# The Action of Translations on Wavelet Subspaces

The Action of Translations on Wavelet Subspaces by Eric Scott Weber B.A. Gustavus Adolphus College, 1995 M.A. University of Colorado, 1998

A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial ful?llment of the requirements for the degree of Doctor of Philosophy Department of Mathematics 1999

This thesis entitled: The Action of Translations on Wavelet Subspaces written by Eric Scott Weber has been approved for the Department of Mathematics

Lawrence W. Baggett

Arlan Ramsay

Date

The ?nal copy of this thesis has been examined by the signatories, and we ?nd that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.

iii Weber, Eric Scott (Ph.D. Mathematics) The Action of Translations on Wavelet Subspaces Thesis directed by Professor Lawrence W. Baggett

In this thesis we develop two equivalence relations on the collection of all wavelets. The ?rst uses the decomposition of spectral measures, obtained from looking at integral translations on a subspace of L2 (? ). Applications of this equivalence relation to operator interpolation of wavelets is presented. The second equivalence relation is generated by looking at translations by dyadic rationals on the same subspace of L2 (? ).

iv

Dedication

To my wife Jen and my daughter Emily, without whom mathematics would have little meaning.

v

Acknowledgements

I would like to take this opportunity to thank my advisor, Dr. Larry Baggett, for his constant encouragement and guidance. Without his support, none of my ideas would have made sense, and without his help, this thesis would never have happened. Thank you to Dr. Ramsay, who was always willing to listen to my ideas, and make suggestions in return. I would like to thank Carol Deckert. She always knew what was going on and what needed to be done. Thanks to Kathy Merrill, who provided me with some directions to consider in my research. Finally, a special thanks goes out to my fellow graduate students: Doug Norris, who always provided a laugh; Curtis (are you sure that is well de?ned?) Caravone, who kept me on my toes; Jen Courter, who provided encouragement and restraint; and Sharon Scha?er, who allowed me to give her a rough time. They all have made these past four years enjoyable.

Contents

Chapter 1 Introduction 1.1 A Brief Historical Background . . . . . . . . . . . . . . . . . . . . 1.2 The Basic De?nitions . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 1.4 MSF Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Examples of Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Structure of MSF Wavelets 2.1 The Structure of Wavelet Sets . . . . . . . . . . . . . . . . . . . . 2.2 GMRA’s Associated with MSF wavelets . . . . . . . . . . . . . . 2.3 A Mapping Between MSF Wavelets . . . . . . . . . . . . . . . . . 2.4 A Construction Procedure for Wavelet Sets . . . . . . . . . . . . . 3 Operator Interpolation and Core Equivalence 3.1 The Local Commutant . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Operator Interpolation of Wavelets . . . . . . . . . . . . . . . . . 3.3 A Necessary Condition for Operator Interpolation . . . . . . . . . 3.4 The Wavelet Connectivity Problem . . . . . . . . . . . . . . . . . 1 1 2 5 7 9 10 10 14 16 18 24 24 28 33 36

vii 4 The Wavelet Multiplicity Function 4.1 A Description of The Core Representation . . . . . . . . . . . . . 4.2 An Explicit Formula . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Examples of the Wavelet Multiplicity Function . . . . . . . . . . . 5 On the Translation Invariance of Wavelet Subspaces 5.1 A New Classi?cation . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Characterization of M∞ . . . . . . . . . . . . . . . . . . . . . . 5.3 A Characterization of Mn . . . . . . . . . . . . . . . . . . . . . . 5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 38 43 47 49 49 51 53 56

Appendix A Decomposition of Projection Valued Measures A.1 Canonical Projection Valued Measures . . . . . . . . . . . . . . . A.2 Cyclic Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Proof of the PVM Decomposition Theorem . . . . . . . . . . . . . A.4 Application of the Theorem to Unitary Representations . . . . . . 60 61 66 71 76

Bibliography

80

Figures

Figure 3.1 Plot of the Interpolated Wavelet . . . . . . . . . . . . . . . . . . . 33

Chapter

1

Introduction

1.1

A Brief Historical Background A wavelet is a function that generates an orthonormal basis of L2 (? ) by

“stretching” and “shifting” the original function. More speci?cally, this wavelet basis is generated from dilates by powers of 2 and translates by integers of the original function. This basis can be thought of as a non-trigonometric Fourier basis, as many of the applications of wavelets involve time-frequency analysis. The ?rst wavelet was introduced by Haar over 80 years ago, although he did not call his function a wavelet. The concept lay dormant until the early 1980’s, when Grossman, Morlet, and some others began using the concept of a wavelet in doing geological research. Then in 1988, Ingrid Daubechies published her landmark paper Orthonormal Bases of Compactly Supported Wavelets, which gave an algorithm for numerically e?cient methods for applying wavelets to real problems. Indeed, Daubechies’ algorithms play the same role for wavelet analysis as the Fast Fourier Transform plays for Fourier Analysis. Since the publication of Daubechies’ paper, both the theory and the applications of wavelet analysis has ?ourished. In particular, recently there has been an explosion in the analysis of wavelets using Functional Analytic tools. It is many of these tools that are used here in examining and classifying wavelets. Indeed, as the title suggests, we shall look at the way translation operators (bilateral shift op-

2 erators of in?nite multiplicity) “act” on wavelets. In this process, two equivalence relations on the collection of all wavelets are developed. Recently, it was conjectured independently by Guido Weiss and David Larson that the collection of wavelets are path connected in the norm topology of L2 (? ). Much of this work was motivated by this conjecture. The concept of a wavelet has been generalized extensively: to ? n , to a separable abstract Hilbert space, using a dilation factor other than 2, and in other ways. For the purposes of this thesis, unless explicitly described, wavelets shall be considered on ? , with a dilation factor of 2. 1.2 The Basic De?nitions A wavelet ψ ∈ L2 (? ) is a function such that the following set forms an orthonormal basis for L2 (? ): {ψj,k (x) = √ 2 ψ (2j x ? k ) : j, k ∈ }.

j

For historical reasons, sometimes the function ψ is called the mother wavelet. We wish to develop a framework that will permit the use of operator theory and functional analysis to analyze wavelets. Let U be a unitary system, i.e. a (countable) collection of unitary operators on L2 (? ) that contains the identity. Then a complete wandering vector f ∈ L2 (? ) for U is a function such that the collection {Uf : U ∈ U} forms an orthonormal basis for L2 (? ). The collection of all complete wandering vectors for U will be denoted by W (U ). Consider the unitary operators on L2 (? ), de?ned by T f (x) = f (x ? 1) and √ Df (x) = 2f (2x) and the unitary system UD,T = {D n T l : n, l ∈ }. It is clear that DnT l ψ = hence, the following de?nition. √ n 2 ψ (2n x ? l),

3 De?nition 1.1. A function ψ ∈ L2 (? ) is a wavelet if it is a complete wandering vector for the unitary system UD,T . The collection of all wavelets will be denoted by W (D, T ). These operators D and T have the following commutation relation: T n D j f (x) = D j f (x ? n) √ j = 2 f (2j (x ? n)) j √ j = T 2 n 2 f (2j x) = Dj T 2 n f (x) Central to the theory of wavelets is the concept of a Generalized Multiresolution Analysis, abbreviated GMRA, which is a sequence {Vj }j ∈ of subspaces of L2 (? ) that satisfy the following conditions: (1) Vj ? Vj +1, (2) (3)

j∈

j

(1.1)

Vj is dense in L2 (? ), Vj = { 0 } ,

j∈

(4) D (Vj ) = Vj +1 (5) V0 is invariant under T n for n ∈ . Much of the classical wavelet theory was built upon the notion of a Multiresolution Analysis, abbreviated MRA, which is the same as a GMRA, except condition (5) becomes: (5a) there exists a scaling function φ such that {T k φ : k ∈ orthonormal basis for V0 . Sometimes φ is called the father wavelet, so named since a wavelet can be “built” from a scaling function. The construction of a wavelet from a scaling function is not in the scope of this thesis; see [7] or [19]. } forms an

4 Theorem 1.1. Let ψ be a wavelet. De?ne subspaces of L2 (? ) by Vj = span{Dn T l ψ : n, l ∈ , n < j }. These subspaces form a GMRA which is associated to ψ in the sense that the translates of ψ form an orthonormal basis for V1 ∩ V0⊥ . The subspace V1 ∩ V0⊥ has a special name, it is called the wavelet subspace, and is denoted by W0 . An entire “dual” sequence of subspaces can be generated in this manner. De?ne Wj to be the perpendicular compliment of Vj in Vj +1 so that Vj +1 = Vj ⊕ Wj . Thus, L2 (? ) can be decomposed into the follow direct sum: L2 (? ) = V0 ⊕∞ j =0 Wj (1.3) (1.2)

Proof of Theorem 1.1. It is clear from the de?nition of the Vj ’s that they satisfy conditions (1), (2), and (4) above. Furthermore, by equation 1.2, it is clear that Wj = span{Dj T l ψ : l ∈ }, so that the translates of ψ form an orthonormal basis of V1 ∩ V0⊥ , as required. To establish condition (3), it su?ces to show that limj →?∞ Pj f = 0 for f ∈ L2 (? ), where Pj is the projection onto Vj . Indeed, by the containment relation, if f ∈ Vj for all j , then Pj f = f so that the limit would be f . By de?nition,

j ?1 ∞

Pj f =

n=?∞ l=?∞

f, D n T l ψ D n T l ψ.

By Parseval’s theorem,

j ?1

Pj f Let on

2

2

∞

=

n=?∞ l=?∞

| f, Dn T l ψ |2 .

> 0 be given. Since the coe?cients for f are a square-summable sequence , there exists a negative N ∈ PN f

2 N ?1

such that

∞

=

n=?∞ l=?∞

| f, Dn T l ψ |2 < .

5 It follows that for m < N , Pm f

2

< , and hence the limit is 0.

Now, to establish condition (5), note that since W0 is generated by translates of ψ , W0 is invariant under translations. Since Vj = Dj V0 and Vj +1 = Vj ⊕ Wj , it follows that Wj = Dj W0 . By the commutation relation (1.1), T Wj = T Dj W0 = D j T 2 W0 = D j W0 = Wj , whence Wj is invariant under translations. The translation invariance of V0 is then established by the decomposition given in (1.3). Corollary 1.1. If ψ is a wavelet, then the associated GMRA has subspaces Wj given by Wj = span{Dj T l ψ : l ∈ }. By the de?nition of GMRA, since V0 is invariant under translations, V0 admits a unitary representation of the group of integers. De?nition 1.2. The representation of the integers given by integral translations acting on V0 is called the core representation. If ψ and η are wavelets whose core representations are unitarily equivalent, then they are core equivalent. Not all wavelets are core equivalent. The core representation generates a wavelet multiplicity function; two wavelets are core equivalent if and only if their multiplicity functions are equal. An equivalence relation on the collection of all wavelets is generated by the multiplicity function, which is presented in chapter 4. 1.3 The Fourier Transform There is much overlap between Wavelet Analysis and Fourier Analysis. Indeed, as mentioned above, wavelets can be thought of as non-trigonometric Fourier series. The techniques used in time-frequency analysis are similar in both Fourier Analysis and Wavelet Analysis. This thesis, however, examines properties of wavelets as opposed to their applications.

j

6 Thus, Fourier Analysis is used as a tool to investigate properties of wavelets. In fact, the Fourier-Plancherel transformation will be used extensively in this thesis. Let F denote the Fourier transform on L2 (? ), which is de?ned for f ∈ L1 (? ) ∩ L2 (? ) as: F f (ω ) = ?(ω ). f (x)e?ixω dx = f

?

Note that for this de?nition of the Fourier transform, Plancherel’s theorem becomes: Theorem 1.2 (Plancherel’s Theorem). If f, g ∈ L2 (? ), then f, g =

1 2π

?, g f ?.

We shall also consider the Fourier transform of operators on L2 (? ); if B ∈ ?. ? is F B F ?1. Note that Bf = B ?f B(L2 (? )), its Fourier transform, denoted by B ? and D ? . As is common practice, a multiplication In particular, we shall discuss T operator on L2 (? ) is de?ned by Mf = gf for some bounded measurable function g . Such an M will be denoted by Mg . Additionally, Mg is a bounded operator if g ∈ L∞ (? ). ?: To calculate T F T f (ξ ) = = = by substituting x ? 1 = y = e?iξ f (y )e?iyξ dy T f (x)e?ixξ dx f (x ? 1)e?ixξ dx f (y )e?i(y+1)ξ dy

? ? ?

?

= e?iξ F f (ξ ) ? = Me?i· . Hence, F T = Me?i· F , so T

7 ?, For D F Df (ξ ) = = Df (x)e?ixξ dx √ 2f (2x)e?ixξ dx f (y )e?i 2 ξ

y

?

? √ = 2

?

dy 2

by substituting 2x = y

ξ 1 f (y )e?iy 2 dy =√ 2 ? ξ 1 = √ Ff( ) 2 2

= D ?1 F f (ξ ) ? = D ?1 = D ? . Thus, F D = D ?1 F , so that D 1.4 MSF Wavelets In this section we introduce a special class of wavelets. These wavelets exist in abundance and have much more structure than an arbitrary wavelet. This structure will be presented in chapter 2. Because of this structure, this type of wavelet can be fairly easily constructed. As such, these wavelets are ideal for examples and, as will be shown in chapter 3, constructing new wavelets. We begin with a proposition. Proposition 1.1. Let f ∈ L2 (? ). Then {T l f, l ∈ if and only if ?(ξ + 2πn)|2 = 1 a.e. ξ. |f

n∈

} forms an orthonormal set

(1.4)

Proof. Assume that the set is orthonormal. By the orthonormality of T l f , we

8 have δl,0 = T l f, f 1 ?ilξ ? ?(ξ ) e f (ξ ), f 2π 1 ?(ξ )f ?(ξ )dξ e?ilξ f = 2π 2π 1 ?(ξ + 2πn)f ?(ξ + 2πn)dξ = e?ilξ f 2 π n∈ 0 = 1 = 2π

2π 0 n∈

?(ξ + 2πn)|2 e?ilξ dξ |f

Note that the function in 1.4 is 2π periodic, so that the last integral gives the lth Fourier coe?cient. However, the lth coe?cient is 0 unless l is 0, thus it is constant. Furthermore, if we integrate the function in 1.4 over [0, 2π ], we get ?, which is 2π by Plancherel’s theorem and by virtue of f the 2-norm squared of f being normal. Hence, the 2π periodic function in 1.4 is identically 1. The converse follows by reversing this process. ?(ξ )| ≤ 1 a.e. ξ . Corollary 1.2. If ψ is a wavelet, |ψ ?|2 must integrate to 2π , it follows that As a consequence of this, since |ψ ? must have measure at least 2π . It is not clear that a wavelet the support of ψ exists such that the support of its Fourier transform has measure exactly equal to ?|2 is 1 on its support. As 2π . If this were to happen, then we must have that |ψ mentioned above, these wavelets actually exist in abundance and have a special role in the theory. ?| = χW for some set W ? ? , then De?nition 1.3. If ψ is a wavelet such that |ψ ψ is called a Minimally Supported Frequency wavelet, or MSF wavelet. The set W is called a wavelet set. In chapter 2, we shall study the properties of MSF wavelets by studying the properties of wavelet sets.

9 1.5 Examples of Wavelets

Example 1.1. The simplest example of a wavelet basis is the Haar basis, generated by the wavelet ψ : ? ? ? ? 1, 0≤x< 1 ; ? 2 ? ? ? ψ (x) = ?1, 1 ≤ x < 1; 2 ? ? ? ? ? ? ? 0, elsewhere. This wavelet is associated to an MRA. The proof that the dilates and translates of this function form an orthonormal basis for L2 (? ) is elementary but some of the calculations are actually quite involved. The proof will not be given here. Example 1.2. The next example is the Shannon wavelet, which is also an MRA wavelet. It is a MSF wavelet whose wavelet set is: W = [?2π, ?π ) ∪ [π, 2π ). The proof that this is a wavelet set will be given in section 2.1 Example 1.3. This example is due to Journ? e; it is a non-MRA wavelet. It is also a MSF wavelet whose wavelet set is: W = [? 32π 4π 4π 32π , ?4π ) ∪ [?π, ? ) ∪ [ , π ) ∪ [4π, ). 7 7 7 7

The fact that this is a wavelet set will be shown in section 2.1; the fact that it is a non MRA wavelet will be proven in section 4.3.

Chapter

2

The Structure of MSF Wavelets

In chapter 1, we introduced the concept of a minimally supported frequency wavelet. This chapter ?rst completely characterizes wavelet sets, then explores the added structure that that is inherent to such wavelets. We show that the MSF wavelets presented in chapter 1 are in fact wavelets. We shall conclude this chapter with a procedure for constructing wavelet sets. 2.1 The Structure of Wavelet Sets Since MSF wavelets are de?ned by their wavelet sets, we shall ?rst examine the structure of wavelet sets. In this chapter, ψ is a MSF wavelet, with wavelet ?| = χW . If W is a wavelet set, then its associated wavelet will set W , so that |ψ be denoted by ψW . By the de?nition of a wavelet, the set {T l ψ : l ∈ } forms an orthonormal set; indeed, by Theorem 1.1, this set is an orthonormal basis for W0 . Hence, by ?(ξ ) : l ∈ } is the calculation in section 1.3 and Plancherel’s theorem, { √1 e?ilξ ψ 2π an orthonormal basis for W0 . De?nition 2.1. Let E, F ? ? ; E and F are said to be 2π translation congruent if there exists a measurable partition En of E such that the collection {En + 2πn : n ∈ } forms a measurable partition of F , modulo null sets, i.e. F = ∪n∈ (En + 2πn),

11 where the union is disjoint. Such an equivalence will be denoted by E ?2π F . Let us begin with a lemma regarding translation congruence. Lemma 2.1. The functions { √1 e?ilξ χE (ξ ) : l ∈ 2π and only if E ?2π [0, 2π ). If. Clearly, if E ?2π [0, 2π ), the functions are all normal. Furthermore, e?ilx dx = δl,0 ,

E

} form an orthonormal set if

hence, the functions are orthogonal. Thus, they are orthonormal. Only If. It su?ces to show that E ?2π [0, 2π ) if {e?ilξ χE (ξ ) : l ∈ } forms

an orthogonal set. Now, by the discussion preceding de?nition 2.1, this set is the Fourier Transform of {T l F ?1 χE : l ∈ Proposition 1.1, |χE (ξ + 2kπ )|2 = 1

k∈

}, which is an orthonormal set. By

for almost every ξ ∈ ? . This equation gives a partition of [0, 2π ) that e?ects a translation congruence with E . Indeed, since the sum is 1, for ξ ∈ [0, 2π ), there exists a unique integer k such that ξ + 2kπ ∈ E . Hence, for every k set Ik = {ξ ∈ [0, 2π ) : ξ + 2kπ ∈ E }. Furthermore, since the 2π translates of [0, 2π ) cover the real line, all of E is covered by the translates of Ik . It follows that E ?2π [0, 2π ). Lemma 2.2. Let f ∈ L2 (? ) and let E = supp(f ). Then, the set {eilx f (x) : l ∈ } is an orthonormal basis for L2 (E ) if and only if the following two conditions hold: (1) E ?2π [0, 2π ], (2) |f (x)| =

√1 2π

for a.e.x ∈ E .

12 Proof. For convenience, we shall show that {e?ilx f (x)} is an orthogonal basis for L2 (E ) if and only if condition (1) and |f (x)| = 1 are satis?ed. The su?ciency of the two conditions is clear. Necessity. If {e?ilx f (x)} is an orthogonal basis for L2 (E ), then for any g ∈ L2 (E ), g (x) =

l∈

cl e?ilx f (x). Hence, there exists an h ∈ L2 ([0, 2π )) such that g = hf ;

l∈

indeed, h(x) =

cl e?ilx . Suppose that there exists a set F ? E of non-zero

measure and an integer k such that F + 2kπ ? E . Since L2 (E ) is invariant under M

e?i 2k ·

1

, e?i 2k x f (x) = g (x)f (x)

1

for some g ∈ L2 ([0, 2π )). Hence, on the set E , e?i 2k = g (x). If x ∈ F , then x + 2kπ ∈ E , and we have: g (x)f (x + 2kπ ) = g (x + 2kπ )f (x + 2kπ ) = e?i 2k (x+2kπ) f (x + 2kπ ) = e?iπ e?i 2k f (x + 2kπ ) = ?1g (x)f (x + 2kπ ) a contradiction, since then f (x + 2kπ ) must be 0. This shows that E is 2π translation congruent to a subset of [0, 2π ). By Proposition 1.1,

k∈

x 1

x

|f (ξ + 2kπ |2 = 1, whence E is 2π translation

congruent to all of [0, 2π ). Furthermore, the same equation yields that |f | is identically 1 on E . Lemma 2.3. Let E ? ? . Then, ⊕n∈ L2 (2n E ) = L2 (? ) if and only if the collection {2n E : n ∈ } is a measurable partition of ? , modulo null sets. If. Suppose that the dilates of E form a partition of the real line. Then, it is clear that the spaces L2 (2n E ) and L2 (2k E ) are orthogonal if n = k , since the

13 sets are disjoint. If f ∈ L2 (? ) then by virtue of the dilates of E covering ? , f=

n∈

f χ2n E . Each of the summands is in the corresponding L2 (2n E ), hence

the direct sum is all of L2 (? ). Only If. Now suppose that ⊕n∈ L2 (2n E ) = L2 (? ). For integers n and k such that n = k , if 2n E ∩ 2k E has non-zero measure, then by choosing F to be a subset of the intersection with ?nite (but non-zero) measure, χF is contained in both L2 (2n E ) and L2 (2k E ), which contradicts the assumption that L2 (2n E ) and L2 (2k E ) are perpendicular. Furthermore, if the union of the 2n E ’s does not cover the entire real line, then, choose F a subset of the complement of ∪n∈ 2n E that has ?nite, but nonzero measure. It follows that, under these assumptions, χF ∈ L2 (? ), but χF is not an element of the direct sum, which contradicts the original assumptions. The following theorem completely characterizes wavelet sets, ?rst presented in [9]. Theorem 2.1. If W ? ? , then W is a wavelet set if and only if the following two conditions hold: (1) W ?2π [0, 2π ), (2) the collection {2n W : n ∈ } is a partition of ? . If. If W is 2π -translation congruent to [0, 2π ), then by Lemma 2.2, ψW has or?W (ξ ) : n ∈ thonormal translates. Moreover, span{e?ilξ ψ } = L2 (W ). Clearly

?W . By Lemma 2.3, these subD n (L2 (W )) = L2 (2n W ), which contains D n T l ψ spaces are orthogonal and their direct sum is all of L2 (? ), which shows that the dilates and translates of ψW are orthonormal, and form a complete basis. Only If. If W is a wavelet set, then by de?nition, {e?inξ χW (ξ )} is an orthogonal set, so again by Lemma 2.2, W ?2π [0, 2π ); indeed, W0 = L2 (W ). Furthermore,

14 L2 (? ) = ⊕n∈ D n W0 , whence Lemma 2.3 proves condition (2). We are now in a position to prove that examples 1.2 and 1.3 from chapter 1 are indeed wavelets, by proving that the sets are in fact wavelet sets. Example 1.2. Recall that this example had wavelet set: W = [?2π, ?π ) ∪ [π, 2π ). We wish to show that this set satis?es the conditions of Theorem 2.1. It is clear that this set satis?es condition (2), and, since [?2π, ?π ) + 2π = [0, π ), that it also satis?es condition (1). Example 1.3. Recall that this example had wavelet set: W = [? 32π 4π 4π 32π , ?4π ) ∪ [?π, ? ) ∪ [ , π ) ∪ [4π, ). 7 7 7 7

π π To see that this set satis?es condition (1), note that [? 32 , ?4π ) + 6π = [ 10 , 2π ); 7 7 π π π π π [?π, ? 47 )+2π = [π, 10 ); and [4π, 32 ) ? 4π = [0, 47 ), which, together with [ 47 , π) 7 7

indeed partition the interval [0, 2π ). To establish the fact that the dilates of W partition the real line, it su?ces to show that there are intervals [?2α, ?α) and [β, 2β ) that are contained in the

π π π π dilates of W. Indeed, 1 [? 32 , ?4π ) = [? 87 , ?π ), so that the interval [? 87 , ? 47 ) is 4 7 π 8π contained in W ∪ 1 W . Since this set is symmetric about 0, it follows that [ 47 , 7) 4

is also contained in W ∪ 1 W. 4 Finally, note that the above calculation also shows that the dilates of W are disjoint. Hence, condition 2 is satis?ed. 2.2 GMRA’s Associated with MSF wavelets Now that we have examined the structure of wavelet sets, we can move on to studying the structure of the MSF wavelets themselves. Since a wavelet is closely

15 related to a GMRA structure, it is natural to examine the added structure of a GMRA that is associated to a MSF wavelet. As we have seen in the proof of Theorem 5.2, the associated GMRA is quite nice. Indeed, W0 = L2 (W ), so that since Wj = D j W0 , we have Wj = L2 (2j W ). By equation (1.3), V0 = ⊕j<0 D j W0 . Hence, for an MSF wavelet, V0 = L2 (E ) where E = ∪j<0 2j W . Notice that since Vj = L2 (2j W ), Vj is invariant under multiplication by e?in· . Hence, it follows from the calculation in chapter 1 that Vj is invariant under integral translations for all j ∈ . A GMRA arising from a wavelet ψ has this

invariance property if and only if ψ is a MSF wavelet; this fact will be proven in chapter 5. Suppose now that ψ is not only an MSF wavelet, but an MRA wavelet. Then, there exists a scaling function φ ∈ V0 such that the set {T l φ : l ∈ } forms an orthonormal basis for V0 , whence {e?ilξ φ(ξ )} forms an orthogonal basis for L2 (E ). It follows by Lemma 2.3 that E ?2π [0, 2π ). This, clearly, is an if and only if statement, i.e. we have proven the following proposition. Proposition 2.1. Suppose ψW is a MSF wavelet with wavelet set W . Then, ψW is an MRA wavelet if and only if E = ∪j<0 2j W is 2π translation congruent to [0, 2π ). Hence, we can now prove the statements from the examples 1.2 and 1.3 in chapter 1 concerning whether the wavelets are associated to an MRA. Example 1.2. A routine computation shows that for W = [?2π, ?π ) ∪ [π, 2π ), E = [?π, π ), which is indeed 2π translation congruent to [0, 2π ). Example 1.3. Recall that W = [? 32π 4π 4π 32π , ?4π ) ∪ [?π, ? ) ∪ [ , π ) ∪ [4π, ). 7 7 7 7

16 Note ?rst that since W is symmetric, E will be symmetric also. Dividing the last

π π interval of W by powers of 2 yields: [2π, 16 ) ? E ; [π, 87 ) ? E ; and [ π , 4π ) ? E . 7 2 7 π π Furthermore, dividing the third interval of W by 2 yields [ 27 , 2 ) ? E . Combining π 4π π this with the [ π , 4π ) shows that [ 27 , 7 ) ? E , so that in fact, [0, 47 ) ? E . Now, 2 7 π π [2π, 16 ) ? 2π = [0, 27 ), which is already in E , hence, E is not 2π translation 7

congruent to [0, 2π ), and thus the Journ? e wavelet is not an MRA wavelet. 2.3 A Mapping Between MSF Wavelets For the purposes of this section, E , and F will denote wavelet sets. By Theorem 2.1, if E , F are two wavelet sets, then E ?2π F , and there is a mapping σ : E → F such that σ (x) ? x = 2πkx for some kx ∈ . Note that kx may depend on x. This σ may be extended to be a measurable bijection of ? onto itself in the following manner. Again by Theorem 2.1, {2j E } forms a partition of the real line. Hence, for x ∈ ? , there exists a y ∈ E and an integer j such that x = 2j y . De?ne σ (y ) = 2j σ (2?j y ) = 2j σ (x). Clearly, we now have that σ : ? → ? . Furthermore, σ maps the dilates of E to the dilates of F . Indeed, if x ∈ 2n E , then 2?n x ∈ E , so that σ (x) = 2n σ (2?n x), and since σ (2?n x) ∈ F , 2n σ (2?n x) ∈ 2n F . By virtue of σ mapping E onto F , it follows that σ maps 2n E onto 2n F . This establishes that σ is a bijection. Finally, σ is measurable; clearly it is measurable on E , and is also measurable on 2n E . If H ? ? is measurable, then H = ∪n∈ (H ∩ 2n E ), and σ (H ∩ 2n E ) is measurable, thus σ (H ) is measurable. Proposition 2.2. If E and F are two wavelet sets, and σ : ? → ? is de?ned as above, then σ is a measurable, measure preserving, 2-homogeneous bijection. Proof. We have already demonstrated the fact that σ is measurable and a bijection. We shall show next that σ is 2-homogeneous, i.e. that σ (2x) = 2σ (x) for

17 almost all x. By the discussion above, if x ∈ 2n E , then 2x ∈ 2n+1 E ; but also σ (x) ∈ 2n F , so we have that 2σ (x) ∈ 2n+1 F and σ (2x) ∈ 2n+1 F . Since σ is a bijection, these must be equal. Finally, we need to show that σ is measure preserving. Indeed, σ is measure preserving from E to F , since it is given by translations. On the dilates of E it is also measure preserving. Let G ? 2n E ; then σ (G) = 2n σ (2?n G). Note that 2?n G ? E so that the measure of σ (2?n G) is the measure of 2?n G. It follows that 2n σ (2?n G) has the same measure as G, whence, σ is measure preserving.

π π π 4π Example 2.1. Let E and F be wavelet sets given by E = [? 83 , ? 43 ) ∪ [ 23 , 3 ), π π π 8π and let F = [? 43 , ? 23 ) ∪ [ 43 , 3 ). Then, σ is given by: ? ? ? ?ξ + 4π, ξ ∈ [? 8π , ? 4π ) 3 3 σ (ξ ) = ? ? ?ξ ? 2π, ξ ∈ [ 2π , 4π ) 3 3 π π π 8π π 4π π π , ? 43 )) = [ 43 , 3 ) and σ ([ 23 , 3 )) = [? 43 , ? 23 ). Note that σ ([? 83

These mappings can have somewhat varied properties. Indeed, some of them have the property that σ 2 is the identity. Others have the property that σ 2 not only is not the identity, but also is not a 2π translation on E . De?nition 2.2. If σ is such that σ 2 is the identity, then σ is called involutive. Example 2.2. The σ from example 2.1 is involutive. It su?ces to show that σ 2

π 8π is the identity on E , since σ is 2-homogeneous. Consider σ on F . If y ∈ [ 43 , 3 ),

then

1 y 2

π 4π ∈ [ 23 , 3 ). Hence, σ (y ) = 2σ ( 1 y ) = 2[ 1 y ? 2π ] = y ? 4π . So, for 2 2

π π x ∈ [? 83 , ? 43 ), σ 2 (x) = σ (x + 4π ) = x + 4π ? 4π = x. A similar computation π 4π shows that σ 2 is the identity on [ 23 , 3 ), so it is indeed involutive.

These mappings will be of particular interest in chapter 3. An example of a σ whose square is not given by 2π translations on E will be given in section 3.4.

18 2.4 A Construction Procedure for Wavelet Sets In section 2.2, we showed that V0 = L2 (E ), where E = ∪j<0 2j W . It follows that W = 2E \ E . Hence, under the right conditions, E has the property that 2E \ E is a wavelet set. Notice that in the case of starting with a wavelet set, E has measure 2π . As in [24], we make the following de?nition. De?nition 2.3. If E ? ? is a set that has measure 2π and is such that 2E \ E is a wavelet set, then E will be called a generalized scaling set. We wish to characterize generalized scaling sets. As a direct result of the de?nition, since W has measure 2π and 2E has measure 4π , E must be contained in 2E . By Theorem 2.1, 2E \ E must have the property that (1) 2E \ E ?2π [0, 2π ) and (2) the dilates of 2E \ E form a partition of ? . In [13], the following 2π periodic function is de?ned on ? :

∞

Dψ (ξ ) =

j =1 n∈

?(2j (ξ + 2πn))|2 . |ψ

This is called the wavelet dimension function. However, we can de?ne Df for any f ∈ L2 (? ). It follows that Df is measurable, since it is a sum of measurable functions, and is, in fact, ?nite almost everywhere. Indeed, integrating over [0, 2π ) yields:

2π 0 2π ∞

Df (ξ )dξ = =

0 ∞

?(2j (ξ + 2πn))|2 dξ |f

2π 0

j =1 n∈

?(2j (ξ + 2πn))|2 dξ |f

j =1 n∈

by Monotone Convergence Theorem, since everything is non-negative,

∞ 2(n+1)π 2nπ

=

j =1 n∈

?(2j ω )|2dω |f

19 by substituting ξ + 2nπ = ω ,

∞

=

j =1 ∞

?

?(2j ω )|2dω |f

=

j =1

1 ?(ξ )|2dξ |f 2j ?

by substituting 2j ω = ξ , = f

2

.

This dimension function satis?es the following consistency equation, introduced in [5], when ψ is a wavelet: ξ ξ Dψ (ξ ) + 1 = Dψ ( ) + Dψ ( + π ). 2 2 Consider the right hand side of the equation for any f ∈ L2 (? ): ξ ξ Df ( ) + Df ( + π ) = 2 2

∞ j =1 n∈ ∞

?(2j ( ξ + 2πn))|2 |f 2 ?(2j (( ξ + π ) + 2πn))|2 |f 2

+

j =1 n∈ ∞

=

j =1 n∈ ∞

?(2j ?1 (ξ + 2π (2n)))|2 |f ?(2j ?1 (ξ + 2π + 2π (2n)))|2 |f ?(2j (ξ + 2π (2n)))|2 |f ?(2j (ξ + 2π (2n + 1)))|2 |f ?(2j (ξ + 2πn))|2 + |f

n∈

+

j =1 n∈ ∞

=

j =0 n∈ ∞

+

j =0 n∈ ∞

=

j =1 n∈

?(ξ + 2nπ )|2 |f

= Df (ξ ) +

n∈

?(ξ + 2nπ )|2 |f

since we have both the even and odd multiples of 2π . Proposition 1.1 and this calculation prove the following theorem.

20 Theorem 2.2. If f ∈ L2 (? ), then Df satis?es the consistency equation if and only if f has orthonormal translates. Lemma 2.4. Let E ? ? have ?nite measure, and let W = 2E \ E . Then {2j W : j ∈ } forms a partition of the real line if the following two conditions

hold: (1) E ? 2E , and (2) E contains a neighborhood of 0. Proof. First note that since W = 2E \ E , 2j W = 2j +1 E \ 2j E . Indeed, if x ∈ 2j W , then 2?j x ∈ W so that 2?j x ∈ 2E but not E . It follows that x ∈ 2j +1E but not 2j E , which shows that 2j W ? 2j +1 E \ 2j E . Now, for the reverse containment, let x ∈ 2j +1 E \ 2j E . By the reasoning above, 2?j x ∈ 2E but not E , so that x ∈ 2j W . This shows that 2j +1 E \ 2j E ? 2j W . We shall ?rst show that the dilates of W are disjoint. Note that by assumption, E ? 2E , from which it follows that E ? 2m E for m ≥ 0, since, by induction, 2j E ? 2j +1 E . Suppose, by way of contradiction, that 2j W ∩ 2k W = ? for j > k . Then, there exist x, y ∈ W such that 2j x = 2k y ; further, there exist z, z ∈ E such that 2z = x and 2z = y , with 2z, 2z ∈ / E . This implies that 2m z is not in E for any m > 0. Indeed, if 2z ∈ / E , then 2m z ∈ / 2m?1 E , and E ? 2m?1 E , whence 2m z ∈ / E. By the equation above, 2j ?k z = z , a contradiction to the previous statement. Next, we wish to show that E ? ∪j<0 2j W by showing that for x ∈ E , there exists a j such that 2j x ∈ W . Suppose to the contrary, there existed a set F ? E of non-zero measure such that for all x ∈ F , 2j x is not in W = 2E \ E for any integer j . This could only happen if 2j F ? E . Without loss of generality, we may assume that F ? (α, 2α) for some α, or the negative of that interval, so that the dilates of F are disjoint. However, the union of all of these dilates has in?nite measure, a contradiction of the assumption that E has ?nite measure.

21 Furthermore, since by de?nition, 2j W ? E for negative j , we have in fact that E = ∪j<0 2j W . Now, if E contains a neighborhood of 0, for any x ∈ ? , there exists an integer j such that 2j x ∈ E , so that the union of the 2j E ’s covers the entire line. But, by above, the dilates of W contain E , and hence a neighborhood of 0. It follows that the dilates of W cover the real line. We are now in the position to prove the following characterization of generalized scaling sets. Theorem 2.3. A set E of measure 2π is a generalized scaling set if and only if the following conditions hold: (1) E ? 2E ; (2) E contains a neighborhood of 0; (3) m(ξ ) =

k∈

χE (ξ + 2kπ ) satis?es the consistency equation.

If. Conditions (1) and (2) assure that the dilates of W = 2E \ E form a partition of the real line by Lemma 2.4. Furthermore, in the proof of Lemma 2.4, we showed that E = ∪j<0 2j W . Hence, m(ξ ) de?ned in condition (3) is equal to ? = χW , and so Theorem 2.2 shows that the collection {e?inξ χW (ξ )} Df (ξ ) where f is an orthogonal set. Thus, W ?2π [0, 2π ), and the conditions of Theorem 2.1 are satis?ed. Only If. If E is a generalized scaling set, then we have seen that condition (1) holds. To see condition (2), note that if E did not contain a neighborhood of 0, then for any α ∈ ? , (?α, α) ∩ E c is not empty. If x is in the intersection, then x is not in any of the negative dilates of W , since x is not in E . For α su?ciently small, anything in this intersection would also not be in the non-negative dilates of W , a contradiction.

22 Furthermore, if ψW is a MSF wavelet, then

∞

DψW (ξ ) =

j =1 n∈ ∞

j 2 |ψ? W (2 (ξ + 2πn))|

=

j =1 n∈ ∞

χW (2j (ξ + 2πn)) χ2?j W (ξ + 2πn)

j =1 n∈

= =

n∈

χE (ξ + 2πn)

j<0 χ2j W

where again E = ∪j<0 2j W . Note that

= χE since the dilates of W are

disjoint. This combined with Theorem 2.2 show that condition (3) is satis?ed. The general procedure is to begin with a function m on [?π, π ) that satis?es the consistency equation, and then moving pieces of the support of m by multiples of 2π so that the conditions of Theorem 2.3 are satis?ed. In practice, we use the interval [?π, π ) instead of [0, 2π ) so that symmetry about 0 may be exploited. Example 2.3. We begin with the following function on [?π, π ), de?ned symmetrically about 0 by: ? ? ? π ? 2, ξ ∈ [0, 27 ) ? ? ? ? m(ξ ) = 1, ξ ∈ [ 2π , 4π ) ∪ [ 6π , π ) 7 7 7 ? ? ? ? ? ? ?0, elsewhere. A routine calculation shows that this function satis?es the consistency equation. We wish to construct the set E . First include in E the middle interval of

π 4π the support of m, [? 47 , 7 ). Next, we shall move the other parts of the support π π π of m: translate [?π, ? 67 ) by 2π to get [π, 87 ), and similarly translate [ 67 , π ) by π ?2π to get [? 87 , ?π ). Now, we shall move the intervals where m is 2: translate π π , 0) by ?2π to get [? 16 , ?2π ), and analogously translate the the interval [? 27 7

23

π π interval [0, 27 ) by 2π to get [2π, 16 ). Thus, 7

E = [?

8π 4π 4π 8π 16π 16π , ?2π ) ∪ [? , ?π ) ∪ [? , ) ∪ [π, ) ∪ [2π, ). 7 7 7 7 7 7

This set E clearly satis?es the ?rst two conditions. Furthermore, since the function

k∈

χE (ξ + 2kπ ) is 2π periodic, we have that this function equals m(ξ ) above,

and hence satis?es the consistency equation. Therefore, by Theorem 2.3, E is a generalized scaling set. By calculating 2E \ E , we get W = [? 32π 4π 4π 32π , ?4π ) ∪ [?π, ? ) ∪ [ , π ) ∪ [4π, ) 7 7 7 7

which is the Journ` e wavelet set, as in example 1.3.

Chapter

3

Operator Interpolation and Core Equivalence

This chapter presents the concept of the Local Commutant of a unitary system. This construction yields a parameterization of all wandering vectors for a unitary system. When applied to wavelet theory, it yields operator interpolation of wavelets, a procedure for constructing new wavelets from a suitable collection of MSF wavelets. This construction is not complete; it cannot always be done, and it is unknown whether this construction produces all wavelets. Operator interpolation is due to Dai and Larson, presented in full depth in [6]. Our contribution to the theory is that the concept of core equivalence gives a necessary condition for operator interpolation to occur. This new result also has relevance to the wavelet connectivity problem. For this chapter, a MSF wavelet ψW shall be thought of as a wavelet such ?W = χW . Note that this is a stronger condition than the original de?nition, that ψ ?W | = χW . i.e. |ψ 3.1 The Local Commutant As de?ned in chapter 1, a unitary system is a collection of unitary operators V on L2 (? ) that contains the identity. There is no assumption on the algebraic properties of this collection. Also de?ned in the introduction is the idea of a complete wandering vector for V , which is a vector f ∈ L2 (? ) such that the

25 collection {V f : V ∈ V} is an orthonormal basis for L2 (? ). Recall that the collection of all wandering vectors for V is denoted by W (V ). The main object of study in this chapter is the local commutant, which is a collection of operators that commutes with operators in V “at” a vector f . De?nition 3.1. The Local Commutant of V at f is a subset of B (L2 (? )) de?ned to be: Cf (V ) = {S ∈ B (H ) : SV (L2 (? )) ? V S (f ) = 0 ? V ∈ V}. We shall denote by Cψ (D, T ) the local commutant at the wavelet ψ of the unitary system {Dn T l : n, l ∈ }. The local commutant gives a parameterization of all wandering vectors for a unitary system in the following sense. Theorem 3.1. Let ψ ∈ W (V ). Then η ∈ W (V ) if and only if there exists a unitary U ∈ Cψ (V ) such that Uψ = η . If. If U is a unitary in Cψ (V ), then for V ∈ V , UV ψ = V Uψ = V η . Hence, U sends the orthonormal basis {V ψ : V ∈ V} to {V η : V ∈ V}, which must also be an orthonormal basis by virtue of U being unitary. Hence, it follows by de?nition that η ∈ W (V ). Only If. If η ∈ W (V ), then both {V ψ } and {V η } are orthonormal bases. De?ne UV ψ = V η ; it follows that U is unitary. Furthermore, under this de?nition, Uψ = η , so that UV ψ = V η = V Uψ , whence U ∈ Cψ (V ). Clearly, Cψ (V ) is closed under scalar multiplication, and a simple calculation shows that it is also closed under addition; hence, Cψ (V ) is a linear space. It is not necessarily, however, an algebra, for it may not be closed under multiplication. Additionally, it is not necessarily closed under adjoints. These properties depend

26 on V , but do not depend on ψ , facts that are proven in [6]. The local commutant does have the following topological property. Proposition 3.1. The local commutant Cψ (V ) is weakly closed. Proof. Suppose that Bn ∈ Cψ (V ) is a sequence of operators that converges to B . Then, if x, y are arbitrary elements of L2 (? ) and V ∈ V , then V Bx, y = lim V Bn x, y

n→∞ n→∞

= lim Bn V x, y = BV x, y

For the unitary system {Dn T l : n, l ∈

}, the local commutant is not

closed under either multiplication or adjoints, facts we shall prove in this chapter. However, unitaries in the local commutant at a wavelet preserves the GMRA structure in the following sense. Proposition 3.2. Let ψ be a wavelet, let U ∈ Cψ (D, T ) be a unitary operator, ?j where the Vj ’s are the GMRA associated to ψ and let η = Uψ . Then, UVj = V ?j ’s are the GMRA associated to η . and the V ? j . By de?nition, {Dj T l ψ : l ∈ Proof. It su?ces to show that UWj = W {D j T l η : l ∈ } and

? j , respectively. But also } form orthonormal bases for Wj and W

by de?nition, UD j T l ψ = Dj T l η , from which the proposition follows. In practice, given two wavelets ψ and η , it is di?cult to explicitly calculate the unitary in Cψ (D, T ) which maps ψ to η whose existence is guaranteed by Theorem 3.1. Fortunately, if ψ and η are both MSF wavelets, then this unitary can be calculated.

27 Recall that in section 2.3 we proved that if E and F are wavelet sets, then there exists a bijection σ of ? such that σ (E ) = F . De?ne an operator on L2 (? ) by Uσ f (x) = f ? σ ?1 (x); this is in fact a unitary operator since σ is a measure preserving bijection. Proposition 3.3. Let ψE and ψF be MSF wavelets. De?ne a unitary operator U by U = Uσ . Then U ∈ CψE (D, T ) is such that UψE = ψF . ?E and that Uσ ψ ?E = Proof. We shall prove that U = Uσ commutes with D n T l at ψ ?E (ξ ) = ?F . The second statement follows from the fact that σ (E ) = F . Indeed, Uσ ψ ψ ?F . Uσ χE (ξ ) = χE (σ ?1 (ξ )) = χσ(E ) (ξ ) = χF (ξ ) = ψ Now, to establish the commutation relation, ?E (ξ ) = D ?n T l χE (σ ?1 (ξ )) Uσ D ?n T l ψ √ n = 2 T l χE (2n σ ?1 (ξ )) √ n = 2 T l χE (σ ?1 (2n ξ )) since σ ?1 is 2-homogeneous = 2 e?ilσ (2 ξ ) χE (σ ?1 (2n ξ )) √ n ?1 n = 2 e?ilσ (2 ξ ) χF (2n ξ ). √

n

?1 n

Calculating the other way, ?E (ξ ) = D ?n T l Uσ ψ 2 T l Uσ χE (2n ξ ) √ n n = 2 e?il2 ξ Uσ χE (2n ξ ) √ n n = 2 e?il2 ξ χE (σ ?1 (2n ξ )) √ n n = 2 e?il2 ξ χF (2n ξ ). √

n

Thus, we need to show that

√ n √ n ?ilσ?1 (2n ξ ) n 2 e χF (2n ξ ) = 2 e?il2 ξ χF (2n ξ ), for

?1 (2n ξ )

which it su?ces to show that e?ilσ

= e?il2

nξ

when 2n ξ ∈ F . But σ ?1

28 is e?ected by a translation by a multiple of 2π on F , so if 2n ξ ∈ F , then σ ?1 (2n ξ ) = 2n ξ + 2kπ for some integer k . Therefore those exponentials indeed agree. When E and F are wavelet sets, the unitary in the local commutant at ψE will be denoted by Uσ . 3.2 Operator Interpolation of Wavelets Operator interpolation provides a way of constructing new wavelets. This can only happen under certain rather strong conditions, but these conditions occur frequently enough to be of interest. The crux of operator interpolation is to build a unitary operator out of other unitary operators in such a way as to assure that this new operator is in the local commutant of some wavelet. It turns out that this interpolated operator generates a von Neumann algebra that is in the local commutant of this wavelet. This ?rst building block is to start with a collection of wavelets that is “isomorphic” to a group of unitary operators in the local commutant. Suppose {Ug : g ∈ G} forms a group of unitary operators, and suppose that every Ug ∈ Cψ (D, T ) for some wavelet ψ . Then, by Theorem 3.1, Ug ψ is a wavelet for all g ∈ G. Let ψg = Ug ψ . We shall say that the collection {ψg : g ∈ G} forms a pre-interpolation family based at ψ . This collection {Ug } can be thought of as a unitary representation of the group G. It follows that the linear span of {Ug } is a ?-algebra that is contained in Cψ (D, T ). Operator interpolation then is the construction of an operator V out of this collection {Ug }, given by V =

g ∈G

Ag Ug

(3.1)

29 which for the right “coe?cient operators” Ag will be a unitary operator in Cψ (D, T ). Let {D, T } denote the commutant of the unitary system {D n T l : n, l ∈ }. Proposition 3.4. If A ∈ {D, T } and U ∈ Cψ (D, T ), then AU ∈ Cψ (D, T ). In other words, Cψ (D, T ) is closed under left multiplication by {D, T } . Proof. We have that AV Dn T l ψ = ADn T l V ψ = Dn T l AV ψ . It follows that if the coe?cient operators Ag ∈ {D, T } above, then V will again be in Cψ (D, T ). The following theorem is a characterization of {D, T } . Theorem 3.2. Analogous to above, let {D, T } denote the commutant of the unitary system {D n T l : n, l ∈ h(2x) a.e.}. Proof. We shall prove that {Mh : h ∈ L∞ (? ), h(x) = h(2x) a.e.} ? {D, T } . The reverse containment shall be demonstrated in the Appendix. First note that T l = Me?ilξ , which clearly commutes with Mh . So, it su?ces to show that D commutes with Mh . Suppose that h ∈ L∞ (? ) and h(x) = h(2x) a.e. Then, ?(ξ ) = DMh f √ ?(2ξ ) ?(2ξ ) = 2h(2ξ )f 2Mh f √ ?(2ξ ) = h(ξ )Df ?(ξ ) = h(ξ ) 2f √ }. Then {D, T } = {Mh : h ∈ L∞ (? ), h(x) =

?(ξ ) = Mh D f as required. De?nition 3.2. If h is a measurable function such that h(x) = h(2x) a.e., then h will be called 2-dilation periodic. We now have a necessary condition for V to be in Cψ (D, T ), but we also want V to be unitary. Suppose that the pre-interpolation family {ψg : g ∈ G} is

30 ?nite. Suppose further that for g ∈ G the operator Ug normalizes {D, T } in the following sense:

? Ug {D, T } Ug = {D, T } .

Then the *-algebra generated by {Ug }∪{D, T } will consist of, before closure, polynomials as in equation (3.1), a fact we shall implicitly prove in Proposition 3.5. The closure of this algebra in the weak operator topology is a von Neumann algebra which is *-isomorphic to the cross-product of {D, T } under the automorphism group induced by {Ug }. This cross-product has special matricial form (see [6], [17]). De?nition 3.3. Suppose that {ψg : g ∈ G} forms a pre-interpolation family based at ψ , and suppose further that all of the corresponding Ug ’s normalize {D, T } . Then {ψg } will be called an interpolation family of wavelets. In the special case when two wavelets ψ and η form an interpolation family, we shall say that ψ and η form an interpolation pair. Proposition 3.5. Suppose that V is an interpolated operator as in equation (3.1), where the operators Ag ∈ {D, T } and the operators Ug normalize {D, T } . Then the von Neumann algebra generated by V is in Cψ (D, T ). Proof. We shall ?rst show that polynomials in V and V ? are contained in Cψ (D, T ). First note that V ? =

g ∈G ? ? Ug Ag . Since the Ug ’s normalize {D, T } , there exist

? ? operators Bg ∈ {D, T } such that Ug Ag = Bg Ug , from which it follows that

V ? ∈ Cψ (D, T ). Furthermore, by the same reasoning, V 2 ∈ Cψ (D, T ), since V 2 is a sum with terms Ag Ug Ag Ug . But the Ug Ag can be rewritten as Bg,g Ug , so that V 2 can be written in the form above. It follows that polynomials in V and V ? are in Cψ (D, T ), and then Proposition 3.1 shows that the von Neumann algebra generated by V is in Cψ (D, T ).

31 In practice, operator interpolation is done when the interpolation family is “cyclic”, i.e. the group {Ug } is isomorphic to a ?nite cyclic group. Furthermore, the interpolation family consists of MSF wavelets for the following reason. Proposition 3.6. The unitary Uσ between two MSF wavelets normalizes {D, T } . Proof. If A ∈ {D, T } , then A = Mh for some essentially bounded 2-dilation

? periodic function h. We need to show that Uσ Mh Uσ = Mg for some essentially

bounded 2-dilation periodic function g . Claim: g = h ? σ , i.e. Mh Uσ = Uσ Mh?σ ? ∈ L2 (? ). Then and h ? σ is 2-dilation periodic. Let f ?(ξ ) = h(ξ )Uσ f ?(ξ ) = h(ξ )f ?(σ ?1 (ξ )), Mh Uσ f but ?(ξ ) = Mh?σ?1 f ?(σ ?1 (ξ )) = h(σ (σ ?1 (ξ )))f ?(σ ?1 (ξ ) = h(ξ )f ?(σ ?1 (ξ )). Uσ Mh?σ?1 f Finally, h ? σ (2ξ ) = h(2σ (ξ )) = h ? σ (ξ ) since σ is 2 homogeneous and h is 2-dilation periodic. If σ 2 is the identity on ? , it follows that U 2 = I where U = Uσ . Thus, if E and F are wavelet sets such that σ is involutive, ψE and ψF form an interpolation pair. Example 3.1. The two wavelets from example 2.1 from chapter 2 form an in2 terpolation pair, since Uσ = I , and Proposition 3.6 shows that Uσ normalizes

{D, T } . If ψE and ψF form an interpolation pair, then the interpolated operator is given by V = Mh1 I + Mh2 Uσ

32 where, by Theorem 3.2, h1 and h2 are essentially bounded and 2 dilation periodic. Applying this operator yields ?E = h1 ψ ?E + h2 ψ ?F . Vψ As mentioned above, V is an element of a cross product which is *-isomorphic to a matricial algebra. This gives the following coe?cient criterion, which dictates that V is unitary if and only if the matrix ? ? h2 ? ? h1 ? ? h2 ? σ ?1 h1 ? σ ?1

(3.2)

is unitary almost everywhere. Since σ ?1 is 2-homogeneous and the hi ’s are 2dilation periodic, and the dilates of E partition ? , it su?ces to check this condition only on E . We shall now present an example of operator interpolation. Example 3.2. Consider the following wavelet sets: E = [? 7π 4π 4π 7π 28π 32π 32π 28π ,? ) ∪ [? , ? ) ∪ [ , ) ∪ [ , ) 7 7 7 7 7 7 7 7 8π 4π 4π 6π 24π 32π F = [? , ? ) ∪ [ , ) ∪ [ , ) 7 7 7 7 7 7

We have that σ is given by: ? ? ? π π π 6π π 32π ? ξ, ξ ∈ [? 77 , ? 47 ) ∪ [ 47 , 7 ) ∪ [ 28 , 7 ) ? 7 ? ? ? σ (ξ ) = ξ ? 2π, ξ ∈ [ 6π , 7π ) 7 7 ? ? ? ? ? ? ?ξ + 8π, ξ ∈ [? 32π , ? 28π ) 7 7 It can be veri?ed that this σ is involutive. Construct h1 and h2 as follows: 1 h1 = χE ∩F + √ χ[? 32π ,? 28π )∪[ 6π , 7π ) 7 7 7 7 2 1 h2 = √ χ[? 8π ,? 7π ) ? χ[ 24π , 28π ) 7 7 7 7 2

33 ?= We can extend h1 and h2 so that they are 2-dilation periodic. In order for ψ ?E + h2 ψ ?F to be a wavelet, we must verify that h1 and h2 satisfy the coe?cient h1 ψ criterion of the matrix in 3.2. As remarked above, it su?ces to show this condition on E . Clearly, on E ∩ F the matrix is unitary since it is in fact the identity

π 7π π 7π π π there. Now, on [ 67 , 7 ), σ ?1 (ξ ) = ξ ? 2π , so that σ ?1 ([ 67 , 7 )) = [? 87 , ? 77 ). π 7π If ξ ∈ [ 67 , 7 ), then h1 (ξ ) = h2 ? σ ?1 (ξ ) = 1 √ . 2 π 7π Since [ 67 , 7) = 1 24π 28π [ , 7 ) 4 7 1 √ 2

1 and h2 is 2-dilation periodic, h2 (ξ ) = ? √ . Finally, h1 ? σ ?1 (ξ ) = 2 π π π π π 7π [? 87 , ? 77 )= 1 [? 32 , ? 28 ). Hence, on [ 67 , 7 ), 4 7 7 ? ? ? ? 1 1 h2 ? ? √2 ? √2 ? ? h1 ?=? ? ? 1 1 √ √ h2 ? σ ?1 h1 ? σ ?1 2 2

since

which is unitary. A similar computation shows that the matrix is unitary on

π π [? 32 , ? 28 ). 7 7

Here is a graph of the interpolated wavelet in the frequency domain.

1

-5 p

-3 p

-p

p

3p

5p

Figure 3.1: Plot of the Interpolated Wavelet

3.3

A Necessary Condition for Operator Interpolation The main new result of this section is that if a unitary operator in the local

commutant of a wavelet has all of its powers back in the local commutant, then

34 it is necessary for all of the associated wavelets to be core equivalent. This result gives a necessary condition for operator interpolation to occur, as well as a result concerning the interpolated wavelet. We begin with a lemma. Lemma 3.1. Let ψ be a wavelet, let U ∈ Cψ (D, T ) be a unitary operator, let η = Uψ , and let V be another operator such that V U is also in Cψ (D, T ). Then, V is in Cη (D, T ). Proof. We need to show that V D n T l η = D n T l V η for all n, l ∈ . We have: V Dn T l η = V Dn T l Uψ = V UD n T l ψ = Dn T l V Uψ = D n T l V η.

Theorem 3.3. Let U be a unitary operator such that U n ∈ Cψ (D, T ) for all n ∈ . Then, we have a sequence of wavelets, U n (ψ ), and they are all core

equivalent. Proof. Each wavelet can be associated to a GMRA; let ψ (i) denote the wavelet U i (ψ ), and let V0i denote the V0 core space for the wavelet ψ (i) . We need to construct an intertwining operator between V0 and V0 . Let Y denote the closed subspace spanned by ∪i∈ (V0i )⊥ . Lemma 3.2. If x ∈ Y , then U i T n (x) = T n U i (x). Proof of Lemma. It su?ces to establish the lemma for a generating vector of Y . If x ∈ ∪i∈ (V0i )⊥ , then x ∈ (V0i0 )⊥ for some i0 . Let η denote the wavelet ψ (i0 ) . We have:

∞ (i)

x=

j =0 k ∈

x, D j T k η D j T k η.

35 By Lemma 3.1, U i ∈ Cη (D, T ). Hence:

∞

U T (x) = U T = Ui

i

n

i

n j =0 k ∈

x, D j T k η D j T k η x, D j T k η D j T 2 x, D j T k η D j T 2

j n+k

∞ j =0 k ∈ ∞

η

= =T

j =0 k ∈ ∞ n

j n+k

U iη

x, D j T k η D j T k U i η x, D j T k η D j T k η

=T U

j =0 k ∈ ∞ n i j =0 k ∈

= T n U i (x). as required. ? and Since (V0 )⊥ ? Y for all i, we have Y ⊥ ? V0 . Write V0 = Y ⊥ ⊕ Y ? (i) . De?ne S : V0 → V0 V0 = Y ⊥ ⊕ Y

(i) (i) (i) (i)

by S = PY ⊥ + U i PY ?.

(1)

Proposition 3.2 shows that U (V0 ) = V0 . Since U ∈ Cψ(i) (D, T ) for all i the subspace Y is invariant under U . Indeed, U (Y ) = Y . Therefore, Y ⊥ is also invariant under U . Analogously, both Y and Y ⊥ are invariant under T . Note that U i maps V0 unitarily onto V0 and leaves Y ⊥ ?xed, hence U i maps ? unitarily onto Y ? (i) . This proves that S is a unitary operator. Furthermore, as Y noted above, all of these subspaces in question are invariant under translation, hence by Lemma 3.2 shows that S commutes with translations. Therefore, S is the required intertwining operator. The following corollaries are direct consequences of Theorem 3.3. Corollary 3.1. If F = {ψg : g ∈ G} is an interpolation family of wavelets, then they are all core equivalent.

(i)

36 Corollary 3.2. If A ? Cψ (D, T ) is a von Neumann algebra, then all wavelets that are parameterized by unitary operators in A are core equivalent. Corollary 3.3. If ψ is an operator interpolated wavelet, interpolated from the interpolation family F , then ψ is core equivalent to all of the wavelets in F . 3.4 The Wavelet Connectivity Problem The wavelet connectivity problem is the question of whether the collection of all wavelets forms a path connected subset of the unit sphere of L2 (? ) with respect to the norm topology. Using the unitary operators in a von Neumann algebra is useful for solving this problem since the unitary group of a von Neumann algebra is path connected in the operator topology (see [17]). Hence, of particular interest is when the local commutant contains a von Neumann algebra. By Theorem 3.2, a necessary condition for a von Neumann algebra to be in the local commutant is that all of the associated wavelets are core equivalent. Hence, as far as the local commutant parameterizing connected subsets of wavelets, it can only parameterize those that are core equivalent. A su?cient condition for a von Neumann algebra to be in the local commutant would be of great importance, as it may yield new path connected subsets of the collection of wavelets. However, the “converse” of Theorem 3.3 is false, i.e. if ψ and η are core equivalent, then the unitary in Cψ (D, T ) that maps ψ to η does not necessarily have all of its powers back in Cψ (D, T ). Lemma 3.3. Suppose E and F are wavelet sets, and Uσ is the unitary in the

2 is also in the local commutant at ψE if and local commutant at ψE . Then, Uσ

only if σ 2 is given by 2π translations on E .

2 Proof. In Proposition 3.3, we proved that Uσ ∈ Cψ ?E (D, T ) is equivalent to the ex-

ponentials e?ilσ

?2 (2n ξ )

= e?il2

nξ

agreeing on σ 2 (E ). It follows that this is equivalent

37 to σ 2 being given by 2π translations on E .

π 15π De?ne the wavelet set E = [? π , ?π ) ∪ [ 15 , 4 ), let F = ?E , which is 4 8 8

also a wavelet set. Proposition 2.1 shows that both ψE and ψF are associated to MRA’s, hence they are core equivalent, as will be shown in chapter 4. A routine calculation shows that: ? ? ? π 18π ? ξ ? 2π, ξ ∈ [? π , ?π ) ∪ [ 17 , 8 ) ? 4 8 8 ? ? ? σ (ξ ) = ξ ? 4π, ξ ∈ [ 15π , 17π ) 8 8 ? ? ? ? ? ? ?ξ ? 6π, ξ ∈ [ 18π , 30π ) 8 8

π 273π , 128 ) ? E . Then, A ? 2π = [ π , 17π ) ? [ π , π ) ? F , hence, Consider A = [ 17 8 8 128 8 4

σ (ξ )|A = ξ ? 2π . Now let’s calculate σ 2 (ξ )|A = σ (ξ )|A?2π . Note that 16(A ? 2π ) =

π [2π, 17 ), so then if ξ ∈ A, σ 2 (ξ ) = σ (ξ ? 2π ) = 8 1 σ (16(ξ 16

? 2π )) =

1 [(16(ξ 16

?

. Hence, Lemma 3.3 shows that U 2 given by σ 2 cannot be 2π ) ? 4π ] = ξ ? 2π ? π 4 in Cψ ?E (D, T ). As a result of this example we have the following corollary. Corollary 3.4. If ψ is a wavelet, then Cψ (D, T ) is not necessarily closed under multiplication or adjoints. Proof. The fact that U 2 ∈ / Cψ (D, T ) shows that Cψ (D, T ) is not closed under

? multiplication. Lemma 3.1 shows that in the example above, Uσ ∈ CψF (D, T ),

but Uσ is not.

Chapter

4

The Wavelet Multiplicity Function

In section 1.2, a GMRA is built from a wavelet ψ . This GMRA has the property that V0 is invariant under integral translations. As such, there exists a representation of the integers on V0 . Recall that this representation is called the core representation, and that two wavelets are core equivalent if their core representations are unitarily equivalent. These core representations, however, are not always equivalent. Indeed, we shall show that ψ is an MRA wavelet, if and only if the core representation is equivalent to the left regular representation. We have already seen that there are MRA and non-MRA wavelets, whence they are not core equivalent. The goal of this chapter is to develop a method for determining exactly when two wavelets are core equivalent. Much of the material presented here requires some standard, and some nonstandard concepts from abstract harmonic analysis. These concepts will be presented here without proof; the proofs will be given in the Appendix. Additionally, we will refer to Haar measure on [0, 2π ), by which we mean Haar measure on the circle S 1 transferred to [0, 2π ). 4.1 A Description of The Core Representation The ?rst step in our goal of exploring core equivalence is to use an extension of Stone’s theorem of classical harmonic analysis which gives a description of a

39 unitary representation. In this theorem, G denotes the dual group of G, i.e. the group of characters on G. Theorem 4.1. Suppose π is a unitary representation of a locally compact abelian group G. Then there exists a projection valued measure p on G such that πx =

G

ξ (x)dp(ξ )

for all x ∈ G. Outline of Proof. The proof naturally utilizes the Spectral Theorem from Functional Analysis. The ?rst step is to prove that L1 (G) is a commutative Banach ?-algebra and that π extends to a ?-representation of L1 (G), so that there exists a projection valued measure on the spectrum of L1 (G). Then, for f ∈ L1 (G), πf is given by an integral against this projection valued measure using the Gelfand Transform. The next step is to show that the spectrum of L1 (G) can be associated with G, so that the projection valued measure can be pushed on to G. The ?nal step is to show that πx = lim π (Lx fU ), where fU is an approximate identity. This yields that πx is given by the integral above. A complete proof can be found in [11]. This next theorem describes projection valued measures in terms of canonical projection valued measures. This theorem provides a method of distinguishing two representations up to unitary equivalence. The proof will be given in the appendix. Theorem 4.2. Let p be a projection valued measure on a Borel space (S, B). There exists a ?nite measure ν and a measurable function m : S → {0, 1, 2, 3, . . . , ∞} such that (1) there exists a unitary operator U from H onto

2 ⊕∞ j =1 L (Ej , ν |Ej , j

) ⊕ L2 (E∞ , ν |E∞ , l2 ( ))

40 where Ej = m?1 (j ), and (2) U intertwines p with the canonical projection valued measure. Moreover, if ν and m are another ?nite measure and function, respectively, satisfying properties (1) and (2) above, then ν ≡ ν and m = m a.e. ν . Remark 4.1. Throughout this chapter, we shall refer to the multiplicity function which may take on the value 0. This only makes sense when the representation in question, the core representation, is thought of as a sub-representation of some larger representation. Here the larger representation is, naturally, the representation of given by translations acting on all of L2 (? ). In general, as shall be shown

in the proof of Theorem 4.2, given a projection valued measure, the multiplicity function attaining the value 0 has no sensible meaning. In this decomposition, we have two “parameters” that describe a representation up to unitary equivalence. Indeed, if π and π are two representations with associated probability measures ν , ν and multiplicities m and m respectively, then π is unitarily equivalent to π if and only if ν is equivalent to ν and m = m almost everywhere. In the case of core representations, the multiplicity function alone completely describes the representation. We begin with the following proposition. Proposition 4.1. The measure from Theorem 4.2 is absolutely continuous with respect to Lebesgue measure for the core representation of any wavelet. Proof. We shall supply the proof that is presented in [5]. By the calculation in chapter 1, taking the Fourier transform of T n yields multiplication by e?inξ . The representation of ?) ? restricted to the subspace H = {f ∈ L2 (? ) : supp(f

[0, 2π )} is equivalent to the left regular representation. The left regular representation has a probability measure that is equivalent to Haar (Lebesgue) measure on

41 [0, 2π ), so that the representation of on all of L2 (? ) has a probability measure

that is equivalent to Lebesgue measure. It follows that any sub-representation, in particular the core representation, has a probability measure that is absolutely continuous with respect to Lebesgue measure. Theorem 4.3. Two wavelets ψ and η are core equivalent if and only if their multiplicity functions agree a.e. (with respect to Lebesgue measure). Proof. Proposition 4.1 shows that the measures for ψ and η , denoted by ν and ν , are absolutely continuous with respect to Lebesgue measure. We shall show that if ψ and η have multiplicity functions that agree, then ν and ν must be equivalent. Suppose, by contradiction, that the multiplicities agree a.e. but there exists a set F of non-zero Lebesgue measure such that ν (F ) = 0 but ν (F ) = 0. The proof of Theorem 4.2 shows that the measure ν is “supported” exactly where the multiplicity function is supported. Hence, the multiplicity function for η is nonzero on F but the multiplicity function for ψ is 0 on F , a contradiction to the assumption that they agree a.e. We have seen that the core representation can be described by integration against a projection valued measure, which in turn is described by a multiplicity function. Thus, we have the following de?nition. De?nition 4.1. If ψ is a wavelet, then there exists a multiplicity function associated to it which describes the core representation. This function is called the wavelet multiplicity function. The wavelet multiplicity function generates an equivalence relation on all wavelets, one of two to be presented in this thesis. This equivalence relation extends the concept of MRA versus non-MRA wavelets. Indeed, the collection of all MRA wavelets forms an equivalence class under this relation.

42 Theorem 4.4. If ψ is a wavelet, and m is its associated multiplicity function, then ψ is a MRA wavelet if and only if m ≡ 1. Proof. We shall prove in the Appendix that m ≡ 1 if and only if a representation is equivalent to the left regular representation. Suppose that the core representation is equivalent to the left regular representation and that U : L2 ( ) → V0 is an intertwining operator. Then the function χ{0} is a cyclic vector for the left regular representation. Furthermore, Lz χ{0} = χz is perpendicular to χ{0} , so that in fact this cyclic vector generates an orthonormal basis of L2 ( ). It follows that {ULz χ{0} : z ∈ } forms an orthonormal basis for V0 , whence Uχ{0} is a scaling function. Conversely, if there exists a scaling function φ, then for f ∈ V0 , f =

z∈

cz T z φ. De?ne an operator V f =

z∈

cz Lz χ{0} . A routine computation

shows that V : V0 → L2 ( ) is a unitary operator. Furthermore, V intertwines the core representation with the left regular representation: V T lf = V T l

z∈

cz T z φ = V

z∈

cz T l T z φ = V

z∈

cz T l + z φ cz Lz χ{0}

z∈

=

z∈

cz Ll+z χ{0} =

z∈

cz Ll Lz χ{0} = Ll

= Ll V f hence, the core representation and the left regular representation are unitarily equivalent. There are, in fact, an uncountable number of wavelet multiplicity functions; these functions can take on any ?nite value, and also can be unbounded. These facts are outside the scope of this thesis.

43 4.2 An Explicit Formula In this section we will derive an explicit formula for the wavelet multiplicity function in terms of the wavelet itself. Indeed, as it turns out, the wavelet multiplicity function coincides with the wavelet dimension function. We begin by decomposing the core representation into cyclic subrepresentations. Cyclic representations have multiplicity functions that take on only the values 0 and 1, and the multiplicity function of a direct sum of representations is the sum of the individual multiplicity functions. For j > 0, let ψj = D ?j ψ .

⊥ Let g1 = ψ1 , and let Y1 be the cyclic subspace generated by g1 . Let g2 = PY ψ, 1 2

and let Y2 be the cyclic subspace generated by g2 . Recursively de?ne gj to be the projection of ψj onto the perpendicular complement of ⊕n<j Yn , and Yj to be the cyclic subspace generated by gj . By de?nition, each Yj determines a cyclic subrepresentation, with cyclic vector gj . Proposition 4.2. (1) V0 = ⊕j>0 Yj , (2) m =

∞ j =1 mj ,

where mj is the multiplicity function of the cyclic repre-

sention on Yj . Proof. By de?nition, the Yj ’s are orthogonal and are subspaces of V0 . Since the translations of the ψj ’s spans V0 , it su?ces to show that they are contained in this direct sum. But note that ψ1 is in Y1 , and then ψ2 can be written as g2 + f2 , where f2 ∈ Y1 since g2 is obtained by a projection. By the recursive de?nition of the gj ’s, we get that ψj is in the direct sum, and item 1 is established. Item 2 follows from the general fact that the multiplicity function for a representation is the sum of the multiplicity functions for orthogonal cyclic subrepresentations.

44 Since gj is a cyclic vector, it generates a positive de?nite function, pj (l) = T l gj , gj . By Bochner’s theorem, there exists a measure ?j whose Fourier-Stieltjes transform is pj . Since ? is absolutely continuous with respect to Lebesgue measure, the measure ?j is also. Let hj be the Radon-Nikodym derivative of ?j with respect to Lebesgue measure. Since the subrepresentation on Yj is cyclic, mj takes on only the values 0 and 1; in fact, mj = χsupp(hj ) . Furthermore, since m = have then that m =

∞ j =1 χsupp(hj ) . ∞ j =1 mj

we

Proposition 4.3. Let hj be as above. Then: hj (ξ ) = 2π gj (ξ ) Proof. We have:

2π 0 2

(4.1)

e

?inξ

2π

hj (ξ )dλ =

0

e?inξ d?j (ξ )

= ?j (?n) = pj (?n) = T ?n g j , g j = 2π

2π

?

e?inξ gj (ξ )gj (ξ )dλ

=

0

e?inξ 2π gj (ξ ) 2 dλ

By de?nition, gj (x) = ψj (x) ? wj (x) (4.2)

where wj is the unique element in ⊕k<j Yk such that gj ⊥ ⊕k<j Yk . Additionally, since wj can be expressed in terms of the translates of the gk ’s, by taking the Fourier Transform of both sides of 4.2, we get: ?j (ξ ) ? gj (ξ ) = ψ

k<j

ηj,k (ξ )gk (ξ )

45 where the ηj,k are 2π -periodic measurable functions. Proposition 4.4. If ηj,k is as above, then ηj,k (ξ ) = ψj (ξ ), gk (ξ ) gk (ξ )

2

where this is interpreted to be 0 when the denominator is 0. Proof. First notice that since ηj,k gk ∈ L2 (? ), ηj,k (ξ ) gk (ξ )

2

∈ L1 ([0, 2π ]). Fur⊥

thermore, ηj,k gk ∈ Yk , indeed, ηj,k gk is the function such that ψj ? ηj,k gk ∈ Yk . Hence, ηj,k gk , e?in· gk = ψj , e?in· gk . Hence,

2π 0

ηj,k (ξ ) gk (ξ ) 2 e?inξ dλ =

?

ηj,k (ξ )gk (ξ )gk (ξ )e?inξ dλ

= ηj,k gk , e?in· gk = ψj , e?in· gk = =

0

?

ψj (ξ )gk (ξ )e?inξ dλ ψj (ξ ), gk (ξ ) e?inξ dλ

2π

This gives us that gj (ξ ) = ψj (ξ ) ?

k<j

ψj (ξ ), gk (ξ ) gk (ξ )

2

gk (ξ ).

(4.3)

Since the inner product in equation 4.3 is invariant under 2π translations, we have: gj (ξ ) = ψj (ξ ) ?

k<j

ψj (ξ ), uk (ξ ) uk (ξ )

(4.4)

where uk (ξ ) = if the norm is non-zero. gk (ξ ) gk (ξ )

2

46 As we have mentioned above, the multiplicity function is the sum of the multiplicity functions for the cyclic subspaces Yj , each of which is the characteristic function of the support of hj . Hence, by equation 4.1, the multiplicity function is precisely the number of non-zero sequences gj (ξ ). Now, let us examine more closely equation 4.4. Note that g1 = ψ1 , so g1 (ξ ) = ψ1 (ξ ) for almost all ξ , and u1 is the normalization of that vector. Furthermore, g2 (ξ ) = ψ2 (ξ ) ? ψ2 (ξ ), u1(ξ ) u1(ξ ) which is the Gram-Schmidt orthogonalization of ψ1 (ξ ) and ψ2 (ξ ), with u1 (ξ ) and u2 (ξ ) being normalized. By the recursive de?nition of the gj ’s, equation 4.4 is actually the Graham-Schmidt orthogonalization of the ψj (ξ )’s. Hence, hj (ξ ) = 0 if and only if ψj (ξ ) is in the linear span of the previous ψn (ξ )’s. Therefore, m(ξ ) is the number of linearly independent vectors in the collection {ψj (ξ )}, i.e. the dimension of the subspace spanned by those vectors. Let ψ be a wavelet on L2 (? ), and recall that the wavelet dimension function is given by:

∞

Dψ (ξ ) =

j =1 k ∈

?(2j (ξ + 2πk ))|2 . |ψ

+

Theorem 4.5. Let ψ be a wavelet, and let m : [?π, π ] → multiplicity function. Then m(ξ ) = Dψ (ξ ).

be its associated

Proof. It is shown in [13] that the dimension function is integer valued, and in fact is the dimension of a subspace of l2 ( ). Let Ψj (ξ ) be a sequence on given

by Ψj (ξ )[k ] = ψ (2j (ξ + 2πk )). It is also shown that Dψ (ξ ) is the dimension of the subspace of l2 ( ) spanned by {Ψj (ξ ) : j > 0}. We have that both the dimension function and the multiplicity function describe the dimension of some subspace of l2 ( ). The spanning vectors are

47 di?erent; however, the subspaces are the same. Indeed, Ψj (ξ )[k ] = ψ (2j (ξ + 2πk ))

j j ?j (ξ ), ?(2j (ξ + 2πk )). Hence, Ψj (ξ ) = 2 2 where as ψj (ξ )[k ] = ψj (ξ + 2πk ) = 2 2 ψ ψ

and so the spaces spanned by them are the same. Corollary 4.1. The wavelet multiplicity function is ?nite almost everywhere. Proof. The wavelet dimension function is ?nite almost everywhere, which was shown in section 2.4. Corollary 4.2. The wavelet multiplicity function satis?es the following consistency equation: ξ ξ m(ξ ) + 1 = m( ) + m( + π ). 2 2 Proof. We have already demonstrated in chapter 2 that the wavelet dimension function satis?es the consistency equation. It then follows from Theorem 4.5. Remark 4.2. The consistency equation is ?rst introduced in [5], where the wavelet multiplicity function is also ?rst introduced. In that paper the consistency equation is derived by looking at the representation of the integers on the subspace V1 , and then compared to the representation on V0 . The main idea is that the left side of the consistency equation describes the action of on V0 plus W0 , where

the representation is equivalent to the left regular representation. The right side of the equation is a result of the fact that the representation on V1 is described by conjugating the translation operators by the dilation operator. 4.3 Examples of the Wavelet Multiplicity Function

Example 4.1. We have already seen that the Journ? e wavelet is a non-MRA wavelet. Recall that it is a MSF wavelet whose wavelet set is: W = [? 32π 4π 4π 32π , ?4π ) ∪ [?π, ? ) ∪ [ , π ). ∪ [4π, ) 7 7 7 7

48 Using Theorem 4.5, it is easy to calculate the multiplicity function of this wavelet. ? ? ? ? 2, 0 ≤ x < 1 ; ? 2 ? ? ? m(ξ ) = 1, 1 ≤ x < 1; 2 ? ? ? ? ? ? ?0, elsewhere. Example 4.2. The interpolated wavelet in example 3.2 has the same multiplicity function as above. This follows from corollary 3.3.

Chapter

5

On the Translation Invariance of Wavelet Subspaces

We have seen in chapter 4 that the action of the integers on V0 generates an equivalence relation whose equivalence classes consist of wavelets whose multiplicity functions are the same. This equivalence relation hinges on the fact that V0 is invariant under integral translations. A natural question is: are there other groups of translations under which V0 is invariant? We will answer this question by looking at groups of translations by dyadic rationals. This analysis generates a new equivalence relation on the collection of all wavelets. For the purposes of this chapter, denote by Tα the unitary operator T f (x) = f (x ? α). When n is an integer, T n = Tn . A similar calculation to that in chapter 1 shows that Tα = Me?iα· . For ease of notation, we shall say that a set E ? ? is self-similar with respect to α ∈ ? if there exists a set F of non-zero measure such that both F and F + α are subsets of E . Recall that if G, H are two subsets of ? , then G is 2π translation congruent to H if there exists a measurable partition Gn of G such that the collection {Gn + 2nπ : n ∈ } forms a partition of H , modulo sets of measure zero. 5.1 A New Classi?cation As mentioned above, we shall consider groups of translations by dyadic

m : m ∈ rationals. In particular, we will consider the groups Gn = {T 2 n

}, and

50 the group G∞ = {Tα : α ∈ ? }. Denote by Ln the collection of all wavelets whose subspace V0 is invariant under Gn . Note that these collections are nested: L0 ? L1 ? L2 . . . ? Ln ? Ln+1 ? . . . ? L∞ We can then de?ne an equivalence relation whose equivalence classes are given by Mn = Ln ? Ln+1, with M∞ = L∞ . Hence, Mn is the collection of all wavelets such that V0 is invariant under Gn but not Gn+1 . The goal of this chapter is to characterize these equivalence classes, while showing that several of them are not empty. In general, V0 can be quite complicated in structure. Indeed, it may not even be generated by translations of a ?nite number of functions. Hence, we wish to restrict our analysis to W0 . Recall that Wj ’s are de?ned by Vj +1 = Vj ⊕ Wj . Clearly, Wj = span{D j T l ψ : l ∈ f=

k∈

}. Furthermore, if f ∈ W0 , then we can write

? = hψ ? for ck T k ψ , so taking the Fourier transform of both sides yields f

some h ∈ L2 ([0, 2π )). Lemma 5.1. Let r < n, and let p = n ? r . Then Vr (resp. Wr ) is invariant under Gk if and only if Vn (resp. Wn ) is invariant under Gk+p . f ∈ Vr , then Proof. By de?nition, f ∈ Vr if and only if D p f ∈ Vn . Hence, if f , T m k

2

Dpf , DpT m f ∈ Vn . However, k

2

Dp T m f (x) = 2 2 T m f (2p x) k k

2 2

p

m ) 2k m = D p f (x + k+p ) 2 = 2 f (2p x +

p 2

=T

m 2k+p

D p f (x)

m 2k+p

hence, Vn is invariant under Gk+p . Likewise, if f and T and D?p T

m 2k+p

f are in Vn , then D ?p f

f are in Vr , whence Vr is invariant under Gk .

51 By lemma 5.1, another way to describe Mn is that ψ ∈ Mn if n is the largest integer such that V?n is invariant under integral translations. If ψ ∈ Mn , we shall say ψ has the translation invariance of order n property. Theorem 5.1. V0 is invariant under Gn if and only if W0 is invariant under Gn . If. Suppose that W0 is invariant under Gn . Then, by Lemma 5.1, for k > 0, Wk is invariant under Gn+k , whence V0⊥ = ⊕∞ k =0 Wk is invariant under Gn . If follows that V0 is invariant under Gn . Only If. Suppose V0 is invariant under Gn . Then, again by Lemma 5.1, V1 is invariant under Gn+1 , and hence Gn . Since V1 = V0 ⊕ W0 , it follows that W0 is also invariant under Gn . 5.2 A Characterization of M∞ Wavelets in M∞ are those which have the property that W0 is invariant under translations by any real number. By taking the Fourier transform, this is equivalent to W0 being invariant under multiplication by e?iα· for any α. We have already seen in section 2.2 that W0 = L2 (W ) which is clearly invariant under multiplications by exponentials. These conditions are actually equivalent, as illustrated by the next theorem. Theorem 5.2. Let ψ be a wavelet. Then, the following are equivalent: i) ψ is a MSF wavelet, ii) the subspace V0 is invariant under translations by all real numbers, iii) the subspaces Vj of the corresponding GMRA are invariant under integral translations.

52 iv) the subspaces Wj of the corresponding GMRA are invariant under integral translations. Proof. i) ? ii). This was demonstrated in the remarks preceding this theorem. ii) ? iii). Since V0 is invariant under Gn for all n ≥ 0, by Lemma 5.1, V?n is invariant under G0 . iii) ? iv). By de?nition, Vj +1 = Vj ⊕ Wj . If both Vj +1 and Vj are invariant under integral translations, it follows immediately that Wj is also invariant under integral translations. iv) ? i). Let C be the collection of all operators for which W0 is invariant. An easy calculation shows that C is WOT closed. If every Wj is invariant under integral translations, then again by Lemma 5.1, W0 is invariant under Gj for all j . Since ∪n≥0 Gn is dense in G∞ , in the strong operator topology, it follows that W0 is invariant under G∞ . If we take the Fourier transform, then we get that W0 is invariant under multiplication by e?iαξ . The linear span of these operators are dense in the collection {Mh : h ∈ L∞ (? )} with respect to the WOT. It follows that W0 is invariant under multiplication by any L∞ (? ) function. ?). First note Next, we wish to show that W0 = L2 (E ), where E = supp(ψ ?(ξ ) has maximal ?(ξ )} forms an orthonormal basis for W0 , ψ that since {e?inξ ψ ? is contained in the ? ∈ W0 , then the support of f support in the sense that if f ?. Thus, W0 ? L2 (E ). support of ψ Let g (ξ ) be a compactly supported simple function, whose support F is contained in E . De?ne En = {ξ :

1 n?1

?(ξ ) > ≥ψ

1 }, n

and de?ne Fn = F ∩ En .

Since g is a simple function, it is uniformly bounded by some constant M . Let > 0 be given. Choose an N such that λ(∪n>N Fn ) <

g ? χ∪n≤N Fn . ψ M

, and de?ne h0 to be

?(ξ ) = g (ξ ) on ∪n≤N Fn , so that h0 ψ ? ? g < . Since W0 Then, h0 (ξ )ψ

53 is closed, g ∈ W0 ; furthermore, the set of all such g ’s is dense in L2 (E ), whence ?(ξ )} forms an orthonormal basis L2 (E ) ? W0 . Note that this implies that {e?ilξ ψ for L2 (E ). Hence, by Lemma 2.2, E is 2π translation congruent to [0, 2π ). Furthermore, Wj ⊥ Wk , so that 2j E ∩ E is a set of measure zero, and since ⊕j ∈ Wj is dense in L2 (? ), ∪j 2j E = ? , whence the dilates of E partition ? . It follows by Theorem 2.1 that E is a wavelet set, and ψ is a MSF wavelet. Corollary 5.1. The equivalence class M∞ can be characterized in the following two ways: (1) M∞ = ∩∞ n=0 Ln , (2) M∞ is precisely the collection of all MSF wavelets. Proof. By Theorem 5.2, V0 is invariant under Gn for all n if and only if V0 is invariant under translations by all reals. This is equivalent to ψ being an MSF wavelet. 5.3 A Characterization of Mn Suppose ψ is a wavelet that is in L1 . If T m f ∈ W0 , then by taking the Fourier 2 ? ∈ W0 , and vice versa, so W0 is invariant under translations by Transform, e?i 2 · f half integers if and only if W0 is invariant under multiplication by e?i 2 · . We shall proceed with the analysis in the frequency domain. ?(ξ ) : h ∈ L2 ([0, 2π ))}. Suppose that ξ ∈ E = We can describe W0 by {h(ξ )ψ ?). If W0 is invariant under multiplication by e?i 2 ξ , then for m = 1, supp(ψ

1 ?(ξ ) = g (ξ )ψ ?(ξ ) e?i 2 ξ h(ξ )ψ m m m

54 for some g ∈ L2 ([0, 2π )). Let k be an odd integer. Then, ?(ξ + 2kπ ) = g (ξ + 2kπ ) ψ ?(ξ + 2kπ ) g (ξ ) ψ

1 ?(ξ + 2kπ ) = e?i( 2 )(ξ +2kπ) h(ξ + 2kπ ) ψ 1 ?(ξ + 2kπ ) = ?e?i 2 ξ h(ξ ) ψ

?(ξ + 2kπ ) = ?g (ξ ) ψ ? cannot have both ξ and ξ + 2kπ in its support. We This calculation shows that ψ have proven the next theorem. ?) is not self Theorem 5.3. Let ψ be a wavelet. Then ψ ∈ L1 only if E = supp(ψ similar with respect to any odd multiple of 2π . ?) = ? , then ψ ∈ M0 . Corollary 5.2. If supp(ψ Corollary 5.3. If ψ is compactly supported, then ψ ∈ M0 . Theorem 5.3 extends to Ln . ? is not Theorem 5.4. Let ψ be a wavelet. Then ψ ∈ Ln only if the support of ψ self similar with respect to any odd multiple of 2j π for all j = 1, 2, . . . , n. Proof. Let ψ ∈ Ln . Hence,

1 ?(ξ ) = g (ξ )ψ ?(ξ ) e?i 2n ξ h(ξ )ψ

for some g ∈ L2 ([0, 2π )). Let 1 ≤ j ≤ n, and let k be an odd integer. Then, by a similar computation, ?(ξ + 2j kπ ) = g (ξ + 2j kπ ) ψ ?(ξ + 2kπ ) g (ξ ) ψ = e?i( 2n )(ξ +2

1 j kπ )

?(ξ + 2j kπ ) h(ξ + 2j kπ ) ψ

k 1 ?(ξ + 2j kπ ) = e?i 2n?j π e?i 2n ξ h(ξ ) ψ k ?(ξ + 2j kπ ) = e?i 2n?j π g (ξ ) ψ

as above.

55 The converse of Theorem 5.4 also holds. ?) be such that it is not self Theorem 5.5. Let ψ be a wavelet and let E = supp(ψ similar with respect to any odd multiple of 2j π for j = 1, 2, . . . , n. Then ψ ∈ Ln . Proof. It su?ces to show that ?(ξ ) = g (ξ )ψ ?(ξ ) e?i 2n ξ ψ for some g ∈ L2 ([0, 2π )). ? ? [0, 2π ) be de?ned by ξ ∈ F ? if there exists an integer kξ such that Let F ? , choose an integer mξ such that ξ + 2mξ π ∈ E ξ + 2kξ π ∈ E . For each ξ ∈ F in the following manner: 1) if ξ ∈ E , choose mξ = 0; if not, choose mξ = min{k > 0 : ξ + 2kπ ∈ E }, else choose mξ = max{k < 0 : ξ + 2kπ ∈ E . Let F = {ξ + 2mξ π : ξ ∈ F }. Note that F ? E ; furthermore, F is 2π translation ? ? [0, 2π ). Hence, congruent to F e?i 2n ξ χF (ξ ) = g (ξ ) on F , where g (ξ ) ∈ L2 (F ) and is 2π periodic. Thus, for ξ ∈ F ,

1 ?(ξ ) = g (ξ )ψ ?(ξ ). e?i 2n ξ ψ 1 1

Now, for almost any ξ ∈ E ? F , there exists a ξ ∈ F and an integer lξ such that ξ ? ξ = 2lξ π . Moreover, by hypothesis, lξ is an even multiple of 2n , since E is not self similar with respect to any odd multiple of 2j π . Hence, since e?i 2n ξ is 2n π periodic, we have that for ξ ∈ E ? F , ?(ξ ) = e?i 2n (ξ +2lξ π) ψ ?(ξ + 2lξπ ) e?i 2n ξ ψ ?(ξ + 2lξπ ) = e?i 2n ξ ψ ?(ξ + 2lξπ ) = g (ξ )ψ ?(ξ ). = g (ξ )ψ This completes the proof.

1 1 1 1

56 We have established the following characterization of the Mn ’s. Corollary 5.4. Mn is the collection of all wavelets ψ such that the support of ? is not self similar with respect to any odd multiples of 2k π , for k = 1, 2, . . . , n ψ but is self similar with respect to some odd multiple of 2n+1 π . 5.4 Examples In this section, we will present wavelets that are in the ?rst four equivalence classes, with the last being in M0 but it is not an MRA wavelet, and hence cannot be compactly supported. This example will show that this new equivalence relation is skewed with respect to the equivalence relation given by the wavelet multiplicity function. The tool used to generate these wavelets is operator interpolation; in particular, these wavelets will be interpolated between two MSF wavelets. Recall that, from chapter 3, if W1 and W2 are two wavelet sets such that σ is involutive, then h1 ψW1 + h2 ψW2 is the Fourier transform of a wavelet if h1 and h2 are measurable, essentially bounded, 2-dilation periodic functions, and the matrix ? ? (5.1)

h2 ? ? h1 ? ? ?1 ?1 h2 ? σ h1 ? σ

is unitary almost everywhere. Note that the interpolated wavelet ψ has the prop?) ? W1 ∪ W2 . Further, note that since σ on W1 is given by erty that supp(ψ translations by integral multiples of 2π , σ completely describes the self similarity of W1 ∪ W2 with respect to 2π . In the following examples, σ will always be involutive. Example 5.1. We shall now present an example of a wavelet in M1 , which by ?) being not self similar with respect corollary 5.4 is equivalent to E = supp(ψ

57 to any odd multiples of 2π , but does have self similarity with respect to some multiple of 4π . Consider the following two wavelet sets:

W1 = [?

4π 6π 24π 32π 8π 4π ,? ) ∪ [ , ) ∪[ , ) 7 7 7 7 7 7 2π 3π 24π 30π 8π 4π W2 = [? , ? ) ∪ [ , ) ∪ [ , ) 7 7 7 7 7 7 31π 32π 60π 62π ∪[ , )∪[ , ) 7 7 7 7 ? ? ? ?ξ, ξ ∈ W1 ∩ W2 ? ? ? ? σ (ξ ) = ξ ? 4π, ξ ∈ [ 30π , 31π ) 7 7 ? ? ? ? ? ? ?ξ + 8π, ξ ∈ [ 4π , 6π ) 7 7

A routine calculation shows:

π 31π π 3π π 3π π 6π This σ is involutive. Indeed, since σ ([ 30 , 7 )) = [ 27 , 7 ) and [ 27 , 7 ) = 2[ 47 , 7 ), 7 π 31π , 7 ), σ 2 (ξ ) = σ (ξ ? 4π ) = 1 σ (2(ξ ? 4π )) = 1 (2ξ ? 8π + 8π ) = ξ . A for ξ ∈ [ 30 7 2 2 π 6π similar computation shows that σ 2 is the identity on [ 47 , 7 ).

Construct h1 and h2 as follows: 1 h1 = χW1 ∩W2 + √ χ[ 4π , 6π )∪[ 30π , 31π ) 7 7 2 7 7 1 h2 = √ χ[ 2π , 3π ) ? χ[ 60π , 62π ) 7 7 7 7 2 We need to check the condition of the matrix in equation 3.2. It su?ces to verify that the matrix is unitary on W1 . Clearly, on W1 ∩ W2 the

π 31π matrix is unitary, indeed it is the identity there. On [ 30 , 7 ), h1 = h2 ? σ ?1 = 7 1 √ . 2

π 31π 1 Furthermore, if ξ ∈ [ 30 , 7 ), h2 (ξ ) = h2 (2ξ ) = ? √ . Finally, σ ?1 (ξ ) = ξ ? 4π ∈ 7 2 pi 3π π [ 27 , 7)= 1 [ 4π , 67 , hence h1 ? σ ?1 (ξ ) = 2 7 ? 1 √ . 2 1 √ 2 1 ?√ 2

Thus, the matrix is simply: ? ? ?

? √2 ?

1 √ 2

1

58 which is unitary as required. A similar computation shows that the matrix is also

π 6π unitary on [ 47 , 7 ).

Example 5.2. Here we give an example of a wavelet in M2 , which by corol?) being not self similar with respect to any lary 5.4 is equivalent to E = supp(ψ odd multiples of 2π or 4π , but does have self similarity with respect to some multiple of 8π . Consider the following two wavelet sets:

W1 = [?8π, ?

16π 14π 8π 112π ) ∪ [? , ?π ) ∪ [? ,? ) 15 15 15 15 8π 14π 16π 112π ∪[ , ) ∪ [π, )∪[ , 8π ) 15 15 15 15 14π π 112π W2 = [?8π, ? ) ∪ [? ,? ) 15 15 2 8π 14π 225π 224π ∪[ , )∪[ , 8π ) ∪ [ , 15π ) 15 15 30 15 ? ? ? ? ξ, ξ ∈ W1 ∩ W2 ? ? ? ? σ (ξ ) = ξ ? 8π, ξ ∈ [ 112π , 225π ) 15 30 ? ? ? ? ? ? ?ξ + 16π, ξ ∈ [? 16π , ?π ) 15

A routine calculation shows:

As in example 5.1, σ is involutive, and de?ne h1 and h2 analogously: 1 h1 = χW1 ∩W2 + √ χ[? 16π ,?π)∪[ 112π , 225π ) 15 15 30 2 1 h2 = √ χ[? 8π ,? π ) ? χ[ 224π ,15π) 15 2 15 2 These functions satisfy 3.2. Example 5.3. We shall now present an example of a wavelet in M3 .

59

32π 30π 16π 480π ) ∪ [? , ?π ) ∪ [? ,? ) 31 31 31 31 16π 30π 32π 480π ∪[ , ) ∪ [π, )∪[ , 16π ) 31 31 31 31 30π π 480π W2 = [?16π, ? ) ∪ [? ,? ) 31 31 2 16π 30π 32π 31π 960π ∪[ , ) ∪ [π, )∪[ , 16π ) ∪ [ , 31π ) 31 31 31 2 31 W1 = [?16π, ? Then, σ is given by: ? ? ? ? ξ, ξ ∈ W1 ∩ W2 ? ? ? ? σ (ξ ) = ξ ? 16π, ξ ∈ [ 480π , 31π ) 31 2 ? ? ? ? ? ? ?ξ + 32π, ξ ∈ [? 32π , ?π ) 31 Again, as in example 5.1, σ is involutive; analogously de?ne h1 and h2 as: 1 h1 = χW1 ∩W2 + √ χ[? 32π ,?π)∪[ 480π , 31π ) 31 31 2 2 1 h2 = √ χ[? 16π ,? π ) ? χ[ 960π ,31π) 31 2 31 2 Example 5.4. In this example we shall construct a non-MRA wavelet in M0 . In fact, we have already constructed an example of such a wavelet. The interpolated wavelet in example 3.2 is just such a wavelet. Recall that the σ showed a self-similarity with respect to 2π so that it is in M0 ; additionally, we noted in example 4.2 that it is a non-MRA wavelet.

Appendix

A

Decomposition of Projection Valued Measures

The decomposition theorem describes how a projection valued measure can be written in terms of canonical projection valued measures. This description is unique up to unitary equivalence. Throughout this appendix, (S, B) will denote a measurable space, and we will assume that for any measure ν on S , L2 (ν ) is separable. This thesis assumes previous knowledge of what a projection valued measure is. The following de?nitions will be needed. De?nition A.1. Two projection valued measures, p and p , with Hilbert spaces H and H , are unitarily equivalent if there exists a unitary operator U : H → H such that for all E ∈ B, UpE = pE U . De?nition A.2. A projection valued measure p is cyclic if there exists a vector x ∈ H such that the collection {pE x : E ∈ B} has dense span in H . If y ∈ H is any vector, then denote by C (y ) the cyclic subspace generated by y , i.e. C (y ) = span{pE y : E ∈ B}. A subspace K ? H is cyclic if there exists a vector x ∈ H such that C (x) = K . A cyclic subspace M is maximal if for any cyclic subspace K such that M ? K implies that M = K . The next lemma will be useful later.

61 Lemma A.1. If p and p are unitarily equivalent, and p is cyclic, then p is also cyclic. Proof. If p is cyclic, then there exists a vector x ∈ H such that {pE x} has dense span in H . Since p and p are unitarily equivalent, there exists a unitary operator U : H → H ; it follows that {UpE x} has dense span in H . Finally, since U is an intertwining operator, UpE x = pE Ux, this Ux is a cyclic vector for p . The decomposition theorem can be used as a way of determining when two unitary representations (of the same group) are unitarily equivalent. In particular, as described in chapter 4, the multiplicity function is used to determine when two wavelets are core equivalent. This appendix provides a proof of the decomposition theorem, as well as how it relates to unitary representations. For notational purposes, if H and H have unitarily equivalent projection valued measures, we shall write H H . In some cases, one or both of the

projection valued measures will be assumed to be canonical, if no other measure is speci?cally mentioned. A.1 Canonical Projection Valued Measures

We begin with the de?nition of a canonical projection valued measure. Let K be a separable Hilbert space. De?nition A.3. Let ν be a measure on S , and let H = L2 (ν, K ). Then for E ∈ B, de?ne a projection valued measure p by pE = MχE . This is the canonical projection valued measure. Proposition A.1. If ν is a ?nite measure, then the canonical projection valued measure on L2 (ν ) is cyclic.

62 Proof. Since ν is ?nite, χS ∈ L2 (ν ); this is a cyclic vector for the canonical projection valued measure. Indeed, if f ∈ L2 (ν ) is a simple function, so that f =

n i=1 ci χEi ,

then f =

n i=1 ci pEn χS .

It follows that the span of {pE χS : E ∈ B}

is dense in L2 (ν ), as required. The next theorem provides a description of operators that “commute” with canonical projection valued measures. It will be used to prove Theorem 3.2, as well as the main theorem of the appendix. Theorem A.1. Let ν be a ?nite measure on a Borel space (S, B), and let K and K be Hilbert spaces. Suppose A is a bounded operator from L2 (ν, K ) into L2 (ν, K ) that satis?es ApE = pE A for every E ∈ B, where p denotes the canonical projection valued measure on L2 (ν, K ) and p denotes the canonical projection valued measure on L2 (ν, K ). Then, for ν almost all x ∈ S there exists a bounded operator Ax from K into K such that for all f ∈ L2 (ν, K ) we have [A(f )](x) = Ax (f (x)) for almost all x. That is, A is a multiplication operator from L2 (ν, K ) into L2 (ν, K ). Further, if A is unitary, then for ν almost all x, Ax is unitary, whence K and K have the same dimension. Proof. For the purpose of this proof, let V be a countable dense collection of functions in L2 (ν, K ); more speci?cally, let the collection {kn } be a countable dense set in K and let the constant functions fn (x) = kn be in (but not all of) V . Let V be the complex rational span of V . It follows that for all x ∈ S , {f (x) : f ∈ V} is countably dense in K . Since A intertwines the canonical projection valued measures, it follows that for f ∈ L2 (ν, K ) supported on E , Af is also supported on E . Hence, since A is bounded, there exists an M such that [A(f )](x) 2 dν ≤ M 2

E E

f (x) 2 dν.

(A.1)

63 Claim. For f ∈ L2 (ν, K ), [A(f )](x) ≤ M f (x) for ν almost all x. Suppose there exists a set F of ν non-zero measure such that the inequality in the claim does not hold. Then [A(f )](x) 2 dν >

F F

M 2 f (x) 2 dν,

which contradicts the inequality in equation A.1. For each f ∈ V , there exists a set Ef of measure zero such that the inequality in the above claim holds. Since V is countable, E = ∪f ∈V Ef is a set of measure zero for which, if x ∈ / E , the inequality holds for all f uniformly. Thus, for x not in E , f (x) → [A(f )](x) is a bounded, complex rational linear transformation from a dense set in K into K , denote it by Ax . This can be extended to all of K , and then extended to a complex linear transformation. It follows that for f ∈ L2 (ν, K ), [A(f )](x) = Ax (f (x)). Now suppose that A is unitary. Let {kn } be a countable dense collection in K and as above let gn (x) = kn ; include in V the preimage of {gn } under A (this does not change the arguments above). Thus, we now have that V (x) is dense in K while AV (x) is dense in K . Hence, Ax is onto for almost all x. Finally, by an argument similar to the claim above, [Ax ]f (x) = f (x) for almost all x, whence Ax is unitary almost everywhere. We have the following partial converse to Theorem A.1. Proposition A.2. Suppose that ? and ? are two ?nite measures on S , with canonical projection valued measures p and p . Then any multiplication operator from L2 (?, K ) → L2 (? , K ) intertwines (not necessarily unitarily). In particular, p and p are unitarily equivalent if and only if ? ≡ ? . Proof. Suppose that B : L2 (?, K ) → L2 (? , K ) is given by Bf (x) = g (x)f (x). It is clear that pE gf = χE gf = gχE f = gpE f . Now, suppose that ? and ? are

64 equivalent. Then there exists a Radon-Nikodym derivative, call it h, with the property that h is non-zero almost everywhere. It follows that U : L2 (?, K ) → √ L2 (? , K ) given by Uf = hf is a unitary operator which, by above, intertwines the canonical projection valued measures. Conversely, if p and p are equivalent, then there exists an intertwining operator. Suppose that ?(E ) = 0. Then pE χS = 0, whence 0 = UpE χS = pE UχS . It follows that UχS is supported almost everywhere ? since U is unitary, χS is a cyclic vector for p and UχS is cyclic for p . Hence, pE UχS is supported on a set of ? measure 0. This argument is symmetric in ? and ? . Recall the statement of the Theorem 3.2. Theorem A.2. Let {D, T } denote the commutant of the unitary system {D n T l : n, l ∈ }. Then {D, T } = {Mh : h ∈ L∞ (? ), h(x) = h(2x) a.e.}. We ?rst need the next proposition. Proposition A.3. The von Neumann algebra A generated by {D, T } contains all multiplication operators Mh for h ∈ L∞ (? ). Proof. A routine exercise from Functional Analysis shows that a sequence of multiplication operators Mhn converges to Mh in the weak operator topology if and only if the hn ’s converge to h pointwise and boundedly. This fact will be used here in several cases. From the computation in 1.1, D ?j T n D j = T 2 n , so A contains all operators of the form Me?i2j nξ , for all integers j . It follows, by virtue of these dyadic rationals being dense in ? and the remark above, that A contains multiplication operators of all real numbers, i.e. Me?iαξ for all α ∈ ? . Standard Fourier Analysis shows that given any interval I , there exists a subcollection of the above exponentials that forms an orthonormal basis for L2 (I ).

j

65 Let f be a twice di?erentiable function with compact support (in the interval I = [?N, N ]). Periodize f with respect to the interval I . Denote the periodized ?; f ? may be uniformly approximated by sums of exponentials. Thus, function by f Mf? ∈ A. Next, periodize f with respect to the interval [?N ? n, N + n], denote ? it by f ∈ A. Note that f is the bounded pointwise limit of ? n . We have that Mf n ? f n , whence Mf ∈ A. Let E be a ?nite interval. Since χE may be approximated by twice di?erentiable functions, MχE ∈ A. A standard Monotone Class argument now shows that any Lebesgue measurable set E has the property that MχE ∈ A. Hence, Mg ∈ A for any simple function g . Finally, Mh ∈ A for any bounded function h by writing h as the pointwise limit of simple functions. We are now in a position to prove Theorem 3.2. Proof. We have already demonstrated that {Mh : h ∈ L∞ (? ), h(x) = h(2x) a.e.} ? {D, T } . We shall now prove the reverse containment. Suppose that B ∈ {D, T } . Then B commutes with every operator in the von Neumann algebra A generated by {D, T }. Proposition A.3 shows that A contains operators of the form MχE for any Borel set E . Hence, B commutes with the canonical projection valued measure, whence, by Theorem A.1, B = Mh for some h ∈ L∞ (? ). It follows from the calculation in chapter 3 that it is necessary for h to be 2-dilation periodic. This completes the proof. We conclude with a lemma which will be useful later. Lemma A.2. Let p be the canonical projection valued measure on L2 (?). Then M ? L2 (?) is an invariant subspace if and only if M = L2 (?|E ) for some set E . Proof. Clearly, if M = L2 (?|E ), M is invariant under p. Conversely, the proof of Theorem 5.2 shows if M is invariant under p, so that M is invariant under

66 multiplication by L∞ (?), then M = L2 (E ) for some set E and some measure. It follows that the measure must be ?|E . A.2 Cyclic Subspaces

This section provides a description of cyclic subspaces, in particular, maximal cyclic subspaces. Additionally, much of the work here is to show the relatively simple sounding statement that every vector is an element of a maximal cyclic subspace. This next lemma describes cyclic subspaces in terms of canonical projection valued measures. Lemma A.3. An invariant subspace K of H is cyclic if and only if there exists a ?nite measure ? on S and a unitary map U : L2 (?) → K that intertwines the canonical projection valued measure with p. That is, U (χE f ) = pE (U (f )) for every Borel set E ? S . Indeed, if x is a cyclic vector of norm 1, then the measure de?ned by ?(E ) = pE x, x satis?es the above condition. If. Proposition A.1 shows that L2 (?) is cyclic under the canonical projection valued measure with cyclic vector χS . Since U intertwines the PVM’s, it follows from Lemma A.1 that K is cyclic for p, with cyclic vector U (χS ). Only If. Suppose p is cyclic, with a cyclic vector x of norm 1. Then ?(E ) = pE x, x is a probability measure. De?ne U : L2 (?) → K by U (χE ) = pE x. This operator is surjective since the functions χE have dense span in L2 (?), and x is a cyclic vector. Furthermore, U is injective since if pE x = 0, then ?(E ) = pE x, x is also 0, whence χE is the zero function. Additionally, U is norm-preserving since χE

2

=

χE d? = ?(E ) = pE x, x = pE x 2 ,

67 so that U is a unitary operator. Finally, by de?nition, U intertwines p with the canonical projection valued measure, since U (χE χS ) = pE ∩S x = pE pS x = pE U (χS ). The measure in Lemma A.3 is absolutely continuous with respect to p in the sense that if pE = 0, then ?(E ) = 0. Indeed, Proposition A.2 shows that if ? is as in Lemma A.3, then the measure ν (E ) = pE x, x is equivalent to ?, where x is a cyclic vector. It is easy to see that ν is absolutely continuous with respect to p, from which it follows that ? p.

Lemma A.4. Let K1 and K2 be cyclic subspaces of H , with cyclic vectors x1 and x2 respectively, and assume that K1 ? K2 . Let ?1 and ?2 be the measures on S as guaranteed by Lemma A.3. Then: (1) ?1 is absolutely continuous with respect to ?2 , (2) the subspace K de?ned by K2 = K1 ⊕ K is cyclic. Indeed, there exists a set F ? S such that pF x2 is a cyclic vector for K1 and pF c x2 is a cyclic vector for K , (3) K1 = K2 if and only if ?1 ≡ ?2 . Proof. For part (1), suppose that ?2 (E ) = pE x2 , x2 = 0. We need to show that ?1 (E ) = pE x1 , x1 = 0. Claim. If ?2 (E ) = 0, then for any y ∈ K2 , pE x2 , y = 0. Since x2 is a cyclic vector for K2 , it su?ces to show that pE x2 , pF x2 = 0 for any set F . But this inner product is the same as pE ∩F x2 , x2 = ?2 (E ∩ F ) = 0, as required.

68 Let > 0 be given; we shall show that |?1 (E )| < . Since x1 ∈ K2 , there

n i=1 ci pEi x2

exist constants ci and sets Ei such that x1 ? |?(E )| = pE x1 , x1

n n

< . Hence,

= pE (x1 ?

i=1 n

ci pEi x2 +

i=1

ci pEi x2 ), x1

n

= pE (x1 ?

i=1 n

ci pEi x2 ), x1 + pE

i=1 n

ci pEi x2 , x1 ci pEi x1

i=1

≤ x1 ?

i=1

ci pEi x2

x1 + pE x2 ,

< as required. To establish part (2), by the Lebesgue decomposition theorem, there exist measures ν0 and ν1 such that ?2 = ν0 + ν1 , ν1 and ?1 are equivalent, and ?1 , ν0 are mutually singular. Thus, there exists a set F such that ν1 (F ) = ν1 (S ) and ν0 (F ) = 0. It follows that ?1 |F is still equivalent to ν1 . Claim. The vector pF x2 is a cyclic vector for K1 . Let f ∈ L2 (?2 ) be the cyclic vector such that Uf = x2 . Then the claim is equivalent to χF f being cyclic for L2 (?1 ). But ?2 |F = ν1 |F + ν0 |F ≡ ν1 ≡ ?1 |F , whence it su?ces to show that χF f is cyclic for L2 (?2 |F ). This is clear, however, since f is cyclic for L2 (?2 ). Now to establish part (3), suppose that K1 = K2 . Then x1 and x2 are cyclic vectors for both K1 and K2 , whence the calculation for part (1) holds with ?1 and ?2 switched. Conversely, if K1 = K2 , then by the proof of part (2)), we must have that pF c = 0, so that ?1 (F c ) = 0 but ?2 (F c ) = 0. As a partial converse to item (2) above, if K1 and K2 are perpendicular cyclic subspaces, with x1 and x2 cyclic vectors, such that ?1 and ?2 are mutually

69 singular, then K = K1 ⊕ K2 is also cyclic with cyclic vector x = x1 + x2 . To prove the converse, let ? = ?1 + ?2 , and let f1 and f2 correspond to x1 and x2 , respectively. It follows that L2 (?) = L2 (?1 ) ⊕ L2 (?2 ). We need to show that f1 + f2 is cyclic for L2 (?). Since ?1 and ?2 are mutually singular, there exists a set F such that χF f = f1 and χF c f = f2 . It follows that f is cyclic, since χF f and χF c f are cyclic for L2 (?1 ) and L2 (?2 ), respectively. This fact will be used in the following lemma. Proposition A.4. A cyclic subspace M is maximal if and only if the measure ? is equivalent to p, in the sense that their sets of measure 0 coincide. If. Suppose that M is not maximal. Then there exists a cyclic subspace K that properly contains M . Let ? and ? be the measures given by Lemma A.3. Lemma A.4 shows that ? is absolutely continuous with respect to ? , but ? is not absolutely continuous with respect to ?. Thus, ? cannot be equivalent to p. Only If. Suppose that ? is not equivalent to p. Then there exists a set E such that ?(E ) = 0 but pE is not 0. Thus, there exists a vector non-zero y such that y is in the range of pE , furthermore, we may choose y to be perpendicular to M . Hence, C (y ) ⊥ M ; additionally, if ? is the measure associated to C (y ), then ? and ? are mutually singular. Indeed, pE y = pS y , whence ? (E ) = ? (S ). Therefore, M ⊕ C (y ) is a cyclic subspace which properly contains M . Lemma A.5. If {Kn } is a sequence of cyclic subspaces, with Kn ? Kn+1 for all n, then K = ∪Kn is a cyclic subspace. Proof. De?ne a sequence of subspaces Mn by Kn+1 = Kn ⊕ Mn . Then K = K1 ⊕∞ n=1 Mn . Let the sequence of probability measures ?n correspond to the subspaces Kn and the sequence of unitary operators Un : Kn → L2 (?n ) as in Lemma A.3. Since

70 Kn ? Kn+1 , by Lemma A.4, ?n ?n+1 , and for each n, there exists a set En such

that χEn χS = χEn is a cyclic vector for L2 (?n ). Furthermore, we have En ? En+1 (up to sets of measure 0), so that Un+1 Mn = L2 (?n+1 |Fn ), where Fn = En+1 \ En . De?ne the measure ? by ? = ?1 +

∞ i=1

?i |Fi .

Claim. The subspace K is unitarily equivalent to L2 (?). Furthermore, χS is a cyclic vector, whence K is cyclic. We have that χFn L2 (?) = Un+1 Mn , whence L2 (?) = L2 (?1 ) from which it follows that K

2 ⊕∞ i=1 L (?n |Fn ),

L2 (?); the unitary operator is gotten by summing

the Un ’s restricted to the Mn ’s. Clearly, χS is a cyclic vector for L2 (?). We are ?nally able to prove the following statement. Proposition A.5. If x ∈ H , there exists a maximal cyclic subspace M ? H containing x. Proof. Let M be the collection of all cyclic subspaces containing x; partially order this collection by set inclusion. This collection is non-empty since clearly C (x) ∈ M. Zorn’s lemma dictates that if every chain in M has an upper bound, then M has a maximal element. As such, let {Mα } be a chain, then M = ∪α Mα is an upper bound. Indeed, by Lemma A.5, it su?ces to show that M = ∪n Mn for some increasing sequence of cyclic subspaces containing x. Let zn be a countable collection which is dense in ∪α Mα . Then, there exists an Mα such that z1 ∈ Mα ; set M1 = Mα . Let j be the smallest integer such that zj ∈ / M1 ; there exists a β > α such that zj ∈ Mβ ; set M2 = Mβ . Note that M1 ? M2 . Continue this process until all zj ’s are contained in the union. This completes the proof.

71 A.3 Proof of the PVM Decomposition Theorem

We will now prove the decomposition theorem. The theorem uses much of the material developed in the last section. Recall that the statement is as follows. Theorem A.3. There exists a ?nite measure ν and a measurable function m : S → {0, 1, 2, 3, . . . , ∞} such that (1) there exists a unitary operator U from H onto

2 ⊕∞ j =1 L (Ej , ν |Ej , j

) ⊕ L2 (E∞ , ν |E∞ , l2 ( ))

where Ej = m?1 (j ), and (2) U intertwines p with the canonical projection valued measure. Moreover, if ν and m are another ?nite measure and function, respectively, satisfying properties 1. and 2. above, then ν ≡ ν and m = m a.e. ν . Proof. The ?rst step is to decompose H into cyclic subspaces, in such a way so that the associated measures are decreasingly absolutely continuous. Let {zi } be an orthonormal basis. Let H1 be the maximal cyclic subspace containing z1 . Let ?1 be the measure associated to H1 by Lemma A.3; note that by Lemma A.4, ?1 ≡ p. If H1 contains all of the zi ’s, then we are done. If not, let j be the smallest integer such that zj is not in H1 and de?ne a new projection

⊥ valued measure by setting p2 = p|H1 ⊥ . Let y2 = P H1 xj and let H2 be the maximal

cyclic subspace containing y2 generated by p2 . Note that H1 ⊥ H2 by de?nition. It follows that xj ∈ H1 ⊕ H2 . Now, let ?2 be the measure associated to H2 ; ?2 ≡ p2 , and since p2 p1 , it follows that ?2 ?1 . If all of the zi ’s are in

H1 ⊕ H2 , then we are done. If not, take the ?rst zj that is not in H1 ⊕ H2 , and generate a maximal cyclic subspace with respect to p3 = p|(H1 ⊕H2 )⊥ . Continue the process until all of the zi ’s are exhausted. This can be done since the collection

72 {zi } is countable. Finally, the direct sum of the Hj ’s contain all of the {zi }’s, so the direct sum is H .

2 We have now that there exists a unitary operator U : H → ⊕∞ i=1 L (?i ) such

that U intertwines p with q = ⊕qi , where the qi ’s are the canonical projection valued measures on L2 (?i ). Furthermore, the measures have the property that ?i+1 ?i . Decompose the measures using the Lebesgue decomposition theorem as follows: let ?i+1 = νi + σi , where σi ≡ ?i and νi ⊥ ?i . De?ne the measure ν0 =

∞ νi i=1 ki 2i ,

where ki normalizes νi , or is 1 if νi is the 0 measure.

Claim. The measures νi are mutually singular, but are all absolutely continuous with respect to ?1 . All of the measures ?i are absolutely continuous with respect to ?1 , and each νi is absolutely continuous with respect to ?i+1 . Since νi+1 is mutually singular to σi+1 , and σi+1 is absolutely continuous with respect to ?i+1 , it follows that νi and νi+1 are mutually singular. It follows by de?nition that νi that νi ?i+1 ≡ σi+1 ⊥ νi+1 . However, we also have

?j ≡ σj ⊥ νj for any j > i, whence the νi ’s are mutually singular.

The claim gives us that ν0 is a ?nite measure on S that is absolutely continuous with respect to ?1 and p. Additionally, there exist sets Ei such that ν0 |Ei ≡ νi . De?ne a measure ν∞ by ?1 ≡ ν0 + ν∞ ; analogously de?ne the set E∞ so that ?1 |E∞ ≡ ν∞ . Let ν ≡ ?1 = ν0 + ν∞ , and de?ne the multiplicity function m by m?1 (j ) = Ej . We need to show that these satisfy the conditions of the theorem. By de?nition, ?1 ≡ ν1 + σ1 ≡ ν1 + ?2 ≡ ν0 + ν∞ , so we get that ?2 ≡

∞ νi i=2 ki 2i

+ ν∞ . Inductively, ?j =

∞ νi i=j ki 2i

+ ν∞ . By Proposition A.2, there exists

73 an intertwining operator such that H

2 ⊕∞ i=1 L (?i ) 2 ⊕∞ j =1 L ( ∞

νi + ν∞ )

i=j

∞ 2 2 ⊕∞ j =1 ⊕i=j L (νi ) ⊕ L (ν∞ ) 2 2 ⊕∞ i=1 iL (νi ) ⊕ ∞L (ν∞ )

where at each step in this calculation, the equivalence of p to the appropriate canonical projection valued measure is maintained. The last step is to simply note that iL2 (νi ) satis?ed. To show uniqueness, suppose there is another measure ν and another multiplicity function m ; let Fi = (m )?1 (i). Then the canonical projection valued measures are equivalent, since they are both equivalent to p; it follows that ν and ν are equivalent since they are both equivalent to p. Furthermore, suppose that there existed integers j and k such that Ej ∩ Fk had non-zero measure. By assumption, there exists an intertwining operator U such that

2 2 ⊕∞ i=1 iL (νi ) ⊕ ∞L (ν∞ ) 2 2 ⊕∞ i=1 iL (νi ) ⊕ ∞L (ν∞ ).

L2 (νi ,

i

). This shows that conditions (1) and (2) above are

By theorem A.1, U is a pointwise operator; as such, if f is supported on Ej ∩ Fk , then so is Uf . But note that f takes values in

j

, whereas Uf takes values in

j

k

.

Thus, we have for almost all x ∈ Ej ∩ Fk , Ux maps only happen if j = k .

to

k

unitarily, which can

Corollary A.1. The measure and multiplicity function from the above theorem completely describe projection valued measures up to unitary equivalence. Proof. Let p and p are projection valued measure on the same measurable space,

74 with measures ν and ν , multiplicity functions m and m , and Hilbert spaces H and H , respectively. Suppose p and p are unitarily equivalent. Then p and p are equivalent measures, whence ν and ν are equivalent measures. It follows that there exists a unitary operator that intertwines the canonical projection valued measures. Thus, the operator is a multiplication operator, and, furthermore, the dimensions match up. Hence, m = m . Suppose, conversely, that ν ≡ ν and m = m , we wish to show that the

2 canonical projection valued measures on ⊕∞ j =1 L (Ej , ν |Ej , 2 and ⊕∞ j =1 L (Fj , ν |Fj , j j

)⊕L2 (E∞ , ν |E∞ , l2 ( ))

) ⊕ L2 (F∞ , ν |E∞ , l2 ( )) are unitarily equivalent. Since

ν ≡ ν and m = m , ν |Ej ≡ ν |Fj , whence, by Proposition A.2, there exists an intertwining operator between L2 (Ej , ν |Ej ,

j

) and L2 (Fj , ν |Ej ,

j

). Then direct

summing these individual operators yields an intertwining operator for the direct sums, as required. The following fact was used in chapter 4. Proposition A.6. The multiplicity function is additive in the following sense: if p and p are projection valued measures, then q = p ⊕ p has measure ν ≡ ? + ? and multiplicity n = m + m . Proof. By Theorem A.3, we have that q is unitarily equivalent to the canonical projection valued measure on

2 ⊕∞ j =1 L (Ej , ?|Ej , j

) ⊕ L2 (E∞ , ?|E∞ , l2 ( ))

2 ⊕∞ j =1 L (Fj , ? |Fj , j

) ⊕ L2 (F∞ , ? |E∞ , l2 ( )),

which is not the decomposition desired. De?ne a partition by Jj,k = Ej ∩ Fk . Then de?ne Jr = ∪j +k=r Jj,k . Note that on Jj,k , n = m + m = j + k , and that on J∞ , n = ∞.

75 Claim. Restrict p and p to the set Jj,k . Then L2 (Jj,k , ?|Jj,k ,

j

) ⊕ L2 (Jj,k , ? |Jj,k ,

j

k

)

L2 (Jj,k , ?|Jj,k + ? |Jj,k ,

k

j +k

).

Let f ∈ L2 (Jj,k , ?|Jj,k , h ∈ L2 (Jj,k , ?|Jj,k + ? |Ej,k

) and g ∈ L2 (Jj,k , ? |Jj,k ,

). De?ne a function

j +k

by ”appending” f (x) to g (x), in order to get a

2

vector of dimension j + k . Using the fact that h(x)

= f (x)

2

+ g (x) 2, this

mapping can be shown to be well-de?ned, injective, surjective, and is isometric by dividing by the constant multiple ?(Ej,k ) + ? (Ej,k ). We only need to show that this unitary intertwines the canonical projection valued measures. We have that for any E ? Ej,k , χE ⊕ χE (f + g ) = χE f + χE g = χE (f + g ), where the left hand side is the canonical projection valued measure on L2 (Ej,k , ?|Ej,k , L2 (Ej,k , ? |Ej,k ,

k j

)⊕

), and the right hand side is the canonical projection valued

j +k

measure on L2 (Ej,k , ?|Ej,k + ? |Ej,k

).

A slight modi?cation would hold if either j , k , or both are in?nity. Thus, we have that

2 ⊕∞ j =1 L (Ej , ?|Ej , j

) ⊕ L2 (E∞ , ?|E∞ , l2 ( ))

2 ⊕∞ j =1 L (Fj , ? |Fj , j

) ⊕ L2 (F∞ , ? |E∞ , l2 ( ))

is equivalent to

∞ 2 ⊕∞ j =1 ⊕k =1 L (Jj,k , ν |Jj,k , j +k

) ⊕ L2 (J∞,k , ν |J∞,k , l2 ( ))

⊕ L2 (Jj,∞ , ν |Jj,∞ , l2 ( )) ⊕ L2 (J∞,∞, ν |J∞,∞ , l2 ( ))

which is in turn equivalent to

2 ⊕∞ r =1 L (Jr , ν |Jr , r

) ⊕ L2 (J∞ , ν |J∞ , l2 ( ))

as required.

76 A.4 Application of the Theorem to Unitary Representations

The goal of this section is to describe the relationship between the PVM theorem and unitary representations of locally compact Abelian groups, in particular, representations of . Since this is the group in question, for simplicity,

in this section we will assume that G is discrete. This has several consequences: ?rst, we have that G, as characters on G, forms an orthonormal basis of L2 (G). Secondly, G is compact, so L∞ (G) ? L2 (G). These facts will greatly simplify the proofs. The next theorem shows that the decomposition theorem completely determines a unitary representation up to unitary equivalence. Recall that for a unitary representation of an abelian group, there exists a projection valued measure, as was shown in chapter 4. Theorem A.4. Let π and π be unitary representations of G, and let ν , ν and m, m be their associated ?nite measures and multiplicity functions respectively. Then, π and π are unitarily equivalent if and only if ν ≡ ν and m = m a.e. ν , i.e. if and only if the projection valued measures are unitarily equivalent. If. If ν ≡ ν and m = m a.e. ν , then there exists a unitary operator U : H → H such that UpE = pE U . It follows that any function integrated against these projection valued measures are intertwined by U . Thus, Uπg = U

G

g (ξ )dp =

G

g (ξ )dp U = πg U,

whence U is an intertwining operator for the representations. Only If. If the representations are unitarily equivalent, then there exists an intertwining operator such that Uπg = U

G

g (ξ )dp =

G

g (ξ )dp U = πg U.

77 We have that U intertwines any function integrated against p and p that is in the closed linear span of G (as functions in L2 (G)). By our assumption that G is discrete, it follows that for any set E , χE is in the closed linear span, so that UpE = U

G

χE dp =

G

χE dp U = pE U,

so that U intertwines p and p . Proposition A.7. Let M ? H be an invariant subspace for the unitary representation π , and denote the subrepresentation by πM . By Theorem 4.2, both π and πM have measures ν and νM and multiplicity functions m and mM , respectively. The multiplicity functions have the property that mM ≤ m, as a result, νM is absolutely continuous with respect to ν . Proof. We have that H

2 ⊕∞ j =1 L (Ej , ν |Ej , j

) ⊕ L2 (E∞ , ν |E∞ , l2 ( )),

and M ? H is unitarily isomorphic to some subspace of this direct sum. If M is invariant under π , then a similar computation as in the proof of Theorem A.4 shows that M is an invariant subspace under p. Thus, if M is the embedding of M in the direct sum above, M ∩ L2 (ν |Ej ) is also invariant. Lemma A.2 shows that this intersection is equivalent to L2 (ν |Ej ∩Fj ). Hence, M

2 ⊕∞ j =1 L (Fj , ν |Fj , j

) ⊕ L2 (F∞ , ν |F∞ , l2 ( )),

where Fj ? Ej . This direct sum is then the desired decomposition, from which the proposition follows. Proposition A.8. A unitary representation π is cyclic if and only if its associated projection valued measure p is cyclic.

78 Only if. If π is cyclic, then there exists a vector x ∈ H such that the linear span of {πg x : g ∈ G} is dense in H . It su?ces to show that, given g ∈ G and there exist constants ci and sets Ei such that

n

> 0,

πg x ?

i=1

ci p E i x

2

< .

Since G is compact, g (ξ ) can be uniformly approximated by a simple function. Hence, for an appropriately chosen simple function,

n n

πg x ?

i=1

ci p E i x =

G

g (ξ )dp ?

G i=1 n

ci χEi dp x ci χEi dp x

i=1 ∞

=

G

g (ξ ) ?

n

≤ g (ξ ) ?

i=1

ci χEi

x

< as required. If. If π is not cyclic, then for any x ∈ H , there exists a y that cannot be approximated by a linear combination of elements of the form πg x. It follows that p cannot be cyclic, since y cannot be approximated by any L∞ function integrated against p, in the sense above. Proposition A.9. A unitary representation is cyclic if and only if the multiplicity function attains only the values 0 and 1, i.e. is the characteristic function of some set. If πM is a cyclic subrepresentation, then m = χE almost everywhere ?,

M ). where E = supp( d? d?

Proof. Proposition A.8 shows that the multiplicity function attains the values 0 and 1, since if π is cyclic, then so is p. Since p is cyclic, Lemma A.3 gives precisely

79 the decomposition in Theorem A.3. Proposition A.7 shows that ?M is absolutely continuous with respect to ?. Furthermore, in the proof of Proposition A.7, we showed that, in this case, M

M that E = supp( d? ). d?

L2 (?|E ) for some set E ; it follows immediately

Proposition A.10. The representation of

?) ? on H = {f ∈ L2 (? ) : supp(f

[0, 2π )} has measure equivalent to Haar (Lebesgue) measure. Proof. We have T = e?iλ dp = Me?i· whence p must be the canonical projection

valued measure. Further, the Fourier transform shows that H is unitarily equivalent to L2 (?, ), where clearly ? is Lebesgue measure. It follows that this is the decomposition as in the theorem. Proposition A.11. A unitary representation π of G has multiplicity function identically 1 and ? is equivalent to Haar measure on G, if and only if π is equivalent to the left regular representation. If. It su?ces to show that the left regular representation has multiplicity identically 1 and measure ? equivalent to Haar measure. Since the left regular representation is cyclic, its multiplicity is 1. Now, for the left regular representation, H = L2 (G), so we wish to show that L2 (G) L2 (G). We have that

{χg : g ∈ G} is an orthonormal basis for L2 (G), so there exists a unitary operator U : L2 (G) → L2 (G) by sending g to itself. Using the Fourier Transform as above, it can be seen that the canonical projection valued measure on G is the projection valued measure for left translations. Only if. We have shown above that the left regular representation has Haar measure and m = 1; both of which determine the representation up to unitary equivalence. It follows that the representation is equivalent to the left regular representation.

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