnwell equations in terms of the entries
of the coefficient matrices. What is interesting, and somewhat surprising, is that the
entries of the coefficient matrices that we found are precisely the coefficients of the
refinement equation for the Daubechies wavelet with minimal support, usually called
D4. The D4 scaling function satisfies the refinement equation
√
√
√
√
1+ 3
3+ 3
3− 3
1− 3
f (x) =
f (2x) +
f (2x − 1) +
f (2x − 2) +
f (2x − 3).
4
4
4
4
The scaling function for our system satisfies the refinement equation
√
√
√
√
3+ 3
1− 3
1+ 3
3− 3
ϕ(2x) +
ϕ(2x − 1) +
ϕ(−2x + 1) +
ϕ(−2x + 2).
ϕ(x) =
4
4
4
4
This is likely due to the dominant role of the Smith-Barnwell equation in formulating the refinement equations. However, every refinement equation whose solution
satisfies the Smith-Barnwell equation does not have the same coefficients. The D4
131
scaling function is one of a continuous family of scaling functions supported on an
interval of length three. It is extremal in this family in that it has the largest possible number of vanishing moments, i.e. it is the smoothest. We also believe that our
scaling function is a member of a large family of scaling functions defined by the parameter α (3.21). Our scaling function is obviously extremal in this family as α =
√
3− 3
6
is the left end point of one of the intervals from which α produces real-valued solutions to the refinement equation. From (3.22) and (3.22), we see that the coeffecient
matrices of the refinement equation defining our scaling function are rather sparse.
It is likely that our scaling function is extremal in its family in the sense that it has
the maximal number of zeros in the coefficient matrices.
132
Chapter 4
Conclusions
In this dissertation we have answered some of the open questions regarding the existence and accuracy of composite dilation wavelets. This relatively new subset of the
theory of wavelets has more questions than answers. In the process of answering the
questions addressed in this document, we have generated several new questions.
In Chapter 2, we proved that MSF wavelets exist in every dimension. We did
so by examining two admissibility conditions. In theorem 2.2, we showed that an
arbitrary lattice and a finite group with fundamental region bounded by hyperplanes
through the origin can generate the necessary sets to support a scaling function. A
natural question that needs to be answered is whether or not this property of the
fundamental region is necessary. If so, the finite groups used for composite dilations
will be classified for MSF composite dilation wavelets. In a similar spirit, the next
step is to classify the MSF composite dilation wavelets arising from finite groups.
Theorem 2.9 proved that 2In is (B, Γ) admissible in every dimension. As a result,
we were able to show in theorem 2.10 that MRA, MSF composite dilation wavelets
exists in every dimension. The number of wavelet generating functions required when
using 2In is 2n − 1. This grows far too rapidly to be useful in applications in high
dimensions. We then showed that we can reduce the number of wavelet generating
133
functions to 1 at the expense of losing the freedom to choose our composite dilation
group and the lattice. It is important to investigate if one can find a more general
reduction in the determinant of the expanding matrix. It is unlikely that we will
proceed to higher dimensions at no cost, but it is plausible that we should be able
to reduce the number of generators to polynomial growth in dimension rather than
exponential growth in dimension.
At the conclusion to the proof of theorem 2.11, we discussed a potential advantage
of composite dilation wavelets. Here, we showed that in every dimension we can
find an MSF composite dilation wavelet with a single wavelet generating function.
This required us to have the order of the composite dilation group, B, grow to the
dimension of the space. In applications, this would result in a significant reduction in
input requirements for any implementation algorithm. It is important that we study
the possibilities in exploiting the group action in implementing composite dilation
wavelets.
There is some movement toward using MSF wavelets in applications. For example,
shearlets are MSF composite dilation wavelets [8]. The MSF wavelets described in
this dissertation generate very coarse orthonormal bases. To be useful in applications,
one would need to develop powerful smoothing techniques. In traditional wavelet
literature, much has been studied on this approach. Now we need to find similar
results for composite dilation wavelets.
In chapter 3, we found the necessary conditions for compactly supported composite
dilation wavelets for L2 (R) with accuracy p and exhibited the first known composite
dilation wavelet with accuracy greater than the Haar type composite dilation wavelets
[13]. This dissertation simply leads us to believe that we can succeed in finding
smoother composite dilation wavelets but has taken only small steps in that direction.
Much work is left to be done. As the dimension increases, the results in [2] and the
necessary equations from the Smith-Barnwell equations create a large set of quadratic
134
equations that must be solved. Is this direct approach the best way to find these
compactly supported functions or is there a better way? Even in one dimension,
there are significant obstacles to solving the necessary equations. Can we find a
Daubechies-like classification of compactly supported composite dilation wavelets?
Can we extend this to even two dimensions?
The success and rapid growth of the field of wavelet analysis is largely due to the
vast applications of wavelets. While mathematically beautiful, for composite dilation
wavelets to catch on with the same fervor, we will have to find their advantages in
applications. Exploiting the group action of the composite dilations and the ability
to rotate and reflect the orientation of the wavelets gives considerable promise to the
application of composite dilation wavelets. Some significant effort in the application
of these orthonormal bases will go a long way toward advancing the subject.
135
Bibliography
[1] Eugene Belogay and Yang Wang. Arbitrarily smooth orthogonal nonseparable
wavelets in R2 . SIAM J. Math. Anal., 30(3):678–697, 1999.
[2] Carlos Cabrelli, Christopher Heil, and Ursula Molter. Accuracy of lattice translates of several multidimensional refinable functions. J. Approx. Theory, 95(1):5–
52, 1998.
[3] H. S. M. Coxeter. Regular polytopes. Second edition. The Macmillan Co., New
York, 1963.
[4] H. S. M. Coxeter. Regular complex polytopes. Cambridge University Press, London, 1974.
[5] Ingrid Daubechies. Ten lectures on wavelets, volume 61 of CBMS-NSF Regional
Conference Series in Applied Mathematics. Society for Industrial and Applied
Mathematics (SIAM), Philadelphia, PA, 1992.
[6] K. Gröchenig and W. R. Madych. Multiresolution analysis, Haar bases, and
self-similar tilings of Rn . IEEE Trans. Inform. Theory, 38(2, part 2), 1992.
[7] L. C. Grove and C. T. Benson. Finite reflection groups, volume 99 of Graduate
Texts in Mathematics. Springer-Verlag, New York, second edition, 1985.
[8] Kanghui Guo, Gitta Kutyniok, and Demetrio Labate. Sparse multidimensional
representations using anisotropic dilation and shear operators. In Wavelets and
136
splines: Athens 2005, Mod. Methods Math., pages 189–201. Nashboro Press,
Brentwood, TN, 2006.
[9] Kanghui Guo, Demetrio Labate, Wang-Q Lim, Guido Weiss, and Edward Wilson.
Wavelets with composite dilations. Electron. Res. Announc. Amer. Math. Soc.,
10:78–87, 2004.
[10] Kanghui Guo, Demetrio Labate, Wang-Q Lim, Guido Weiss, and Edward Wilson.
Wavelets with composite dilations and their MRA properties. Appl. Comput.
Harmon. Anal., 20(2):202–236, 2006.
[11] Eugenio Hernández and Guido Weiss. A first course on wavelets. Studies in
Advanced Mathematics. CRC Press, Boca Raton, FL, 1996.
[12] Jelena Kovačević and Martin Vetterli. Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for Rn . IEEE Trans. Inform. Theory,
38(2, part 2):533–555, 1992.
[13] Ilya Krishtal, Benjamin Robinson, Guido Weiss, and Edward Wilson. Some
simple Haar-type wavelets in higher dimensions. J. Geom. Anal., 17(1):87–96,
2007.
[14] Alexi Savov. Finite groups associated with composite dilation wavelets. Senior
Honors Thesis, 2005. Washington University in St. Louis.
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