Self-affine scaling sets in R^2

97 Pages·2014·1.1 MB·English

by  Xiaoye Fu

MEMOIRS
of the
American Mathematical Society
Volume 233 • Number 1097 (third of 6 numbers) • January 2015
R2
Self-Affine Scaling Sets in
Xiaoye Fu
Jean-Pierre Gabardo
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 233 • Number 1097 (third of 6 numbers) • January 2015
R2
Self-Affine Scaling Sets in
Xiaoye Fu
Jean-Pierre Gabardo
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Fu,Xiaoye,1979-
Self-affinescalingsetsinR2 /XiaoyeFu,Jean-PierreGabardo.
pages cm. – (Memoirs of the AmericanMathematicalSociety, ISSN 0065-9266; volume 233,
number1097)
Includesbibliographicalreferencesandindex.
ISBN978-1-4704-1091-9(alk. paper)
1. Scaling laws (Statistical physics) 2. Wavelets (Mathematics) 3. R (Computer program
language) I.Gabardo,Jean-Pierre,1958- II.Title.
QC174.85.S34F89 2015
515(cid:2).2433–dc23 2014033063
DOI:http://dx.doi.org/10.1090/memo/1097
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Contents
Chapter 1. Introduction 1
1.1. Wavelets and Wavelet Sets 1
1.2. Scaling Sets 2
1.3. Self-Affine Tiles 3
1.4. Main Results 5
Chapter 2. Preliminary Results 9
Chapter 3. A sufficient condition for a self-affine tile to be an MRA scaling
set 15
Chapter 4. Characterization of the inclusion K ⊂BK 19
Chapter 5. Self-affine scaling sets in R2: the case 0∈D 29
5.1. The case A=C 31
1
5.2. The case A=C 33
2
5.3. The case A=C 34
3
5.4. The case A=−C 35
3
5.5. The case A=C 36
4
5.6. The case A=−C 50
4
Chapter 6. Self-affine scaling sets in R2: the case D ={d ,d }⊂R2 53
1 2
6.1. The case A=C 53
1
6.2. The case A=C 57
2
6.3. The case A=C ,−C ,C ,−C 59
3 3 4 4
Chapter 7. Conclusion 81
Bibliography 83
Index 85
iii

Abstract
There exist results on the connection between the theory of wavelets and the
theory of integral self-affine tiles and in particular, on the construction of wavelet
bases using integral self-affine tiles. However, there are many non-integral self-
affine tiles which can also yield wavelet basis. In this work, we give a complete
characterizationof all one and two dimensional A(cid:2)-dilation scaling setsK such that
K is a self-affine tile satisfying BK = (K +d ) (K +d ) for some d ,d ∈ R2,
1 2 1 2
where A is a 2×2 integral expansive matrix with |detA|=2 and B =At.
ReceivedbytheeditorOctober11,2012.
ArticleelectronicallypublishedonMay19,2014.
DOI:http://dx.doi.org/10.1090/memo/1097
2010 MathematicsSubjectClassification. Primary42C15;Secondary52C20.
Key wordsand phrases. (MRA)scalingset,self-affinetile,(MRA)waveletset.
Affiliations at time of publication: Xiaoye Fu, Department of Mathematics, The Chinese
University of Hong Kong, Hong Kong, email: [email protected]; and Jean-Pierre Gabardo,
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1,
Canada,email: [email protected].
(cid:3)c2014 American Mathematical Society
v

CHAPTER 1
Introduction
Our main goal in this work is to characterize all two dimensional self-affine
scalingsetswithasetofassociateddigitsoftheform{d ,d },whered ,d ∈R2but
1 2 1 2
not necessarily in Z2, for an integral expansive matrix A with |detA| = 2. Before
starting the discussion of this topic, we will first introduce some basic concepts
that wil(cid:3)l be used frequently later. The Fourier transform of any function f ∈
L1(Rn) L2(Rn) is defined by
(cid:4)
(1.1) F(f)(ξ)=fˆ(ξ)= e−2πix·ξ f(x) dx,
Rn
where x·ξ is the standard inner product of the vectors x,ξ ∈ Rn. The inverse
Fourier transform will be denotedby F−1. For two me(cid:2)asurable sets E,F ⊆Rn, we
write E ∼=F if their symmetric difference i.e. (F \E) (E\F) has zero Lebesgue
measure. Two(cid:3)measurable sets E,F ⊆ Rn are essentially disjoint if the Lebesgue
measure of E F is zero. The Lebesgue measure of a measurable set K ⊂ Rn is
denoted by |K|.
1.1. Wavelets and Wavelet Sets
Asarelativelynewsubject, waveletanalysiswasintroducedintheearly1980s.
It has attracted much attention of researchers from many different fields including
mathematics, physics, engineering and so on because of its wide range of applica-
tions in signal analysis, image analysis, communications systems and differential
equations. An important problem in wavelet analysis is the construction of var-
ious kinds of wavelet bases obtained by applying dilations and translations to a
particular wavelet.
Suppose A is an n×n real expansive matrix, i.e. a matrix with real entries
whose eigenvalues are all of modulus greater than one. An A-dilation wavelet is a
measurable function ψ ∈L2(Rn) such that the set
(cid:5) (cid:6)
(1.2) |detA|2jψ(Aj ·−k): j ∈Z, k ∈Zn
forms an orthonormal basis for L2(Rn). Around 1987, Mallat [27] introduced the
idea of Multiresolution Analysis (MRA) with associated dilation A = 2 in dimen-
sion one that provides a tool for the construction of wavelets. The construction of
anorthonormalwaveletbasis inL2(R)wasreducedtotheconstructionofascaling
function in a corresponding MRA. Later, Daubechies [9] used this result to con-
struct compactly supported wavelet of arbitrary smoothness. Y. Meyer [28] gave a
similar generalizationofMRAinhigherdimension byconsideringmatrixdilations.
If an A-dilation wavelet ψ ∈ L2(Rn) is associated with an MRA, then ψ is called
an A-dilation MRA wavelet.
1
2 XIAOYEFUandJEAN-PIERREGABARDO
However, not every A-dilation wavelet is associated with an MRA. It has been
known[1,17]that|detA|=2isnecessaryandsufficientfortheexistenceofanMRA
wavelet. Dai, Larson and Speegle [7] proved the existence of a single wavelet of
theform F−1(χ ) for some measurable set Q⊆Rn and thisfor any real expansive
Q
matrix A, where χ is the characteristic function of Q. A measurable set Q⊆Rn
Q
is called an A-dilation wavelet set if F−1(χ ) is an A-dilation wavelet. If an A-
Q
dilationwaveletF−1(χ )isassociatedwithanMRA,thenthesetQ⊆Rn iscalled
Q
an A-dilation MRA wavelet set. Moreover, Dai, Larson and Speegle [7] provided a
criterion for a measurable set Q⊆Rn to be an A-dilation wavelet set.
Proposition 1.1. A measurable set Q ⊆ Rn is an A-dilation wavelet set if
and only if Q satisfies the following two conditions:
(cid:7) (cid:8)
(i) BjQ∼=Rn and BiQ BjQ∼=∅ for i(cid:8)=j,
j∈(cid:7)Z (cid:8)
(ii) (Q+k)∼=Rn and Q (Q+(cid:5))∼=∅ for 0(cid:8)=(cid:5)∈Zn.
k∈Zn
The condition (ii) of Proposition 1.1 is equivalent to the following condition
(cid:9)
χ (ξ+k)=1 for a.e. ξ ∈Rn.
Q
k∈Zn
A measurable subset Q of Rn is called a Zn-tiling set if it satisfies the criterion
(ii). The wavelets of form F−1(χ ) are a particular class of minimally supported
Q
frequency(MSF) wavelets (see [10,18]). They are analogs of Shannon wavelet and
lack smoothness in the Fourier domain. However, this defect is compensated by
good localization properties in the frequency domain and these wavelets may have
good potential for applications just like Shannon wavelet or Haar wavelet. Many
researchersincludingBaggett,MedinaandMerrill[2], Dai, LarsonandSpeegle[8],
Benedetto and Leon [3,5], Gabardo and Yu [13] and so on (e.g. see [4,20,30])
have constructed concrete examples of wavelet sets in Rn.
1.2. Scaling Sets
Suppose A is an n×n real expansive matrix and let B =At. The relationship
betweenscalingfunctionandwaveletfunctionreflectstheconnectionbetweenscal-
ing sets and wavelet sets. Bownik, Rzeszotnik and Speegle [6] clarified the relation
between scaling sets and wavelet sets. A measurable set K is called an A-dilation
scaling set (resp. MRA scaling set) if Q = BK \K is an A-dilation wavelet set
(resp. MRA wavelet set). The A-dilation scaling set introduced by Bownik, Rzes-
zotnik and Speegle [6], can be defined in several equivalent ways. The following
result, see ([6], Proposition 3.2), can also serve as a definition.
Lemma 1.2. A measurable set K ⊆Rn is an A-dilation scaling set if and only
(cid:2)∞
if K = B−jQ for some A-dilation wavelet set Q.
j=1
SinceQ=BK\K, ifK andQareasinthepreviouslemma, itfollowsthatan
A-dilation scaling set can be associated with a unique A-dilation wavelet set and
vice-versa. The following lemmas can be easily obtained from [6].
1. INTRODUCTION 3
Lemma 1.3. A wavelet set Q is an A-dilation MRA wavelet set if and only if
(cid:7)∞
the set K := B−jQ is a Zn-tiling set. Such a set K is called an A-dilation
j=1
MRA scaling set.
Lemma 1.4. Let A be an n×n integral expansive matrix with |detA| = 2. A
measurable set K ⊂Rn is an A-dilation MRA scaling set if and only if
(i) K is a Zn-tiling set,
(ii) K ⊂BK,
(iii) lim χ (B−mξ)=1 for a.e. ξ ∈Rn.
K
m→∞
1.3. Self-Affine Tiles
A self-affine tile in Rn is a measurable(cid:7)set K with positive Lebesgue measure
satisfying the set-valued equation BK = (K +d), where B is an n×n real
d∈D
expanding matrix with |detB| = m is an integer and D = {d ,d ,...,d } ⊆ Rn
1 2 m
i(cid:7)s a set of m digits. It has been shown [19] that the set valued equation BK =
(K+d) has a unique solution among all compact sets, given by
d∈D
(cid:5)(cid:9)∞ (cid:6)
K =K(B,D)= A−jd , d ∈D .
j j
j=1
In the case, where B is an n×n integral expanding matrix with |detB|=m and a
set of m digits D={d ,d ,...,d }⊆Zn, the set K is called an integral self-affine
1 2 m
tile in Rn.
The theory of self-affine tiles has been studied by many authors. Particularly,
Lagarias and Wang [22–24] studied in detail the structure and tiling properties of
self-affinetiles. Theseproblemsareconnectedwiththeconstructionoforthonormal
wavelet basis in Rn. Gr¨ochenig and Madych [15] revealed an interesting connec-
tion between the theory of compactly supported wavelet bases and the theory of
integral self-affine tiles. They showed that Haar-type wavelet bases which can be
constructed from an MRA always have an associated scaling function which is the
characteristic function of an integral self-affine tile K and that such a tile gives
a scaling function if and only if K is a Zn-tiling set in Rn. Gr¨ochenig and Hass
[16], LagariasandWang[21]proved, forexample, thatanydilationAyieldsHaar-
type wavelets in dimension 1 and dimension 2 respectively using Gro¨chenig and
Madych’s result [15]. We should mention that these authors only consider integral
self-affine tiles in their work, i.e. the matrix A is an integral expansive matrix and
the associated digits are in Zn. Gabardo and Yu [13] also contributed their work
to construct wavelet sets using integral self-affine tiles.
To our knowledge, until now, there is no result in the literature on the con-
nection between the theory of wavelets and the theory of non-integral self-affine
tiles. However, there are many wavelets which come from non-integral self-affine
tiles. One interesting example is the well-known Shannon wavelet whose corre-
sponding scaling set is K = [−1,1] with associated dilation factor A = 2. Then
(cid:2) 2 2
BK = (K − 1) (K + 1), where B = At = 2, which shows that K is a self-
2 2
affine tile but not an integral self-affine tile. Gu and Han [17] showed that there