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

1. Introduction

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.

2. Self-similarity

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.

A trivial example is the unit square in (see Figure 1).

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)

where

(2.4)

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

(i.e. is self-similar with respect to ). Furthermore, if is the map defined by for each compact set , the sequence given by converges to in .

Proof. The proof is immediate noting that

□

Examples of Self-Similar Sets:

• Examples in
  • 
  • 
• Examples in
  • 
    
    (see Figure 1).
  • 
    In this case, the attractors are shown in Figure 2.
  
  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

  Then the rectangular parallelepiped is a fundamental domain for the group . A matrix is said to be expansive, if all the eigenvalues have absolute value bigger than . If is expansive, and is a lattice such that , a set of representatives of the quotient is called a full set of digits for that lattice. Note that, since , . A full set of digits always has elements. We have the following result.

  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.

  1. .
  2. has nonempty interior, is the closure of , and .
  3. for all if and only if . In this case, for each .
  4. #.

  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

  then there is no set of digits such that the unique self similar set associated to i.e. is a tile, cf. [LW99]. Note that this matrix has determinant .

  4. Spectral sets

  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

  is an orthonormal basis of . is called a Minimal Supported in Frequency wavelet (MSFW).

  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: The annulus tiles by dilations by but not by translates

  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.

  Let us call

  (5.2)

  and define

  (5.3)

  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)

  We define the set

  (5.6)

  
  Figure 4: tiles by dilations by and by translates

  Observe that:

  • By construction, tiles by dilations by .
  • Furthermore tiles by translations on . For this, we first note that if then

    (5.7)

    This fact allows us to conclude that:

    1. for ,
    2. if ,

    which allows us to rewrite

    

    This shows that is in fact -congruent to .

  Therefore, using Fuglede's Theorem (Theorem 4.1) we have that

  is an orthonormal basis for , and using that is a unitary operator, we conclude that is an orthonormal basis of .

  6. Riesz basis are dense

  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 ,

  The constants and are called Riesz basis bounds.

  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 , such that and is continuous.
  • Choose such that
  • We now select small enough such that:

    (6.1) (6.2) (6.3)

  Let now

  Note that by the choice of , we have that and we are therefore in the previously described situation.

  We define the set as in (5.6), and for , the function by

  where

  Then the function , with satisfies that

  is a Riesz basis for (see [CM06] for the details of the proof).

  7. Acknowledgments

  We thank Gustavo Corach for inviting us to be part of this volume in memory of our fond colleague Pucho Larotonda.

  References

  [Ban91]    C. Bandt, Self-similar sets. V. Integer matrices and fractal tilings of , Proc. Amer. Math. Soc. 112 (1991), 549-562.         [ Links ]

  [BJR99]    Ola Bratteli, Palle E. T. Jørgensen, and Derek W. Robinson, Spectral asymptotics of periodic elliptic operators, Math. Z. 232 (1999), no. 4, 621-650.         [ Links ]

  [BL01]    J. J. Benedetto and M. Leon, The construction of single wavelets in d-dimensions, J. Geometric Analysis (2001), 1-15.         [ Links ]

  [BMM99]    Lawrence W. Baggett, Herbert A. Medina, and Kathy D. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in , J. Fourier Anal. Appl. 5 (1999), no. 6, 563-573.         [ Links ]

  [BS02]    John J. Benedetto and Songkiat Sumetkijakan, A fractal set constructed from a class of wavelet sets, Inverse problems, image analysis, and medical imaging (New Orleans, LA, 2001), Contemp. Math., vol. 313, Amer. Math. Soc., Providence, RI, 2002, pp. 19-35.         [ Links ]

  [BS04]    J. J. Benedetto and S. Sumetkijakan, Tight frames and geometric properties of wavelet sets, preprint, 2004.         [ Links ]

  [CHM04]    C. Cabrelli, C. Heil, and U. Molter, Self-Similarity and Multiwavelets in Higher Dimensions, Memoirs of the American Mathematical Society, vol. 170, American Mathematical Society, Providence, RI, USA, 2004.         [ Links ]

  [CM06]    C. Cabrelli and U. Molter, Density of the set of generators of wavelet systems, Constructive Approximation. To appear, 2006. (arXiv math. CA/0509321).         [ Links ]

  [DLS97]    Xingde Dai, David R. Larson, and Darrin M. Speegle, Wavelet sets in , J. Fourier Anal. Appl. 3 (1997), no. 4, 451-456. MR 98m:42048         [ Links ]

  [DLS98]    _________ , Wavelet sets in . II, Wavelets, multiwavelets, and their applications (San Diego, CA, 1997), Amer. Math. Soc., Providence, RI, 1998, pp. 15-40. MR 99d:42054         [ Links ]

  [Fug74]    Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101-121. MR MR0470754 (57 #10500)         [ Links ]

  [GM92]    K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of , IEEE Trans. Inform. Theory 38 (1992), 556-568.         [ Links ]

  [Hut81]    J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747.         [ Links ]

  [ILP98]    Eugen J. Ionascu, David R. Larson, and Carl M. Pearcy, On wavelet sets, J. Fourier Anal. Appl. 4 (1998), no. 6, 711-721.         [ Links ]

  [JP91]    Palle E. T. Jorgensen and Steen Pedersen, An algebraic spectral problem for , C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 7, 495-498.         [ Links ]

  [JP92]    P. E. T. Jorgensen and S. Pedersen, Spectral theory for borel sets in of finite measure, J. Funct. Anal. 107 (1992), 72-104.         [ Links ]

  [JP98a]    Palle E. T. Jorgensen and Steen Pedersen, Dense analytic subspaces in fractal -spaces, J. Anal. Math. 75 (1998), 185-228.         [ Links ]

  [JP98b]    _________ , Local harmonic analysis for domains in of finite measure, Analysis and topology, World Sci. Publishing, River Edge, NJ, 1998, pp. 377-410.         [ Links ]

  [JP99]    _________ , Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl. 5 (1999), no. 4, 285-302.         [ Links ]

  [LW95]    J. C. Lagarias and Y. Wang, Haar type orthonormal wavelet bases in , J. Fourier Anal. Appl. 2 (1995), 1-14.         [ Links ]

  [LW96]    _________ , Haar bases for and algebraic number theory, J. Number Theory 57 (1996), 181-197.         [ Links ]

  [LW97]    _________ , Integral self-affine tiles in . II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997), 83-102.         [ Links ]

  [LW99]    _________ , Corrigendum and addendum to: Haar bases for and algebraic number theory, J. Number Theory 76 (1999), 330-336.         [ Links ]

  [Pot97]    A. Potiopa, A problem of Lagarias and Wang, Master's Thesis, Siedlce University, Siedlce, Poland (Polish), 1997.         [ Links ]

  [PW01]    Steen Pedersen and Yang Wang, Universal spectra, universal tiling sets and the spectral set conjecture, Math. Scand. 88 (2001), no. 2, 246-256.         [ Links ]

  [Rud64]    Walter Rudin, Principles of mathematical analysis, Second edition, McGraw-Hill, New York, 1964. MR MR0166310 (29 #3587)         [ Links ]

  [SW98]    Paolo M. Soardi and David Weiland, Single wavelets in n-dimensions, J. Fourier Anal. Appl. 4 (1998), no. 3, 299-315.         [ Links ]

  [Tao04]    Terence Tao, Fuglede's conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2-3, 251-258. MR MR2067470         [ Links ]

  [Wan02]    Yang Wang, Wavelets, tiling, and spectral sets, Duke Math. J. 114 (2002), no. 1, 43-57.         [ Links ]

  [Zak96]    Victor Zakharov, Nonseparable multidimensional Littlewood-Paley like wavelet bases, Preprint, 1996.         [ Links ]

   

  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