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Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932versão On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.2 Bahía Blanca jul./dez. 2005
Geometric Methods in Wavelet Theory
C. Cabrelli and U. Molter
To the memory of our teacher and colleague Pucho Larotonda, who taught us much more than Mathematics.
The authors are partially supported by Grants: PICT 15033, CONICET, PIP 5650, UBACyT X058 and X108
Abstract: In this paper we present an overview of how geometric methods can be successfully used to solve problems in Analysis. We will focus on self-similar objects and use their structure to construct frames, Riesz bases and wavelet bases in with a single generator function. Further, we show that the generating functions for these systems are dense in
.
Key words and phrases. Self-Similarity, Wavelets, Affine Systems, Wavelet Sets, Riesz basis wavelets
2000 Mathematics Subject Classification. Primary: 42C40, Secondary: 42C30
In this article we review some geometric methods that have proven to be very successful in different contexts and we show their application to wavelet construction (sections 5 and 6).
We describe the notion of self-similarity as developed by Hutchinson [Hut81] and its characterization in terms of contraction mappings in general metric spaces. We show that when these results are applied to affine functions in the euclidean space using a fixed expanding matrix, they can produce a self-similar tiling of the space. In other words, given an expansive matrix and an admissible lattice
, it is in general the case that there exists a self-similar set associated to
that tiles the plane by
-translates.
Tilings by lattice translates are associated to local Fourier orthonormal bases, which leads to the notion of spectral sets and to the "Fuglede Conjecture", as we describe in section 4.
When the property of tiling by -translates is linked to the self-similarity of the tile by an expansive matrix, a beautiful construction of wavelet bases is obtained. This is the theory of wavelet sets (section 5).
Finally, a careful choice of the expanding matrix and the lattice in the wavelet set construction, allows to approximate any function in by a generator of a Riesz wavelet system. We develop this density result in section 6.
Throughout this section the concept of self-similarity in a very general way will play a fundamental role. In , a similarity is a function of the type
where
is a point in
and
is a
isotropic matrix, i.e. all eigenvalues have the same magnitude. The name reflects the fact that this transformation copies objects into similar objects, just smaller, bigger or simply translated or rotated, depending on the matrix
and the point
.
In this sense, an object could be called self-similar if you can write it as the finite union of copies of itself, i.e.
![r B = ⋃ Ai(B) + bi. i=1](/img/revistas/ruma/v46n2/2a0818x.png)
![2 ℝ](/img/revistas/ruma/v46n2/2a0819x.png)
![PIC](/img/revistas/ruma/v46n2/2a0824x.png)
Figure 1: Self-Similar square
Not so trivial, but still simple, is the well known middle third Cantor set, which we will denote by . If you look at the Cantor set, you realize, that you can write it as the union of two shrunken copies of itself, one in the
interval and the other one in the
interval.
Both examples are examples of compact sets in . They also share the property, that they are the (almost) disjoint union of smaller copies of themselves. This allows to virtually see the self-similarity.
We will also be interested in self-similar functions, or measures. Self-similar objects are interesting for us, since the self-similarity allows us to recover information of the whole object by looking only at some part, since one can think that the properties are "translated" into each part. Therefore self-similar objects may be easier to study than "arbitrary" objects. This is one of the reasons why we are interested in finding and characterizing self-similar objects. The fact that they are simple to describe and in many cases they are dense, in the sense that they approximate arbitrary objects, makes this characterization extremely useful.
One of the main tools to construct self-similar objects, will be by resort to the fixed point theorem or Banach contraction principle. Since we will try to apply it to very different scenarios, we will state it here in its most general way. The proof can be found in may textbooks, such as [Rud64].
Theorem 2.1 (Generalized Banach Contraction Principle). Let be a complete metric space, and let
be a function such that there exists a metric
which is equivalent to
, for which there exist
, and
such that for all
(2.1)
Then there exists a unique point in
, such that
.
2.1. The space of compact sets in . To describe the setting for the theory, we start defining an appropriate structure for the space of non-empty compact sets in
. Precisely, we define
by
(2.2)
We want to transform into a metric space, and therefore we need to define an appropriate metric on
. We would like that this metric takes the similarity of the shapes into account. For example, a point
and a segment
should not be close with the appropriate distance.
One distance that performs this task in a reasonable way, is the so-called Hausdorff distance.
Definition 2.2. Let be non-empty compact sets, then the Hausdorff distance between
and
is
(2.3)
The following Theorem is straightforward (see for example [Hut81].
Theorem 2.3. The space of all non-empty compact subsets of
equipped with the Hausdorff distance is a complete metric space.
3. Attractors and self-similar sets
The following theorem, due to Hutchinson ([Hut81]), is a key result in the theory of self-similar sets and provides a simple way to construct them.
Theorem 3.1. Let be
contraction-mappings in
, with contraction factors
. There exists a unique non-empty compact set
satisfying
![m A = ∪i=1si(A)](/img/revistas/ruma/v46n2/2a0856x.png)
![A](/img/revistas/ruma/v46n2/2a0857x.png)
![s1,...,sm](/img/revistas/ruma/v46n2/2a0858x.png)
![s : H → H](/img/revistas/ruma/v46n2/2a0859x.png)
![s(B) = ⋃m si(B), i=1](/img/revistas/ruma/v46n2/2a0860x.png)
![B0 ⁄= ∅](/img/revistas/ruma/v46n2/2a0861x.png)
![{B } k k∈ℕ](/img/revistas/ruma/v46n2/2a0862x.png)
![B = s(B ) k k-1](/img/revistas/ruma/v46n2/2a0863x.png)
![A](/img/revistas/ruma/v46n2/2a0864x.png)
![(H, d ) H](/img/revistas/ruma/v46n2/2a0865x.png)
Proof. The proof is immediate noting that
![dH(s(A), s(B)) ≤ (1m≤aix≤m ci) dH (A,B).](/img/revistas/ruma/v46n2/2a0866x.png)
Examples of Self-Similar Sets:
![PIC](/img/revistas/ruma/v46n2/2a0879x.png)
Figure 2: Self-Similar parallelogram and twin dragon attractors
Focusing on the application we have in mind, we will look at a particular case of Theorem 3.1 which is satisfied by all our examples so far.
For this, let be a full rank lattice, i.e.
with
any invertible matrix. Let
be a set of generators for the lattice
, i.e., independent vectors such that
![Γ = {m1 γ1 + ⋅⋅⋅ + md γd : mi ∈ ℤ}.](/img/revistas/ruma/v46n2/2a0886x.png)
![P = {x1γ1 + ⋅⋅⋅ + xdγd : 0 ≤ xi < 1}](/img/revistas/ruma/v46n2/2a0887x.png)
![d ℝ ∕Γ](/img/revistas/ruma/v46n2/2a0888x.png)
![d×d M ∈ ℝ](/img/revistas/ruma/v46n2/2a0889x.png)
![1](/img/revistas/ruma/v46n2/2a0890x.png)
![d×d M ∈ ℝ](/img/revistas/ruma/v46n2/2a0891x.png)
![Γ](/img/revistas/ruma/v46n2/2a0892x.png)
![M Γ ⊂ Γ](/img/revistas/ruma/v46n2/2a0893x.png)
![Γ ∕M (Γ )](/img/revistas/ruma/v46n2/2a0894x.png)
![M Γ ⊂ Γ](/img/revistas/ruma/v46n2/2a0895x.png)
![detM ∈ ℤ](/img/revistas/ruma/v46n2/2a0896x.png)
![∣det M ∣](/img/revistas/ruma/v46n2/2a0897x.png)
Proposition 3.2. Let be an expansive matrix and let
be a lattice such that
,
, and
be a full set of digits. There always exists a non-empty compact set
satisfying
(3.1)
The following properties of will be useful [Ban91], cf. also [GM92]. For a general description see also [CHM04]. If
, we will denote by
the
-dimensional Lebesgue measure of
.
Lemma 3.3. Let be as in (3.1). Then the following statements hold.
In other words, part (c) above says that if , then
is a tile in the sense that the
-translates
cover
with overlaps of measure zero.
A long-standing open problem was the question of whether for each dilation matrix there exists a full set of digits
such that the corresponding attractor
is a tile. Lagarias and Wang proved that this is the case if
or if
[LW95], [LW96], [LW97]. Potiopa [Pot97] however showed that if
and
![⌊ 0 1 0 0 ⌋ | | M = | 0 0 1 0 | , ⌈ 0 0 - 1 2 ⌉ - 1 0 - 1 1](/img/revistas/ruma/v46n2/2a08134x.png)
![D = {d ,d } 1 2](/img/revistas/ruma/v46n2/2a08135x.png)
![A](/img/revistas/ruma/v46n2/2a08136x.png)
![(M, D)](/img/revistas/ruma/v46n2/2a08137x.png)
![-1 -1 A = M A + d1 ∪ M A + d2](/img/revistas/ruma/v46n2/2a08138x.png)
![M](/img/revistas/ruma/v46n2/2a08139x.png)
![2](/img/revistas/ruma/v46n2/2a08140x.png)
Let a measurable set such that
. The set
is called a spectral set if there exists a discrete set
such that the set of exponential functions
(4.1)
is an orthonormal basis of . In this case,
is called the spectrum of
. Many results on spectral sets can be found in the work of Jörgensen et. al ([JP91, JP92, JP98a, JP98b, JP99, BJR99]), Wang ([Wan02, PW01]) and others.
In 1974 Fuglede ([Fug74]) proved the following theorem.
Theorem 4.1. is a spectral set with spectrum
if and only if
tiles
by translations on
(the dual lattice of
).
He further conjectured that his theorem was still true, if one removes the condition that the spectrum of has to be a lattice. Terence Tao ([Tao04]) proved in 2003, that - at least for dimension
- this is false.
However, Fuglede's Theorem allows us to add an additional equivalence to item (c) of Lemma 3.3 above:
Proposition 4.2. Let be an expansive matrix and
. Let
be a lattice such that
and let
be a full set of digits. We have
(4.2)
5. Minimal supported (in frequency) wavelets
For a lattice , an expansive matrix
such that
,
, and
we consider the set
(5.1)
The question we are addressing now, is for which ,
is an orthonormal basis for
. Such a function will be called a wavelet.
The following Theorem, whose proof is straightforward, characterizes those wavelets whose Fourier transform has support with smallest possible measure.
Theorem 5.1. Let and
be as before, and let
be such that
tiles
(i.e.
is a tile for
, the dual lattice of
)
tiles
, i.e.
Then if is such that
we have
![{ j∕2 j } F = ∣det(M )∣ ψ(M x - γ) : j ∈ ℤ, γ ∈ Γ](/img/revistas/ruma/v46n2/2a08187x.png)
![L2( ℝd)](/img/revistas/ruma/v46n2/2a08188x.png)
![ψ](/img/revistas/ruma/v46n2/2a08189x.png)
5.1. Construction of . Therefore, in order to obtain MSFW, we need to construct a set
that satisfies the conditions above. Sets of this type are called wavelet sets and have been studied by many groups of researchers ([BL01, BMM99, BS02, BS04, DLS97, DLS98, ILP98, SW98, Wan02, Zak96]. We will illustrate the construction given by Benedetto [BL01, BS02, BS04] for a particular case, which will be useful in the next section.
Assume that with
and that
, for
. (The choice of
is only to simplify the construction.) Set
(see Figure 3).
![Figure 3](/img/revistas/ruma/v46n2/2a08198x.png)
Figure 3: The annulus tiles by dilations by
![M](/img/revistas/ruma/v46n2/2a08199x.png)
![Γ](/img/revistas/ruma/v46n2/2a08200x.png)
This set tiles by dilation by , but not by
-translates. We need to fill the hole.
We define
where is the vertex of the cube
that lies in the
-quadrant.
It will be convenient to use the notation for the intersection of the set
with the
-quadrant. With this notation, note that
(5.4)
Here denotes the usual translation by
in
. Therefore we have that
(5.5)
![PIC](/img/revistas/ruma/v46n2/2a08209x.png)
Figure 4:
![Q](/img/revistas/ruma/v46n2/2a08210x.png)
![M](/img/revistas/ruma/v46n2/2a08211x.png)
![Γ](/img/revistas/ruma/v46n2/2a08212x.png)
Observe that:
This fact allows us to conclude that:
which allows us to rewrite
![pict](/img/revistas/ruma/v46n2/2a08237x.png)
This shows that is in fact
-congruent to
.
Therefore, using Fuglede's Theorem (Theorem 4.1) we have that
![{ } -1-- 2πiγkω d γd∕2e χQ( ω) : k ∈ ℤ](/img/revistas/ruma/v46n2/2a08241x.png)
![KQ](/img/revistas/ruma/v46n2/2a08242x.png)
![d∕2 DM f := m f(M ⋅)](/img/revistas/ruma/v46n2/2a08243x.png)
![{ } F = ∣det(M )∣j∕2ψ(M jx - γk) : j ∈ ℤ, k ∈ ℤd](/img/revistas/ruma/v46n2/2a08244x.png)
![2 d L (ℝ )](/img/revistas/ruma/v46n2/2a08245x.png)
In this section we will outline, how we can use the construction of the previous section, to answer an open question posed by D. Larson. For details, we refer the reader to [CM06].
The question we are going to address now is the following:
Given , and
, does there exist a function
, an expansive matrix
, and a lattice
, such that
and
is a Riesz basis for
?
Here we are relaxing the condition of being an orthonormal basis, to the Riesz basis condition.
We recall that a set is a Riesz basis for a Hilbert space
, if it is complete in
and there exist constants
, such that for every
and every finite sequence of scalars
,
![n n n A ∑ ∣c ∣2 ≤ ∥ ∑ cv ∥2 ≤ B ∑ ∣c∣2. i ii i i=1 i=1 i=1](/img/revistas/ruma/v46n2/2a08262x.png)
![A](/img/revistas/ruma/v46n2/2a08263x.png)
![B](/img/revistas/ruma/v46n2/2a08264x.png)
In order to show, how the previous results can be used to give a positive answer to this question, we need the following result.
Lemma 6.1. Let be a set of finite measure. If
satisfies that
is a Riesz basis for
with bounds
and
and
satisfies that
then
is a Riesz basis for , with Riesz bounds
and
.
If is an invertible matrix and
satisfies that
up to a set of zero measure, with the union being almost disjoint, and
is a Riesz basis for
, then
is a Riesz basis for with the same bounds. (Here
is the dilation operator defined in the previous section.)
Proof. The first assertion is immediate, and the second one follows from the fact that the dilation is a unitary operator in . □
In view of this Lemma, if we are given a function in , we need to find the right lattice and the appropriate dilation matrix. For this we proceed in the following way:
Let now
![Γ = R ℤd and M = R-I . r d×d](/img/revistas/ruma/v46n2/2a08297x.png)
![r](/img/revistas/ruma/v46n2/2a08298x.png)
![R- r ≥ 3](/img/revistas/ruma/v46n2/2a08299x.png)
We define the set as in (5.6), and for
, the function
by
![( |{ ˆg(ω) x ∈ Q ∩ Eα h(ω) := α x ∈ Q \ E . |( α 0 else](/img/revistas/ruma/v46n2/2a08303x.png)
![]() |
Then the function , with
satisfies that
![{ j∕2 j d} F = ∣det(M )∣ ψ(M x - γk) : j ∈ ℤ, k ∈ ℤ](/img/revistas/ruma/v46n2/2a08307x.png)
![L2(ℝd)](/img/revistas/ruma/v46n2/2a08308x.png)
We thank Gustavo Corach for inviting us to be part of this volume in memory of our fond colleague Pucho Larotonda.
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Carlos Cabrelli
Departamento de Matemática,
Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires,
Ciudad Universitaria, Pabellón I,
1428 Capital Federal, ARGENTINA
and CONICET, Argentina
cabrelli@dm.uba.ar
Ursula M. Molter
Departamento de Matemática,
Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires,
Ciudad Universitaria, Pabellón I,
1428 Capital Federal, ARGENTINA
and CONICET, Argentina
umolter@dm.uba.ar
Recibido: 27 de marzo de 2006
Aceptado: 7 de agosto de 2006