WANDERING VECTORS FOR UNITARY SYSTEMS
AND ORTHOGONAL WAVELETS
Xingde Dai
David R. Larson
ii
XINGDE DAI AND DAVID LARSON
1991 Mathematics Subject Classication. Primary 46N99,47N40,47N99 Secondary 47D25,47C05,47D15,46B28.
Author addresses:
Department of Mathematics, University of North Carolina at Charlotte,
Charlotte, NC 28223
E-mail address : xdai@uncc.edu
Department of Mathematics, Texas A&M University, College Station, Texas
77843
E-mail address : drl3533@acs.tamu.edu
Contents
Abstract
1
Introduction
3
Chapter 1. The Local Commutant
7
Chapter 2. Structural Theorems
17
Chapter 3. The Wavelet System hD T i
27
Chapter 4. Wavelet Sets
39
Chapter 5. Operator Interpolation of Wavelets
49
Chapter 6. Concluding Remarks
6.1. Unitary Equivalence
6.2. Higher Dimensional Systems
6.3. Multiresolution Analysis
6.4. A Connection With Some Work of Guido Weiss
6.5. Status of Problems
6.6. Examples
6.7. Acknowledgments
67
67
68
69
69
70
71
74
Appendix: Examples of Interpolation Maps
75
References
85
iii
iv
XINGDE DAI AND DAVID LARSON
Abstract
We investigate topological and structural properties of the set W (U ) of all complete wandering vectors for a system U of unitary operators acting on a Hilbert
space. The special case of greatest interest is the system hD T i of dilation (by 2)
and translation (by 1) unitary operators acting on L2(R) for which the complete
wandering vectors are precisely the orthogonal dyadic wavelets. The method we
use is to parameterize W (U ) in terms of a xed vector and the set of all unitary
operators which locally commute with U at : An analysis of the structure of
this local commutant yields new information about W (U ): The commutant of a
unitary system can be abelian and yet the local commutant of it at a complete
wandering vector can contain non-commutative von Neumann algebras as subsets. This is the case for hD T i: The unitary group of a certain non-commutative
von Neumann algebra can be used to parameterize a connected class of wavelets
generalizing those of Meyer with compactly supported Fourier transform.
Key words and phrases. wavelet, local commutant, wandering vector, von Neumann
algebra..
The rst author was supported in part by a grant from the AFOSR, by a YI grant from the
Linear Analysis and Probability Workshop at Texas A&M University and by a grant provided
by the University of North Carolina at Charlotte
The second author was supported in part
by a grant from the NSF.
1
2
XINGDE DAI AND DAVID LARSON
Introduction
A unitary system is a set of unitary operators U acting on a Hilbert space H
which contains the identity operator I of B(H): A wandering vector for U is a
unit vector x with the property that
U x := fUx : U 2 Ug
is an orthonormal set it is called complete if U x is an orthonormal basis for
H: A wandering vector system will mean a unitary system which has a complete
wandering vector.
Let W (U ) denote the set of complete wandering vectors for a unitary system
U : For W (U ) to be nonempty, the set U must be very special. It must be
countable if it acts separably, and it must be discrete in the strong operator
topology (pointwise convergence) because if U V 2 U and if x is a wandering
vector for U then
p
kU ; V k kUx ; V xk = 2:
Certain other properties are forced on U by the presence of a wandering vector.
One purpose of this paper is to study such properties. Indeed, it was a matter
of some surprise to us to discover that such a theory is viable even in some considerable generality. A more immediate purpose, however, is to study structural
properties of W (U ) for special systems U which are relevant to wavelet theory.
In operator theory, wandering vectors have been studied for groups of unitaries
and semigroups of isometries, particularly those that are singly generated (c.f.
13]). Wavelet theory entails the study of wandering vectors for unitary systems
which are not even semigroups.
In the past ten years wavelet theory has undergone a vast development, and
many dierent aspects of the theory have been studied extensively in the literature. Some of the most frequently studied and used aspects include the following
denition of orthogonal (or orthonormal) wavelet:
An orthogonal wavelet is a unit vectorn (t) in L2 (R ) with Lebesgue measure, such that f2 2 (2n t ; l) : n l 2 Zg
constitutes an orthonormal basis for L2 (R):
3
4
XINGDE DAI AND DAVID LARSON
This is the denition given in, e.g., Chui's book (3], p.4), and it is referred to
in Meyer's book (21], p.28) as the Franklin-Stromberg denition. The simplest
function satisfying this is the Haar wavelet H = 0 21 ) ; 21 1) : Wavelets
having this property (see, e.g., 8]) include those of Stromberg, Meyer, BattleLeMarie and Daubechies. These wavelets also satisfy strong auxiliary regularity
(dierential and moment) properties and time-frequency localization properties
which make them useful in applications. Dilation factors other than 2 (the dyadic
case) have also been studied. Generalizations to Rn with matrix dilations, and
Lp (R ) for other 1 p < 1 are also frequently studied. Representative
articles are contained in the excellent collections 2, 4] and 9].
The term \mother wavelet" is also used in the literature for a function satisfyingnthe above denition of orthogonal wavelet. In this case the functions
nl := 2 2 (2n t ; l) are called elements of the wavelet basis generated by the
\mother." The functions nl will not themselves be mother wavelets unless
n = 0:
The idea of viewing orthogonal wavelets as wandering vectors for dilationtranslation unitary systems is simple and has been used by others (c.f. 11, 12]).
Let T and D be the operators on H = L2 (R) dened by
p
(T f)(t) = f(t ; 1) and (Df)(t) = 2f(2t) f 2 L2 (R) t 2 R:
These are unitary operators, and are in fact bilateral shifts of innite multiplicity, with wandering subspaces L2 0 1] and L2 (;2 ;1] 1 2]) respectively,
considered as subspaces of L2 (R): They fail
to commute, but satisfy the relation
n
2
2
2
T D = DT : For 2 L (R) we have 2 (2n t ; l) = (Dn T l )(t):
The group generated by fD T g is easily computed to be
GroupfD T g = fDn T : n 2 Z 2 D g
where D denotes the set of dyadic rational numbers, and where for real T
denotes the translation unitary
(T f)(t) = f(t ; ):
This can be viewed as a semidirect product of the dyadics by an action of the
integers.
Every countable group G has a representation on a separable Hilbert space for
which it has a wandering vector: just represent G on l2 (G ) by left multiplication,
and note that fgg 2 W (G ) for g 2 G : For a dierent representation (G ) may or may not have a wandering vector. In particular, this fails for
GroupfD T g: To see this, choose n 2 D n f0g n ! 0 so then T ! I
in the strong operator topology (pointwise convergence). So if 2 H then
Tn ! but if were wandering for GroupfD T g then
p
kTn ; k = 2 for all n
INTRODUCTION
5
a contradiction. However, if we consider the subset
UDT := fDn T l : n l 2 Zg
then the property of being a complete wandering vector for UDT is precisely
that of satisfying the denition of orthogonal wavelet. So W (UDT ) is far
from empty. It will sometimes be convenient to use the ordered pair notation
hD T i to abbreviate UDT : Elements of W (UDT ) will be called orthogonal
wavelets.
It is useful for perspective to note that the reversed set UTD = fT n Dl : n l 2
Zg fails to have a wandering vector. To see this choose dyadic k ! 0 k 6= 0
as above. Write lk = pk =2qk and note that (see Lemma 3.2) D;qk Tk =
T pk D;qk : If 2 H then,
kT pk D;qk ; D;qk k = kD;qk Tk ; D;qk k = kTk ; k ! 0:
So if 2 W (UTD ) then orthonormality of UTD implies T pk = I for all but
nitely many k contradicting the assumption that k 6= 0: (We thank Shijin
Lu for this argument.)
Let U be a unitary system. The apparently new idea which we will develop in
this article is an association of W (U ) with the commutant of U in B(H) and
more importantly, with the local commutant (see Chapter 1) of U at a complete
wandering vector. This permits a partial analysis (and theoretically a complete
analysis) of W (U ) using operator-theoretic techniques. The most basic types
of problems seem to be the topological ones: For a given unitary system U is
the set W (U ) closed in the Hilbert space norm topology? Is the linear span of
W (U ) dense in H? Is W (U ) norm-pathwise connected? Work on these issues
lead to other types of operator-theoretic techniques and results. In the case of
wavelets, these techniques do lead to the construction of new ones. The work we
present is intended to be an initial step in a new direction.
We give a number of open problems within the context of our subject matter,
and label by capital Roman letters A, F those which we feel are the most
signicant.
Several graduate students who were enrolled in a topics course on this subject
at Texas A&M University in the spring of 1994 contributed useful examples to
this work. These are credited in context. Others made useful comments. We
thank Eugen Ionascu, Vishnu Kamat, Shijin Lu, Darrin Speegle and Puhong
You for their contributions.
Our basic analysis references for this article are 10, 13, 14, 5, 16, 22] our
basic wavelet references are 3, 8, 20].
We use ] to denote closed linear span. If S is a set of operators, we
use U(S ) to denote the set of unitary operators in S and w (S ) for the von
Neumann algebra generated by S and I: If S is a linear space of operators, a
vector x 2 H is called cyclic for S if S x] = H and x is separating for
S if the map A ! Ax S ! H is injective. The notation S 0 will denote
6
XINGDE DAI AND DAVID LARSON
the commutant of S : the set of operators in B(H) which commute with all
elements in S : In this article Hilbert spaces will be separable.
CHAPTER 1
The Local Commutant
Let S B(H) be a set of operators, and let x 2 H be a nonzero vector. Let
us dene
Cx (S ) := fA 2 B(H) : (AS ; SA)x = 0 S 2 Sg:
We call this the local (or \point") commutant of S at x: It can be a useful
concept, especially when x is a cyclic vector for the linear span of S and S is not
a semigroup. (If S is a semigroup and x is cyclic it reduces to the commutant by
item (ii) below.) It is clearly a linear subspace of B(H) which is closed in the
strong operator topology and the weak operator topology, and it contains the
commutant S 0 of S : In the wavelet case, it turns out that the local commutant
of UDT at a wavelet can contain non-abelian von Neumann algebras as subsets,
and thus has rich structure, while the commutant itself is abelian. We capture
some immediate and useful properties in the form of a lemma. Many of these
generalize analogous properties of the commutant.
Lemma 1.1. If S B(H) is a set and if x 2 H is a vector for which
S x] = H then:
(i) The vector x is separating for Cx(S ):
(ii) If S is a semigroup then Cx (S ) = S 0 :
(iii) If A is an element of Cx (S ) with dense range, then Ax is also cyclic
for S ]:
(iv) Suppose x is also separating for S : Then if S T 2 S with ST 2 S
and TS 2 S and ST 6= TS then neither S nor T is in Cx (S ):
(v) Suppose S = S1S2 where S1 is a semigroup. Then Cx (S ) S10
(vi) If V 2 Cx (S ) is invertible, then
CV x (S ) = Cx(S )V ;1 :
Proof. (i) If A 2 Cx (S ) and if Ax = 0 then for any S 2 S we have
ASx = SAx = 0: So AS x = 0 hence A = 0:
7
8
XINGDE DAI AND DAVID LARSON
(ii) The inclusion \ " is trivial. For \ ", suppose A 2 Cx (S ): Then
for each S T 2 S we have ST 2 S and so
AS(Tx) = (ST)Ax = S(ATx) = SA(Tx):
So since T 2 S was arbitrary and S x] = H it follows that AS = SA:
(iii) Then S Ax = AS x so the conclusion is immediate.
(iv) If S 2 Cx (S ) then since T 2 S we have (ST ; TS)x = 0 so
ST x = TSx which contradicts the fact that x separates S : Similarly, T cannot
be in Cx (S ):
(v) Then S1 S S : Let A 2 Cx (S ) and R 2 S1: For each S 2 S we have
ASx = SAx and also ARSx = RSAx since RS 2 S : So (AR ; RA)Sx = 0
for all S 2 S : Since S x] = H this implies that AR = RA as required.
(vi) We have
CV x (S ) = fA : (AS ; SA)V x = 0 S 2 Sg
= fA : (AV S ; SAV )x = 0 S 2 Sg
= fA : AV 2 Cx (S )g
= Cx (S ) V ; 1 :
If S contains a semigroup S0 with S00 = S and if x is cyclic for S0 in the
sense that S0 x] = H then part (ii) of the above lemma implies that Cx(S )
reduces to simply S 0: This is the case for UDT when x is a scaling function.
(See x6:3:) However, if x is an orthogonal wavelet the structure of Cx (UDT ) is
much richer, as we will see.
The following is included only for perspective. It shows that local commutants
and commutants share an additional special structural property.
Proposition 1.2. Let S B(H) and x 2 H be arbitrary. Then Cx(S ) is
2-reexive in the sense that Cx (S ) I2 is reexive in B(H H2 ):
Proof. We use a duality proof. From 17] it will be enough to show that the
preannihilator of Cx (S ) in C1(H) is generated by operators of rank 2: We
claim in fact that
(Cx (S ))? = spanfS x y] : S 2 S y 2 Hg
where for x y 2 H x y denotes the rank-1 operator dened by
(x y)w := hw yix w 2 H
and
S x y] := S(x y) ; (x y)S:
1. THE LOCAL COMMUTANT
9
For arbitrary A 2 B(H) we have the tracial equation
Tr(AS x y]) = Tr(A(Sx y ; x S y))
= Tr(ASx y) ; Tr(Ax S y)
= hASx yi ; hAx S yi
= hASx yi ; hSAx yi
= h(AS ; SA)x yi:
From this it follows that A 2 Cx (S ) i A is annihilated by all trace class
operators of the form S x y] for S 2 S and y 2 H: So (Cx (S ))? is
the k
k1 -closed linear span of these commutators. Since each of these has rank
2 this proves that Cx(S ) is 2-reexive.
The key to our approach is the following simple result.
Proposition 1.3. Let U be a unitary system in B(H): Suppose 2 W (U ):
Then
W (U ) = fV : V 2 U(C (U ))g:
Moreover, the correspondence
V ! V U(C (U )) ! W (U )
is one-to-one.
Proof. Let V 2 U(C (U )): Let = V : For U 2 U we have
U = UV = V U
since U commutes with V at : Thus U = V U and so U is an orthonormal
basis for H since V is unitary. So 2 W (U ):
Conversely, let 2 W (U ) be arbitrary. Since U and U are orthonormal
bases, there is a unique unitary operator V with
V U = U U 2 U :
Then V = since I 2 U : So V U = UV for all U 2 U : Thus V 2 C (U ):
By Lemma 1.1, separates points of C (U ): Thus the map V ! V is
one-to-one.
Proposition 1.3 shows that if U is a unitary system with W (U ) 6= then
given any 2 W (U ) the entire set W (U ) can be parameterized in a natural
way by the set of unitary operators in the local commutant of U at :
It is an elementary result that if G B(H) is a unitary group, and if x and
y are cyclic vectors for span G then the vector functionals
!x = h
x xi and !y = h
y yi
10
XINGDE DAI AND DAVID LARSON
agree on G i there exists V 2 U(G 0 ) with V x = y: Proposition 1.3 can
be thought of as a special case of the following generalization of this to unitary
systems.
Proposition 1.4. Let U be a unitary system in B(H): Suppose x y 2 H with
U x] = U y] = H: Then
hU1 x U2xi = hU1 y U2 yi
for all U1 U2 2 U if and only if there is a unitary V 2 Cx (U ) with V x = y:
Proof. Fix U1 U2 Un 2 U and 1 2 n 2 C : Then
X
X
X
h i Ui x i Ui xi =
j i hUj Ui x xi
=
ij
X
ij
j i hUj Ui y yi
X
= h
i Ui y
X
i Ui yi:
This shows that the map
V : span(U x) ! span(U y)
P
P
dened by V ( i Ui x) = i Ui y is isometric. Thus since U x] = U y] = H
V extends to a unitary operator. For U 2 U we have V Ux = Uy = UV x:
Thus V 2 Cx (U ):
Proposition 1.4 induces an equivalence relation on the set of cyclic vectors
for span(U ) where in W (U ) constitutes one equivalence class. There is, in
addition, a natural equivalence relation on the set W (U ) itself induced by the
usual notion of the equivalence of group representations (see x6:1).
In certain cases new wandering vectors can be obtained by \interpolating"
between a known pair. The following proposition can be viewed as the prototype
of our results of chapter 5.
Proposition 1.5. Let U be a unitary system, let 2 W (U ) and let V be
the unique unitary operator in C (U ) with V = : Suppose V 2 = I: Then
cos + i sin is in W (U ) for all 0 2:
Proof. Let P = 21 (V + I): Then P is a projection, and is contained in C (U ):
Let
!1 = cos + i sin and !2 = cos ; i sin :
1. THE LOCAL COMMUTANT
11
Then j!ij = 1 so W := !1P + !2(I ; P ) is a unitary operator in C (U ): So
W 2 W (U ): We have W = !1P + !2 (I ; P) and P = 21 (V + I) so
P = 21 V + 12 = 12 ( + ):
Similarly
(I ; P) = 21 ( ; ):
Thus W = 12 (!1 + !2 ) + 12 (!1 ; !2 ) = cos + i sin :
Unitaries V with V 2 = I are called involutions, or symmetries. It turns
out that the involutive case in Proposition 1.5 is not uncommon in dilationtranslation wavelet theory (Chapter 5.) Such pairs ( ) are of course connected
in W (U ) in the Hilbert space metric.
If U is a unitary system which is not a group, and if W (U ) 6= then U is
not even a semigroup.
Lemma 1.6. Let S be a unital semigroup of unitaries in B(H): Suppose
W (S ) 6= : Then S is a group.
Proof. Let 2 W (S ): If S is not a group, let U 2 S such that U ;1 2= S :
Then for each V 2 S hU ;1 V i = h UV i = 0
since UV 2 S and V =
6 U ;1: Hence U ;1 is a non-zero vector orthogonal
to S ] a contradiction.
If U is a unitary system then C (U ) is almost never an algebra, and its set
of unitary operators is almost never a group.
Proposition 1.7. Let U be a unitary system, and suppose W (U )] = H: Then
(i) If C (U ) is an algebra for some 2 W (U ) then C (U ) = U 0 for
every 2 W (U ): In particular, C (U ) is an algebra for all 2 W (U ):
(ii) If U(C (U )) is a semigroup for some 2 W (U ) then U(C (U )) =
U(U 0 ) for every 2 W (U ): In particular, U(C (U )) is a group for
all 2 W (U ):
Proof. Let 2 W (U ) be arbitrary. By Proposition 1.3, there is a unique
V 2 U(C (U )) with = V : Then
C (U ) = C (U )V by Lemma 1.1 (vi). Hence V 2 C (U ):
12
XINGDE DAI AND DAVID LARSON
For item (i), if C (U ) is closed under multiplication then C (U )V C (U )
and it follows that
C (U ) C (U ):
So if A 2 C (U ) then (AU ; UA) = 0 for all U 2 U and for all 2 W (U ):
Since W (U )] = H this implies A 2 U 0: We have shown C (U ) = U 0 : Again
let 2 W (U ) be arbitrary, and let W be the unique unitary in C (U ) = U 0
with = W: Then
C (U ) = C (U )W = U 0:
For item (ii), if U(C (U )) is a semigroup then U(C (U ))V U(C (U )):
Since also U(C (U )) = U(C (U ))V this implies
U(C (U ))
U(C (U )):
If S 2 U(C (U )) then
(SU ; US) = 0 U 2 U 2 W (U )
so as above, S 2 U 0 : The rest is identical to (i).
Proposition 1.8 . Let U be a unitary system, and suppose C (U ) is abelian
for some 2 W (U ): Then C (U ) is abelian for all 2 W (U ):
Proof. Suppose C (U ) is abelian and let 2 W (U ) be arbitrary. Let
V 2 U(C (U )) with = V : Then C (U ) = C (U )V : So V 2 C (U ):
Since V 2 (C (U ))0 and V is normal, V 2 (C (U ))0 : So C (U ) = C (U )V is
abelian.
Example 1.9. Let feng1;1 be an orthonormal basis for a separable Hilbert
space H and let Sen = en+1 be the bilateral shift of multiplicity one. Let
U = fS n : n 2 Zg be the group generated by S: Each en is in W (U ): By
Lemma 1.1 part (ii) and Proposition 1.3,
W (U ) = fV e0 : V 2 U(fS g0 )g:
Here fS g0 coincides with the set of Laurent operators. Let T be the unit circle.
If we represent S on L2 (T) in the usual way by identifying it with the multiplication operator Mz then U(fS g0 ) is identied with (multiplication by) the set
of unimodular functions on T and e0 is identied with the constant function 1:
Then Proposition 1.3 just recovers the well-known fact that the set of complete
wandering vectors for the shift coincides (under this representation)with the set
of unimodular functions on T: In this case W (U ) is clearly a closed, connected
subset of the unit ball of H in the norm topology with dense linear span.
1. THE LOCAL COMMUTANT
13
Example 1.10. Let G be a countable group, let H = l2(G) and let L be
the left regular representation of G on on H: That is, for h 2 G and fg gg2G 2
l2 (G ) dene L(h)fg g = fh;1 g g so writing (g) fg gg2G we have
(L (h1)L (h2 ))(g) = (L (h1 )(h;2 1
))(g)
= ((h;2 1 ))(h;1 1 g)
= (h;2 1 h;1 1g)
= ((h1 h2 );1g)
= (L (h1 h2))(g)
The standard basis for H is feg : g 2 Gg where eg = fgg fgk gk2G : Then
L(h)eg = fgh;1 k gk2G = fhgk gk2G = ehg :
The vectors eg are clearly in W (L (G )): By Lemma 1.1 the local commutant of
L(G ) at eI is just the commutant, (where I denotes the identity element of G
). Since fL (G )g0 is a von Neumann algebra, its group of unitaries is connected
in the norm topology. Since the map V ! V eI is continuous, Proposition
1.3 implies that W (L (G )) is a connected subset of the unit ball of H: Since
eg 2 W (L (G )) g 2 G the set W (L (G )) has dense span.
The algebra w (L(G )) considered above and its commutant are classic in
the theory of von Neumann algebras. See, for instance, x6:7 in 14]. From this
theory we have
fL(G )g0 = w (R (G ))
where R (G ) is the right regular representation of G on H = l2 (G ) dened
by R (h)fg g = fgh g: Moreover, if G is a group which has the property that
the conjugacy class of each element other than the identity is innite (an i.c.c
group) then w (L (G )) and w (R (G )) are factor von Neumann algebras of
type II1 : This is the case, for instance, if G is the free (non-abelian) group on n
generators for n 2: This shows that for some unitary systems U (and perhaps
for many) the structure of W (U ) is at least as complex as the structure of the
unitary group of a type II1 factor.
Example 1.11. Let G be a countable group, and let G0 G be a subset
containing the identity element I: Sometimes it is possible to obtain a faithful
unitary representation of G on l2 (G0) satisfying the requirement that if h 2 G
and g 2 G0 are such that h;1g 2 G0 then (h)eg = ehg : (Where as in the
previous example eg fgg :) Then eI 2 W ((G0 )) trivially.
This \example" is generic. If U is any unitary system on a Hilbert space
H with a complete wandering vector let Ue be the group generated by U in
B(H): Let G = Ue as an abstract group, and let G0 = U a subset of G : Let
K = l2 (G0 ): Dene a unitary operator
W : H ! K by W g = eg g 2 U 14
XINGDE DAI AND DAVID LARSON
making use of the fact that U is an orthonormal basis for H: Dene
: B(H) ! B(K) by (A) = W AW and restrict to G Ue: Then satises the property of the above paragraph,
and is unitarily equivalent to the identity representation of G Ue on H:
Let us call a unitary representation of a group G relative to a unital subset G0 a wandering vector representation of the pair (G G0) if it is faithful
on G and if (G0) has a complete wandering vector. From above, these are
unitarily equivalent to those of Example 1.11. In the context of wandering vector theory, and especially wavelet theory, an abstract question which becomes
rather intriguing is: Given a group G what are the unital subsets G0 which are
allowable in the sense that (G G0) has a wandering vector representation? In
particular, if G is generated as a group by an ordered pair of elements fg1 g2g
and if Gi = Groupfgig is the set
fG1G2 = h1h2 : hi 2 Gig
an allowable subset of G ? Work here may aid in understanding wavelet systems,
in particular. The question is obviously nontrivial in view of wavelet theory. This
question generalizes to ordered n-tuples of generators. As mentioned earlier, we
have the problems: when does W ((G0 )) have dense span, when is it closed,
and when is it connected?
To gain some insight we can abstract the hD T i wavelet system in the
introduction.
Example 1.12. (The abstract one-dimensional system.) Let Sen = en+1 be
the bilateral shift of multiplicity one in Example 1.9. Let
A = S I 2 B(H H)
and let B be any unitary in B(H H) with
B je0 H = (I S)je0 H :
Let UAB := fAnB l : n l 2 Zg: Then U = UA UB where UA and UB are the
groups generated by A and B respectively. We have e0 en 2 W (U ) for each
n 2 Z but except in special cases ep en will not be in W (U ) if p 6= 0:
Note that by Lemma 1.1 (v) if 2 W (U ) then every operator in C (U )
commutes with A: Also, by Lemma 1.1 (iv) if A and B do not commute, then
neither A nor B can ever be in C (U ):
The above example is generic in that it is really a \model" for \one-dimensional"
wandering vector systems such as hD T i: To see this, let K be an arbitrary separable Hilbert space, and let U and V be arbitrary unitary operators in B(K):
1. THE LOCAL COMMUTANT
15
Let
UUV = fU nV l : n l 2 Zg
and suppose W (UUV ) is nonempty. Let 2 W (UUV ): Since the sets
fU n V l : n l 2 Zg and fen el : n l 2 Zg
are orthonormal bases for K and HH respectively, they determine a unitary
operator W : K ! H H such that
WU n V l = en el
for all n l: Then W = e0 e0 and for each n l 2 Z we have
W UW (en el ) = W UU nV l So
= W U n+1V l = en+1 el
= (S I)(en el ):
WUW = S I = A:
Let B = W V W : Then
B(e0 el ) = WV W (e0 el )
= WV U 0V l = WV l+1 = e0 el+1
= (I S)(e0 el )
for each l 2 Z so
B je0 H = (I S)je0 H as required.
As above we have the closure, span and connectedness questions for W (UAB ):
For the wavelet system hD T i Example 4.5(ii) shows that W (UDT ) is not
closed and Corollary 3.17 shows that span W (UDT ) is dense. Are these properties true of general systems of the form UAB ? When is W (UAB ) connected?
We have no counterexample. On the other hand, examples are very hard to
evaluate.
Problem A. Is W (UDT ) connected?
This connectedness problem is perhaps the most important open question in our
theory. A solution may lead to perturbation methods for wavelets, in particular. Construction of a counterexample for some other system UAB may shed
some light on this matter. Based on evidence so far, we conjecture that the
16
XINGDE DAI AND DAVID LARSON
answer is \yes." There are related optimization problems, such as computation
of dist(x W (UDT )) for x 2 H:
Example 1.13. (Twisted Tensor Product). The above example can be nicely
generalized, pointing out the complexity possible in unitary systems having wandering vectors. Let U1 and U2 be unitary systems with wandering vectors on
Hilbert spaces H1 and H2 respectively, and let i 2 W (Ui ) i = 1 2: Let
H = H1 H2 : Let Ue1 = U1 I and let Ue2 be any unitary system in H
leaving the subspace 1 H2 invariant such that the map
U ! U j1 H2
is 1-1 on Ue2 and with
Ue2 j1 H2 = (I U2 )j1H2 :
Then
U = Ue1 Ue2
is a unitary system on H: Clearly 1 2 2 W (U ): By repeating this procedure
one may construct \twisted tensor products" of arbitrary length. The case when
U1 U2 are abelian groups is most relevant to wavelet theory, and models the
higher-dimensional translation-dilation systems on Rn which have been studied.
In this case U2 can correspond to the group generated by the translations in the
n directions, and U1 can correspond to an abelian group of dilation unitaries,
which can be matrix dilations.
Example 1.14. Without additional structural hypotheses, pathological unitary
systems are easily constructed. For instance, if fen g1
n=1 is an orthonormal basis,
let U1 = I and for each n 2 let Un be an arbitrary unitary operator with
Ue1 = en : Then
U = fUn : n 2 Ng
is a unitary system with e1 as a complete wandering vector. For certain (likely
\most") choices of Un one will simply have
W (U ) = fe1 : jj = 1g and Ce1 (U ) = C I:
For instance, if Un is the permutation unitary that interchanges e1 and en
and xes then other basis vectors, then U will have this property.
CHAPTER 2
Structural Theorems
Let U be a unitary system in B(H) and suppose U contains a subset U0 which
is a group such that UU0 = U : This is the situation for the wavelet theory case
U = fDn T l : n l 2 Zg
where U0 = fT l : l 2 Zg: Suppose 2 W (U ): Then U0 W (U ) clearly.
However, U0 will not usually be contained in C (U ): For each U 2 U0 let
VU be the unique unitary in C (U ) with VU = U given by Proposition 1.3.
Let
: U0 ! U(C (U ))
denote the map (U) = VU U 2 U0:
Theorem 2.1. With the above notation, (U0 ) is a group and is a group
anti-isomorphism. The set U0 is contained in a connected subset of W (U ):
Proof. Suppose U1 U2 2 U0: Let S 2 U : Then
(U2 ) (U1 )S = (U2 )S (U1 )
= (U2 )SU1 = SU1 (U2 )
= SU1 U2 = S (U1 U2 )
= (U1 U2 )S:
So (U2 ) (U1 ) agrees with (U1 U2 ) on the orthonormal basis U and
hence they are equal. This shows that (U0 ) is a group and that is
an anti-homomorphism. If U 2 U0 and U 6= I then U 6= since is wandering for U : Hence (U) 6= I: So is one-to-one, as required.
To show that U0 is contained in a component, note that the closure in the
strong operator topology of the span of (U0 ) is the von Neumann algebra
w ( (U0 )) and is contained in C (U ): The unitary group of a von Neumann
algebra is norm connected (c.f. 14]). So as in Example 1.10, continuity of the
17
18
XINGDE DAI AND DAVID LARSON
map V ! V from U(w ( (U0 ))) ! H implies that U(w ( (U0))) is
connected in W (U ): This contains U0 since U = (U) U 2 U0 :
Corollary 2.2. With the above terminology, if U0 is nontrivial, then the
connected components of W (U ) are all nontrivial.
Theorem 2.3. With the above terminology, suppose U0 is abelian. In this
case, if U 2 U(w (U0 )) then U W (U ) = W (U ): The map extends to a
homomorphism of U(w (U0 )) into U(C (U )):
Proof. Let = U: Let
E = U0] = w (U0 )]:
Then E reduces w (U0 ): So UE = E : Suppose W 2 U but W 2= U0:
Then WV1 2= U0 for all V1 2 U0 so W V1 ? V2 for all V1 V2 2 U0 : Hence
WE ? E : More generally, if W1 W2 2 U and W1 U0 6= W2 U0 then
W1 U0 \ W2 U0 = so W1U0 ? W2 U0 and hence W1 E ? W2 E : So if W1 W2 2 U and
W1 U0 6= W2 U0 then since = U 2 E we have W1 ? W2 : On the
other hand, if W1 6= W2 but W1U0 = W2 U0 then W2 = W1 U1 for some
U1 2 U0 U1 6= I: Then U1 ? so UU1 ? U: Since by hypothesis U0
is abelian, so is w (U0) and so UU1 = U1U: Thus U1 ? : Hence
W2 = W1 U1 ? W1 :
We have shown that U is an orthonormal set. We have
U0] = U0 U] = w (U0 )U] = w (U0 )] = E :
So U ] = UU0] = U E ] U ] = H: Thus U is complete. So 2 W (U ):
Next, for each U 2 U(w (U0)) let VU be the unique unitary in C (U ) for
which VU = U that is given by Proposition 1.3, and dene (U) = VU : If
U1 U2 2 U(w (U0 )) let S 2 U be arbitrary. Then as in Theorem 2.1,
(U2 ) (U1 )S = (U2 )S (U1 ) = (U2 )SU1 :
Since SU1 is in the strongly closed linear span of U and since (U2 ) commutes
locally at with each element of U we have
(U2 )SU1 = SU1 (U2 ) = SU1 U2 :
Since U1 U2 2 U(w (U0 )) we have U1 U2 2 U(w (U0 )): Thus
SU1 U2 = S (U1 U2 ) = (U1 U2 )S:
So, as in Theorem 2.1, (U2 U1 ) = (U1 U2 ) agrees with (U2 ) (U1 ) on
an orthonormal basis, so they are equal.
2. STRUCTURAL THEOREMS
19
Remarks 2.4. For the special case of the wavelet system hD T i on L2(R)
Theorem 2.3 implies that if U is a unitary operator in w (T ) then for any
orthogonal wavelet U is also a wavelet. This can also be deduced by a
function-theoretic argument. (c.f. 8].)
Problem B. If 2 W (U ) and 2 E := U0] is there a unitary operator
U in w (U0 ) such that = U ?
In the abelian case, and in particular in the case of the wavelet system hD T i
the answer is yes. (See Corollary 2.17).
Let U U0 be as in Theorem 2.1. Do not assume U0 is abelian. Let
E = U0] and let P = proj (E ):
Lemma 2.5. A 2 C (U ) if and only if (AS ; SA)P = 0 for all S 2 U :
Proof. If (AS ; SA)P = 0 then since P = we have (AS ; SA) = 0:
Conversely, if (AS ; SA) = 0 S 2 U then for all T 2 U0 (AS ; SA)T = A(ST ) ; S(AT ) = (ST)A ; ST A = 0:
So (AS ; SA)U0 = 0 hence (AS ; SA)P = 0:
Lemma 2.6. If 2 W (U ) and if V 2 U(C (U )) with V = then
P = V P V :
Proof. For each S 2 U0 we have S = SV = V S: So U0] = V U0]:
Lemma 2.7. If V 2 U(C (U )) and if V 2 E then
PV = P C (U ) = CV (U ) and P V = V P :
Proof. For S 2 U0 we have
V S = SV 2 SP H P H:
So V P H P H: Let = V : Then P = V P V by Lemma 2.6, so
P P : Let fUng be a sequence in U such that Un U0 \ Um U0 = n 6= m
and n UnU0 = U : Then
Un E ? Um E
W
if n 6= m and n Un E = H: The subspace E has the same property. So
since E E we must have E = E : So P = P : Then Lemma 2.5
implies
C (U ) = CV (U ):
Since P = V P V we have P V = V P :
20
XINGDE DAI AND DAVID LARSON
Let CP (U ) = C (U ) \ fP g0 :
Proposition 2.8. C (U ) is a left module over U 0 and a right module over
CP (U ): In particular, CP (U ) is an algebra.
Proof. Let A 2 C (U ): If B 2 U 0 then for S 2 U BAS = BSA = SBA:
Hence BA 2 C (U ): Now let C 2 CP (U ): Then
(AC)S = ASC = ASCP = ASP C = SAP C = S(AC):
where the fourth equality is via Lemma 2.5. Thus AC 2 C (U ):
Theorem 2.9. CP (U ) is a von Neumann algebra.
Proof.
CP (U ) is an algebra , and is strongly closed. We must show it is
self-adjoint. Suppose A 2 CP (U ): We must show that A V = V A for all
V 2 U : It will suce to show that
hA V W i = hV A Wi
for all V W 2 U :
For U 2 U0 write U (A) = hA Ui: Then
A =
X
U (A)U:
U 2U0
Since P reduces A A E E : Write U (A ) = hA Ui: Then
X
A =
U (A )U:
U 2U0
We have
U (A ) = hA Ui = h AUi = h UAi = hU Ai = U (A):
Now compute:
hA V W i = hV AWi = hV WA
Xi
= hV W
U (A)Ui
= hV W
=
X
U 2U0
U 2U0
X
U 2 U0
U (A)U i
U (A )hV WU i:
2. STRUCTURAL THEOREMS
21
We have WU = W (U ) = (U )W : So
hV W U i = hV (U )W i
= h (U)V Wi
= hV (U) Wi
= hV U Wi
= hU V W i:
And so
X
hA V Wi =
U (A )hU V W i
U 2U0
= h
U 2U0
U (A )U V W i
= hA V Wi
= hV A W i
as required.
Corollary 2.10.
X
U(CP (U ))
is a group.
Corollary 2.11. W (U ) \ E is connected.
Proof. This follows from Lemma 2.7 and Corollary 2.10.
Note that if U0 is abelian, then (U)P = UP for all U 2 U0 : (This can
fail if U0 is nonabelian. For instance, we may have U = U0 is a commutative
group. Then P = I and C (U ) = fUg0 6 U : ) To see this, note that for
commutative U0 for each U V 2 U0 (U)V = V (U) = V U = UV :
So since E = U0] (U)jE = U jE :
Theorem 2.12. Suppose U0 is abelian. Then CP (U ) is abelian.
Proof. Let A B 2 CP (U ): Let U 2 U0: Then
A (U) = AU = UA = UP A = (U)P A = (U)A:
Since A (U) 2 CP (U ) which is an algebra, and since separates CP (U )
this shows that
A (U) = (U)A:
Since A 2 E we have
X
A =
U (A)U
U 2U0
22
XINGDE DAI AND DAVID LARSON
where U (A) = hA Ui: Similarly,
X
B =
U (B)U:
U 2U0
Let V 2 U0 be arbitrary. Compute
hAB V i =
=
=
=
Similarly,
hBA V i =
=
=
X
U 2U0
X
U 2U0
X
U 2U0
X
U 2U0
U (B)hAU V i
U (B)hUA V i
U (B)hA U V i
U (B)U V (A):
X
U 2U0
X
U (A)U V (B)
W 2U0
X
W 2U0
V W (A)W (B)
W (B)W V (A)
where we use the fact that U0 is abelian. This shows that
hAB V i = hBA V i:
So h(AB ; BA) V i = 0 V 2 U0 and since (AB ; BA) 2 U0] this
shows that (AB ; BA) = 0: So since AB ; BA 2 CP (U ) this implies
AB = BA as required.
The following gives a simple but useful structural description of the local
commutant for an important special case.
Proposition 2.13. Let U1 and U0 be unitary groups in B(H) and let U be
the unitary system U = U1U0 = fUV : U 2 U1 V 2 U0g: If 2 W (U ) then
C (U ) = U10 \ fU00 + B(H)P?g:
Proof. First note that
U00 + B(H)P? = fA 2 B(H) : AP 2 U00 P g:
Indeed, if AP 2 U00 P then AP = RP for some R 2 U00 so
A = R + (A ; R)P? 2 U00 + B(H)P?:
2. STRUCTURAL THEOREMS
23
Now suppose A 2 C (U ): Then A 2 U10 by Lemma 1.1 (v). Also, for
any U 2 U0 we have UAP = AUP = AP U: Thus AP 2 U00 and so
AP 2 U00 P : Thus
A 2 U00 + B(H)P?
by the above paragraph. Conversely, if A 2 U10 and A 2 U00 + B(H)P? then
A = B + CP? for some B 2 U00 : For U 2 U0 we have P?U = 0: So for
V 2 U1 U 2 U0 we have
AV U = V AU = V (B + CP?)U = V BU = V UB = V UA:
That is, A 2 C (U ):
Lemma 2.14. Let U0 U1 and U be as in Proposition 2.13 with U1 \U0 = fI g:
Let 2 W
P(U ): (Note that U must be countable.) Then for all A 2 B(H)
the sum U 2U1 UP AP U converges in the strong operator topology to an
element of U10 :
Proof. The operators
fUP AP U ;1 : U 2 U1 g
are uniformly bounded and have mutually orthogonal ranges and mutually orthogonal supports. So any enumeration of U1 leads to a convergent sum, and
the limit is independent of the enumeration.
If V 2 U1 then
X
X
V
UP AP U ;1 = ( (V U)P AP (V U);1 )V:
U 2U1
U 2U1
Since V U is a generic element of the group U1 this shows that V commutes
with the sum, as required.
Lemma 2.15. Let U0 U1 U be as above. Then
X
CP (U ) = f UP SP U : S 2 U00 g:
U 2U1
Proof. Suppose S 2 U00 : Then B := PU 2U1 UP SP U is in U10 by
Lemma 2.14. If U 2 U1 and U 6= I then P UP = 0 so
P B = BP = P SP :
Since P 2 U00 we have P SP 2 U00 P : Thus B 2 CP (U ) using Proposition
2.13 and the rst line of its proof.
Conversely, suppose A 2 CP (U ): Then A 2 U10 and AP = P A:
Also,
2 U00 P : Choose S 2 U00 such that AP = SP : Let B =
P AP
P
U 2U1 UP SP U : Then B 2 C (U ) by the rst paragraph. We have
24
XINGDE DAI AND DAVID LARSON
BP = P SP = P AP = AP :
So B = A: Thus A = B by Lemma 1.1 (i).
Theorem 2.16. If U = U1U0 with U1 U0 groups, with U1 \ U0 = fI g and
with U0 abelian, then
(i) CP (U ) = w ( (U0 )) (ii) w (U0 )P = CP (U )P (iii) CP (U ) is -isomorphic to w (U0)jE (iv) extends to a -homomorphism from w (U0 ) onto CP (U ):
Proof. For U 2 U1 if U 6= I then P UP = 0 so by Lemma 2.15,
CP (U )P = P U00 P :
Since P 2 U00 this is a von Neumann algebra. Theorem 2.12 implies it is
abelian. Note that (U0 ) CP (U ) by Lemma 2.7. So w ( (U0)) CP (U )
and so
CP (U )P w ( (U0))P :
Since P 2 (w ( (U0)))0 and w ( (U0 ))P is a von Neumann algebra,
w ( (U0 ))P is w -closed. The vector is cyclic for w ( (U0 ))jP H so this
compression algebra is a m.a.s.a. So since CP (U )P is abelian, we must have
CP (U )P = w ( (U0))P :
Similarly, P 2 (w (U0 ))0 so w (U0 )P is w-closed and abelian. For U 2 U0
we have UP = (U)P so w (U0 )P f (U )P : U 2 U0 g and hence
w (U0 )P w ( (U0 ))P :
The reverse inequality is similar. Thus w (U0 )P = w ( (U0 ))P : Hence
w (U0 )P = CP (U )P : Item (ii) is proven.
Let be the map dened in Lemma 2.14. That is
X
:=
UP AP U A 2 B(H):
U 2U1
Then is linear, and (A ) = ((A)) A 2 B(H): To prove item (i) it suces
to prove \ ". Suppose B 2 CP (U ): Then
P BP = P AP
for some A 2 w ( (U0 )): By the second paragraph of the proof of Lemma
2.15, A = (A) and B = (B): So
B = A 2 w ( (U0 ))
as required.
2. STRUCTURAL THEOREMS
25
For V 2 U0 V P = (V )P : As above, ( (V )) = (V ): Hence
(V ) = (V ): If V 2 U(w (U0 )) then
(V ) 2 CP (U ) C (U )
by Lemma 2.15, and also (V ) 2 C (U ): Since V is unitary and commutes
with P (V ) is unitary. Also,
(V )P = V P = (V )P :
So by the uniqueness part of Proposition 1.3, (V ) = (V ): Hence (V ) =
(V ) where was previously dened, so extends : Since P 2 U00 is multiplicative on w (U0 ): Thus jw (U0 ) is a -homomorphism. Since
(A) = 0 i P AP = 0 is 1-1 on w (U0 )P : So it induces a -isomorphism
between P w (U0)jE and CP (U ):
Corollary 2.17. With the hypotheses of Theorem 2.16, if 2 W (U ) with
2 E then there is a unitary V 2 w (U0 ) with = V :
Proof. Let W 2 U(CP (U )) with W = : By Lemma 2.7, W 2 CP (U ):
Write W = eiA with A 2 CP (U ) and A = A : By Theorem 2.16 there exists
B = B 2 w (U0 ) with A = (B): Let V = eiB : Then V is unitary, and
(V ) = W: Since
(V ) =
X
U 2U1
UP V P U we have WP = V P so = W = V as required.
Remark 2.18. If U is a unitary system with W (U ) 6= let us use the
term wandering vector multiplier for U to denote a unitary operator V with the
property that V W (U ) W (U ): (That is, we require that V 2 W (U ) for
all 2 W (U ) not simply for a specic as with unitaries in C (U ) and we
do not require that V is in C (U ) for any : ) Every unitary in U 0 is a w.v.
multiplier since U 0 C (U ) for every : In the special case when U = U1 U0
with U1 U0 groups and with U0 abelian, by Theorem 2.3 (see also Remark 2.4)
every unitary V 2 w (U0 ) is a w.v. multiplier. In fact, Corollary 2.17 states
that for a given 2 W (U ) every wandering vector in E can be attained
by acting on by a unitary \multiplier" of this form. So we have a natural
problem.
Problem C. (Factorization problem) Suppose U is a unitary system of the
form U = U1 U0 with U1 U0 groups, U0 abelian. Is every wandering vector
multiplier for U of the form V = V1 V0 with V1 a unitary in U 0 and V0 a
unitary in w (U0 )?
Note that these two types of unitaries commute, so if the answer is yes, then the
set of wandering vector multipliers will be an abelian group if U 0 is abelian. For
26
XINGDE DAI AND DAVID LARSON
the special case of hD T i this question has particular interest. (See Remark
3.7.) Let us refer to unitaries in U 0 as w.v. multipliers of the rst type, and to
unitaries in w (U0 ) as w.v. multipliers of the second type. In the case where
U is a wavelet system, we refer to w.v. multipliers as wavelet multipliers.
CHAPTER 3
The Wavelet System
hD T i
Let T and D be the translation and dilation unitary operators on L2 (R) described in the introduction given by
p
(Tf)(t) = f(t ; 1) and (Df)(t) = 2f(2t):
For f 2 L2(R) we have
p
p
p
(TDf)(t) = T ( 2f(2t)) = 2f(2(t ; 1)) = 2f(2t ; 2) = (DT 2 f)(t)
so TD = DT 2 : As earlier, let
hD T i := UDT := fDn T l : n l 2 Zg:
We will abbreviate
W (D T ) := W (UDT )
for the set of wandering vectors for UDT : These are the orthogonal wavelets.
We will also abbreviate
C (D T) := C (UDT ):
We rst recapitulate some of the results of Section 1 as applied to the system
hD T i:
Lemma 3.1. Let be any xed wavelet for hD T i: Then
(i) W (D T ) = U(C (D T)): The mapping
! U from U(C (D T)) ! W (D T)
(ii)
(iii)
(iv)
is one-to-one and onto.
T 2= C (D T ) and D 2= C (D T):
C (D T ) fDg0:
If 2 W (D T) let V 2 C (D T ) with V = : Then
C (D T ) = C (D T)V :
27
28
XINGDE DAI AND DAVID LARSON
Proof. Item (i) is a special case of Proposition 1.3, and items (ii), (iii) and (iv)
are special cases of Lemma 1.1 parts (iv),(v) and (vi), respectively.
If is an orthogonal wavelet, then T n is an orthogonal wavelet for all
n 2 Z even though T 2= C (D T ): This is simply because UDT T n = UDT :
(See the remark before Theorem 2.1.) More generally, if V is any unitary in
w (T ) then V W (D T) = W (D T): As in Chapter 2, let V = (T) denote
the unique unitary in C (D T) with V = T: Then for each n 2 Z Vn
is the unique unitary in C (D T ) with Vn = T n: In particular, C (D T )
contains the group generated by V : The homomorphism
: Group(T) ! C (D T)
extends uniquely to a homomorphism of
U(w (T )) ! U(C (D T ))
with the property that for each V 2 w (T ) (V ) = V : All this follows
from Theorem 2.1 and Theorem 2.3. See also Remark 2.4.
For 2 R let T denote the unitary of translation by :
(T f)(t) = f(t ; ) t 2 L2 (R):
Lemma 3.2. Let n 2 Z and 2 R: Then
Proof. We have
and
Dn T = T2;n Dn and T Dn = Dn T2n :
(Dn T )(t) = Dn f(t ; )
p
= ( 2)nf(2n t ; )
p
(T2;n Dn f)(t) = Tp2;n ( 2)nf(2n t)
= (p2)n(f(2n (t ; 2;n))
= ( 2)nf(2n t ; ):
This establishes the rst identity. For the second, replace n and with ;n and
; and take adjoints.
Remark 3.3. If f 2 L2 (R) is real valued, then clearly gf is real valued for all
g 2 Group(D T ): Also, if an orthogonal wavelet for hD T i is real valued then
considered as an element of real L2 (R) is a real wavelet because the real span
of (UDT ) is dense in the real L2 -space and the elements of (UDT ) remain
orthonormal. Conversely, if is a real wavelet for hD T i then considered as
3. THE WAVELET SYSTEM
hD T i
29
an element of complex L2(R) it is also a complex wavelet. So Lemma 3.1 (i)
gives a way of parameterizing all real wavelets as well.
Proposition 3.4. Each (real or complex) path-connected component of W (D T)
in (real or complex) L2 (R) is nontrivial.
Proof. The complex case is a special case of Corollary 2.2, with U0 = fT n :
n 2 Zg: The real case is more subtle since it is special to hD T i: Let be a
real wavelet. Regard as a wavelet in complex L2 (R): For 0 t 1 let
V ;V
W := exp(itIm(V )) = exp(t( )):
t
Then Wt is unitary and is contained in w (V )
2
C (D T ): Note that the
matrix coordinate elements of V with respect to the wavelet basis fDn T l :
(n l) 2 Z2g are all real, hence Wt = exp(t( V ;2 V )) has real coordinates with
respect to this basis. So since is a real valued function Wt is real-valued.
Thus the path
(t) := Wt 0 t 1
consists of real wavelets.
Next we will characterize the commutant of fD T g using the FourierPlancherel transformation. This is simple, but is apparently new. It will be
the rst step in our (partial) analysis of C (D T ) and hence W (D T) beyond the theory accomplished abstractly in Sections 1 and 2.
Lemma3.2 shows that DT D;1 = T 2 2 R: This implies that Group(D T)
contains the abelian subgroup fT : is dyadicg: It is easy to see that 01]
is a cyclic vector for the linear span of these dyadic translations. It follows that
the closure of this linear span in the strong operator topology is a maximal
abelian von Neumann subalgebra of L2 (R): (a m.a.s.a.). Denote this by AT :
Then the commutant fD T g0 is contained in AT : This proves that fD T g0
is abelian.
Let F be the Fourier-Plancherel transform on H = L2 (R): Then F is a
unitary transformation in B(H): If f g 2 L1 (R) \ H then
Z
b
(F f)(s) := p1
e;istf(t)dt := f(s)
2 R
and
Z
(F ;1g)(t) := p1
eist g(s)ds:
2 R
Let be an arbitrary real number. Then
Z
e;istf(t ; )dt
(F T f)(s) = p1
2 R
= e;is(F f)(s):
30
XINGDE DAI AND DAVID LARSON
So F TF ;1 g = e;is g: For A 2 B(H) let Ab denote F AF ;1: Thus
Tb = Me;is : (For h 2 L1 we use Mh to denote the multiplication operator
f ! hf f 2 L2 : ) Since
fMe;is : 2 Rg
generates the m.a.s.a.
D(R) = fMh : h 2 L1 (R)g
as a von Neumann algebra, we have
FAT F ;1 = D(R):
Similarly,
Z
p
(F Dn f)(s) = p1
e;ist( 2)n f(2n t)dt
2 R
p ;n 1 Z ;i2;n st
= ( 2) p
e
f(t)dt
2
R
p
= ( 2);n(F f)(2;n s) = (D;n F f)(s):
So Db n = D;n = Dn : Therefore, Db = D;1 = D :
We have
and
b Tbg0
FfD T g0F ;1 = fD
b Tbg0):
F U(fD T g0)F ;1 = U(fD
Theorem 3.5.
b Tbg0 = fMh : h 2 L1(R) and h(s) = h(2s) a.e. g and
fD
b Tbg0) = fMh : jh(s)j = 1 and h(s) = h(2s) a.e. g:
U(fD
b Tbg0 if and
Proof. Since Db = D and D is unitary, it is clear that Mh 2 fD
only if Mh commutes with D: So let g 2 L2 (R) be arbitrary. Then (a.e.) we
have
p
(Mh Dg)(s) = h(s)( 2g(2s)) pand
(DMh g)(s) = D(h(s)g(s)) = 2h(2s)g(2s):
Since these must be equal a.e. for arbitrary g we must have h(s) = h(2s) a.e.
The unimodular condition is necessary and sucient for Mh to be unitary.
Remark 3.6. (An algorithm). Let E = ;2 ;1) 1 2) and for n 2 Z
let En = f2nx : x 2 E g: Observe that the sets En are disjoint and have
union R n f0g: So if g is any uniformly bounded function on E then g extends
uniquely (a.e.) to a function eg 2 L1 (R) satisfying eg(s) = eg(2s) s 2 R by
setting eg(2ns) = g(s) s 2 E n 2 Z and eg(0) = 0: We have kegk1 = kgk1 :
3. THE WAVELET SYSTEM
hD T i
31
Conversely, if h is any function satisfying h(s) = h(2s) a.e., then h is uniquely
(a.e.) determined by its restriction to E: This 1-1 mapping g ! Mg~ from
b Tbg0 is isometric. It is a -isomorphism when one regards
L1 (E) onto fD
1
L (E) as a von Neumann algebra. We will refer to a function h satisfying
h(s) = h(2s) a.e. as a 2-dilation periodic function. This gives a concrete
algorithm for computing a large class of wavelets from a given one:
Given let b = F () choose a real-valued function h 2
1
L (E) arbitrarily, let g = exp(ih) extend to a 2-dilation
periodic function eg as above, and compute g~ = F ;1(eg b):
In the description above, the set E could clearly be replaced with ;2 ;) 2) or with any other \dyadic" set ;2a a) a 2a) for some a > 0:
Remark 3.7. (Wavelet multipliers). In the remark above, a concrete descrip-
tion as multiplication operators is given for the group of wavelet multipliers of
the rst type, via the Fourier transform. It is easily seen, since Tb = Me;is that (w (T))b := fAb : A 2 w (T)g = fMf : f 2 L1 (R) andf is 2-periodicg:
So this gives a characterization of wavelet multipliers of the second type, via
the Fourier transform, also as multiplication operators. So Problem C has the
following function-theoretic subproblem.
Problem C0 . If h is a unimodular function in L1 (R) with the property
that F ;1(hF ()) is a wavelet for every wavelet does h necessarily factor
f = f1 f2 where f1 is 2-dilation-periodic and f2 is 2-translation periodic?
Theorem 3.5, and the algorithm in Remark 3.6 is apparently new to wavelet
theory, although it is simple in nature. We have learned that it was also recently
obtained, completely independently, in 15], for a dierent type of study.
The fact that 2-periodic unimodular functions \multiply" wavelets in the
sense of Remark 3.7 is not new. For instance a proof is contained in 8], in
which it is noted that two orthogonal wavelets are associated with the same
multiresolution analysis i they are related by a unimodular 2-periodic function
in this way.
Remark 3.8. Since the unitary group of a von Neumann algebra is connected
(in this case, more simply, the set fg 2 L1 (E) : jg(s)j = 1 a.e.g is connected)
it follows that for xed the set
fg~ : g 2 L1 (E) jg(s)j = 1 a.e.g
is a norm-path-connected set of orthogonal wavelets. This gives another proof
distinct from Proposition 3.4 that each connected component of W (D T ) is
nontrivial. In Theorem 4.4 we show that V is not contained in fD T g0 except
32
XINGDE DAI AND DAVID LARSON
for special cases. Thus the types of paths implied by Corollary 2.2 and Theorem
3.5 are necessarily dierent.
Now let 2 W (D T) and let P denote the orthogonal projection onto
the translation space
E = spanfT l : l 2 Zg:
(These are the special cases of the E and P from Section 2.) Then P
reduces T: Also, since fT l : l 2 Zg is an orthonormal basis for P H it
follows that the subspaces fDnP H : n 2 Zg are mutually orthogonal and span
H: So E is an innite dimensional wandering subspace for D: As in Section 2
let
CP (D T ) = C (D T ) \ fP g0:
Recapitulating some results of sections 1 and 2 applied to hD T i we have:
Theorem 3.9. With the above notation
(i) A 2 C (D T) if and only if
ADn T l P = Dn T l AP
(ii)
(iii)
(iv)
(v)
for all (n l) 2 Z2 :
If and are orthogonal wavelets, and if V 2 U(C (D T)) with
V = then P = V P V :
If V 2 U(CP (D T)) then PV = P :
If V 2 U(C (D T )) and if V 2 E then V 2 CP (D T ):
If and are wavelets which lie in the same translation space, then
E = E C (D T) = C (D T)
and
CP (D T ) = CP (D T):
(vi) C (D T ) is a left module over fD T g0 and a right module over
CP (D T ):
(vii) If is a wavelet, then
C (D T) = fDg0 \ (fT g0 + B(H)P? ):
(viii)
(ix)
CP (D T) = w (V ) where V = (T):
W (D T) \ E is connected.
Proof. These follow as special cases of items 2.5 ! 2.8 and 2.11, 2.13, 2.16.
3. THE WAVELET SYSTEM
hD T i
33
We next show that if the Fourier transform of a wavelet has full support,
then nonscalar unitaries in fD T g0 always map out of its translation space.
So
U(fD T g0) and U(CP (D T))
act \disjointly" on in this case. We need a lemma.
Lemma 3.10. Let f be a function in L1 (R): Assume that f is 2-periodic
and 2-dilation-periodic. Then f is a constant (a.e.) function.
Proof. If f satises the hypothesis so do Ref and Imf: Hence it will suce
to assume that f is real-valued. Assume that f is not constant (a.e.). Then for
some real number c both sets
E := ft 2 (; ) : f(t) > cg
and
F := ft 2 (; ) : f(t) < cg
have positive Lebesgue measure. Either 0 < m(E) or 0 < m(F ) :
Assume the former. The proof for the latter is similar. Since E has positive
Lebesgue measure it contains points of density (22], p.261). That is, there are
points d0 2 E such that
; h d0 + h]) = 1:
lim m(E \ d0 2h
h !0 +
Fix such a point d0 and let n be large enough so that (d0 ; 2n d0 + 2n ) (; ) and also that
m(E \ (d0 ; 2n d0 + 2n )) > 34 2 2n :
Since f is 2-periodic, we have
m(ft 2 (2nd0 ; 2nd0 + ) : f(t) > cg) = m(E) :
However, since f is 2-dilation-periodic we also have
m(ft 2 (2nd0 ; 2nd0 + ) : f(t) > cg)
= 2nm(ft 2 (d0 ; 2n d0 + 2n ) : f(t) > cg)
= 2nm(E \ (d0 ; 2n d0 + 2n ))
> 1:7
a contradiction.
Theorem 3.11. Let be an orthogonal wavelet whose Fourier transformation
is non-zero a.e. Then
fD T P g0 = C I:
34
XINGDE DAI AND DAVID LARSON
c = Mh
Proof. Let W 2 fD T Pg0 be arbitrary. By Theorem 3.5, W
for some 2-dilation-periodic
function h: Since W commutes with P we have
P
W 2 P H so W = n2ZnT n for some (n) 2 l2 (Z): Thus
X
X
hb = nTbnb = ne;ins b a.e.
n2Z
n2Z
P
Let f be the 2-periodic function given by the sum n2Zne;ins where
convergence is in L2 0 2] with periodic extension to R: It follows that
b = f(s)(s)
b a.e.
h(s)(s)
and since b(s) 6= 0 a.e. we must have h(s) = f(s) a.e. Thus h is 2-periodic
as well as 2- dilation-periodic, and hence constant (a.e.) by Lemma 3.10. So
W 2 C I:
If has compact support, such as the Haar wavelet and Daubechies wavelet,
then b has an analytic extension to the complex plane, so the hypothesis of
Theorem 3.11 are satised.
Corollary 3.12. If b is non-zero (a.e.), let V 2 U(fD T g0) n C I: Then
V 2= E :
Proof. Suppose that V 2 fD T g0 and that V 2 E : Then Theorem 3.9
implies that P V = V P and thus V 2 fD T P g0: So the above theorem
implies that V 2 C I:
The method of multiresolution analysis is important in wavelet theory. For
references, see, for instance 19, 21] or 8].
Corollary 3.13. Let be an orthogonal wavelet with compact support. Sup-
pose that is generated by a multiresolution analysis. Let
V 2 U(fD T g0 ) n C I
and = V : Then is not generated by the same multiresolution analysis.
Proof. Wavelets 1 2 with the same multiresolution analysis are related in
that the Fourier transform of one can be obtained from the Fourier transform of
the other by multiplication by a 2- translation-invariant unimodular function.
(c.f. 8]. See also Notes 3.8 (ii).) Hence E1 = E2 :
The algebra fD T P g0 can be regarded as an operator algebraic unitary
invariant for wavelets. If does not satisfy the hypotheses of Theorem 3.11,
then fD T P g0 need not be trivial. For instance, if E is a wavelet set and E
is the corresponding s-elementary wavelet (see Chapter 4) then Pb = ME a
3. THE WAVELET SYSTEM
hD T i
35
b Tbg0 consists of multiplication operators,
multiplication operator. So since fD
in this case we have
fD T P g0 = fD T g0:
In other cases fD T P g0 may lie strictly between C I and fD T g0: (See
Corollary 3.18, Corollary 5.12 and Example 5.13.)
The methods of chapter 2 yield a generalization of Theorem 3.11.
Theorem 3.14. With the hypotheses of Theorem 2.16, if 2 W (U ) then
fU P g0 = fA 2 U 0 : 9 B 2 w (U0 ) with (A ; B) = 0g:
Proof. Let A 2 U 0: If there exists B 2 w (U0 ) with (A ; B) = 0 then
A = B = (B): So since A (B) 2 C (U ) and is separating for
C (U ) A = (B): Then A = (B ) so (A ; B ) = 0: So A 2 P H
and A 2 P H: It follows that A reduces P H so commutes with P as
required.
Conversely, if A 2 fU P g0 then A 2 C (U ) and AP = P A: Thus by
Theorem 2.16 there exists B 2 w (U0) with A = B:
Corollary 3.15. Let be an arbitrary orthogonal wavelet. Then A 2
fD T P g0 if and only if Ab = Mf where f is a bounded 2-dilation-periodic
function which coincides on the support of b with a 2-translation-periodic
function.
Proof. By Theorem 3.14, fA 2 D T P g0 i there exists B 2 w(T ) with
(A ; B) = 0: But A 2 fD T g0 implies Ab = Mf for some 2-dilation-periodic
L1 -function, and B 2 w (T) implies Bb = Mg for some 2-translationperiodic function, as required. (In case supp(b) = R this implies f = g which
then must be constant by Lemma 3.10, recovering Theorem 3.11.)
To establish density of span(W (D T)) we require a lemma.
Proposition 3.16 . The von Neumann algebra generated by
fMf : f 2 L1 (R) f is 2-periodicg and fMg : g 2 L1 (R) g is 2-dilation-periodicg
is
D := fMh : h 2 L1 (R)g:
Proof. It will be convenient to work with 1-periodic, rather then 2-periodic,
functions. If we let W = D2 then WD = DW and W Meins W = Me2ins so
W fMf : f is 2-periodic gW = fMf : f is 1-periodicg:
36
XINGDE DAI AND DAVID LARSON
So the problems are equivalent.
Let A be the von Neumann algebra generated by the 1-periodic and 2-dilation
periodic multiplication operators. First, suppose k 2 Z k 1 and let E k k+1) be an arbitrary measurable subset. Let n be the (nonnegative) integer
such that 2n 2n+1) contains k k+1) and dene a 2-dilation periodic function
g(s) on R by g(s) = s ; k for s 2 E g(s) = 0 for s 2 2n 2n+1) n E
g(s) = g(2;l s) for s 2 2l 2n 2n+1) l 2 Z and g(s) = 0 for s 0:
Now dene h(s) on 0 1) by h(s) = s and extend 1-periodically to R: Let
r(s) = h(s) ; g(s):
On each interval l l + 1) for l 2 Z l 1 the graph of g(s) is 0 or a
portion of a straight line. The only such interval on which the graph of g(s)
is a portion of a straight line of slope 1 is k k + 1): On each such interval
l l + 1) the graph of h(s) is a straight line of slope 1: So on each interval
l l + 1) l 1 l 6= k the graphs of g and h intersect in at most one point. On
k k + 1) the graphs of g and h coincide on E and dier on k k + 1) n E: On
0 1) the graph of g(s) is piecewise dened by straight lines, none of slope
1 and the graph of h(s) is a straight line of slope 1 so the graphs of g and h
intersect at countably many points on 0 1):
Since h(s) is nonzero except on the integers, the above paragraph shows
that r(s) = 0 on E and at countably many points not in E: Thus the kernel
projection of Mr is ME : So ME 2 A since Mr 2 A:
A construction analogous to the above shows that if k 2 Z with k ;2
and E k k + 1) then ME 2 A:
Next consider measurable subsets of ;1 1): If E is a subset of 2;(k+1) 2;k)
for some k 0 then an argument similar to the above shows that ME 2 A:
The same is true for a subset E of ;2;k ;2;(k+1)):
Every measurable subset of R is a union of countably many sets of the form
E considered above. So MF 2 A for every measurable set F R and hence
A = D:
Corollary 3.17 . The linear span of W (D T ) is dense in L2 (R):
b is nonzero a.e. Let
Proof. Let be any wavelet with the property that (s)
E be the linear span of the operators Mh with h of the form h = fg where f is
unimodular and 2-periodic, and g is unimodular and 2-dilation periodic. Then
Proposition 3.16 implies that D is the strong operator topology closure of E : For
any such h F ;1 (hb) is a wavelet. See Remark 3.6 and 3.7.] Thus F ;1(E b)
is a linear span of wavelets. Since b is nonzero a.e. it is cyclic for D: Thus
E b] = H and so F ;1 (E b)] = H as required.
We conclude with a problem.
Problem D. Suppose and are orthogonal wavelets such that jb(s)j = jb(s)j
a.e. Do there exist a 2-translation-periodic unimodular function h(s) and a
3. THE WAVELET SYSTEM
hD T i
2-dilation-periodic unimodular function g(s) such that
b
b(s) = h(s)g(s)(s)?
37
38
XINGDE DAI AND DAVID LARSON
CHAPTER 4
Wavelet Sets
We say that measurable sets E F are translation congruent modulo 2 if there
is a measurable bijection : E ! F such that (s) ; s is an integral
multiple of 2 for each s 2 E or equivalently, if there is a measurable partition
fEn : n 2 Zg of E such that fEn + 2n : n 2 Zg is a measurable partition
of F: Analogously, dene measurable sets G and H to be dilation congruent
modulo 2 if there is a measurable bijection : G ! H such that for each s 2 G
there is an integer n depending on s such that (s) = 2ns or equivalently,
if there is a measurable partition fGng1
;1 of G such that f2nGng1
;1 is a
measurable partition of H: ( Translation and dilation congruency modulo other
positive numbers of course make sense as well.)
Lemma 4.1. Let f 2 L2 (R) and let E = supp(f): Then f has the property
that feins f : n 2 Zg is an orthonormal basis for L2 (E) if and only if
(i) E is congruent to 0 2) modulo 2 and
(ii) jf(s)j = p12 a.e. on E:
Proof. If E 0 2) mod 2 then clearly
f p12 einsjE : n 2 Zg
is an o.n. basis for L2(E): If this is multiplied by a unimodular function it
remains a basis. This completes the \only if" part.
For the converse, suppose feinsf g is an o.n. basis. Firstly, assume by
way of contradiction that E is not translation congruent to a subset of 0 2)
modulo 2: Then there is a subset F E of nite positive measure such
that F + 2k E for some nonzero integer k: Replacing F with a subset if
necessary, we can assume F \ (F +2k) = and also that f jF and f jF +2k
are bounded. Let G = F (F +2k): Then G f 2 2 L2 (E) so we may expand
G f 2 =
X
39
neinsf
40
XINGDE DAI AND DAVID LARSON
P
with (n ) 2 l2(Z): Let g be the 2-periodic function on R dened by neins
on 0 2) and extended periodically. Then
G f 2 = fg (a.e.) on R:
Since f is nonzero on E this implies G f = E g (a.e.). So since g is 2-periodic,
f(s+2k) = f(s) (a.e.) on F: Let fn = eins f: Then also fn (s+2k) = fn(s)
(a.e.) on F: Hence each fn is orthogonal to the nonzero function
h = F ; F +2k :
This contradicts the fact that ffn : n 2 Zg is an orthonormal basis for L2 (E):
We have proven that E E 0 mod 2 where E 0 is a subset of 0 2):
Let : E 0 ! E be the bijection establishing congruency. Then
fn = eins(f )
is an orthonormal basis for L2 (E 0): Regarding these as functions in L2 0 2]
with support E 0 this implies
Z 2
0
jf j2eils ds = 0 l 2 Z l 6= 0:
Hence jf j must be constant on 0 2): This implies E 0 = 0 2) modulo
a null set, so E 0 2) mod 2 and also that jf j must be constant on E:
Since jf j is a unit vector, this constant is p12 as required.
Let us dene a measurable subset E R to be a wavelet set if p12 E
is the Fourier transform of a wavelet. We will call such wavelets s-elementary
wavelets. (The prex \s" is for \set".) We use the notation
bE := p1 E :
2
The classic example is given by the Littlewood-Paley orthonormal basis f2 n2 b(2n ;
l) : n l 2 Zg with b = bE for E = ;2 ;) 2): (c.f. 8], p.115). This
set E is translation-congruent modulo 2 to 0 2) since ;2 ;) + 2 =
0 ): So since f p12 eils : l 2 Zg is an orthonormal basis for L2 (0 2)) and
since Tb = Meis on R it follows that fTbl b : l 2 Zg is an orthonormal basis
for L2 (E): If f is a function with support in E then Dn f has support in
2;nE: Since Db = D;1 it follows that for each n fDb n Tbl b : l 2 Zg is an
orthonormal basis for L2 (2nE): Since the sets f2nE : n 2 Zg are disjoint and
have union R n f0g it follows that fDb n Tbl b : n l 2 Zg is an orthonormal basis
for L2 (R): That is, is a wavelet. From this argument, it is clear that if E is
any measurable set which is both translation congruent to 0 2) modulo 2
and has the property that f2n E : n 2 Zg is a partition of R (modulo a null set)
then E is a wavelet set. We will prove the converse. But rst, it is convenient
to describe how these properties are essentially related.
4. WAVELET SETS
We say that a measurable subset G partition of R if the sets
R
41
is a 2-dilation generator of a
2nG := f2ns : s 2 Gg n 2 Z
are disjoint and R n n 2nG is a null set. Also, we say that E 2-translation generator of a partition of R if the sets
E + 2n := fs + 2n : s 2 E g n 2 Z
are disjoint and R n n (E + 2n) is a null set.
R
is a
Lemma 4.2. A measurable set E R is a 2-translation generator of a
partition of R if and only if, modulo a null set, E is translation congruent to
0 2) modulo 2: Also, a measurable set G R is a 2-dilation generator of a
partition of R if and only if, modulo a null set, G is dilation congruent modulo
2 to the set ;2 ;) 2):
Proof. Suppose E is a translation generator. For each s 2 E let n(s) be
the unique integer with s ; 2n(s) 2 0 2) and dene
(s) = s ; 2n(s) s 2 E:
Disjointness of the sets (E +2n) implies that is 1-1 on E and the covering
of R implies that is onto 0 2): (Modulo null sets in both cases, of course.)
Conversely, if E 0 2) modulo 2 it is obvious that E is a generator.
Similarly, if G is a 2-dilation generator, then for each s 2 G s 6= 0 let n(s)
be the unique integer such that
2n(s)s 2 ;2 ;) 2)
and dene
(s) = 2n(s)s s 2 G:
Disjointness of f2l G : l 2 Zg implies that is 1-1 and covering implies is
onto ;2 ;) 2): The converse is obvious.
Lemma 4.3. Let E R be a measurable set. Then E is a wavelet set if and
only if E is both a 2-dilation generator of a partition (modulo null sets) of R and
a 2-translation generator of a partition (modulo null sets) of R: Equivalently, E
is a wavelet set if and only if E is both translation congruent to 0 2) modulo
2 and dilation congruent to ;2 ;) 2) modulo 2:
Proof. The \if" part is obvious from Lemma 4.2. For the converse, let E be a
wavelet set. Then
Db n bE = p1 2;n E :
2
42
XINGDE DAI AND DAVID LARSON
Since these are orthogonal the sets f2nE : n 2 Zg must be disjoint. It follows
that
fTbl bE : l 2 Zg = f p12 eils E : l 2 Zg
is an orthonormal basis for L2(E) and hence by Lemma4.1 that E is translationcongruent to 0 2) modulo 2: Since fDb n Tbl bE : n l 2 Zg is a basis the union
of the supports is full, and hence f2nE : n 2 Zg must be a partition of R:
Remark 4.4. If E is a wavelet set, and if f(s) is any function with support E
which has constant modulus p12 on E then F ;1 (f) is a wavelet. Indeed, by
Lemma 4.1 fTbl f :2 Zg is an orthonormal basis for L2 (E) and since the sets
2nE partition R it follows that fDb n Tbl f : n l 2 Zg must be an orthonormal
basis for L2 (R) as required.
Example 4.5. It is usually easy to determine, using the dilation-translation
criteria, whether a given nite union of intervals is a wavelet set. On the other
hand wavelet sets, suitable for testing hypotheses, can be quite dicult to construct. We present some of these, both for usefulness in the sequel, and for
perspective here. Items (ii) ! (ix) appear to be new.
(i) An example due to Journe (c.f. 8], p.136 ) of a wavelet which admits no
multiresolution analysis is the s-elementary wavelet with wavelet set
4 4
32
; 32
7 ;4) ; ; 7 ) 7 ) 4 7 ):
To see that this satises the criteria, label these intervals, in order, as J1 J2 J3 J4
and write J = Ji: Then
16 16 32
J1 4J2 4J3 J4 = ; 32
7 ; 7 ) 7 7 ):
This has the form ;2a ;a) b 2b) so is a 2-dilation generator of a partition
of R n f0g: Then also observe that
fJ1 + 6 J2 + 2 J3 J4 ; 4g
is a partition of 0 2):
set
(ii) The Littlewood-Paley set can be generalized. For any ; < < the
E = ;2 + 2 ; + ) + 2 + 2)
is a wavelet set. Indeed, it is clearly a 2-dilation generator of a partition of
R n f0g and to see that it satises the translation congruency criterion for
; < 0 (the case 0 < < is analogous) just observe that
f;2 + 2 ;2) + 4 ;2 ; + ) + 2 + 2 + 2)g
4. WAVELET SETS
43
is a partition of 0 2): It is clear that E is then a continuous (in L2 (R)norm) path of s-elementary wavelets. Note that
1 b
p
:
lim
=
E
!
2 24)
This is not the Fourier transform of a wavelet because the set 2 4) is not a
2-dilation generator of a partition of R n f0g: So
lim ! E
is not an orthogonal wavelet. (It is what is known as a Hardy wavelet because
it generates an orthonormal basis for H 2(R) under dilation and translation.)
This example demonstrates that W (D T ) is not closed in L2 (R):
(iii) Journe's example above can be extended to a path. For ; 7 7
the set
4 ) 4 + ) 4 + 4 4 + 4 )
J = ; 32
;
4
+
4)
;
+
;
7
7
7
7
is a wavelet set. The same argument in (i) establishes dilation congruency. For
translation, the argument in (i) shows congruency to 4 2 + 4) which is in
turn congruent to 0 2) as required. Observe that here, as opposed to in (ii)
above, the limit of J as approaches the boundary point 7 is a wavelet.
Its wavelet set is a union of 3 disjoint intervals.
(iv) While the Littlewood-Paley and the Journe wavelet sets are symmetric by
reection through the origin (modulo the boundary, which is a null set), the paths
in (ii) and (iii) consist of non-symmetric sets (except at 0). It is noteworthy that
paths of symmetric wavelet sets also exist: For example, consider for 0 3 4
4
2
F = ; 8
3 + 2 ;2) ; 3 ; 2 ; 3 + ) ; ; 3 ; )
4
4
8
2
3 + ) 3 ; 3 + 2) 2 3 ; 2):
We leave to the reader the (easy) verication that F satises the dilation
and translation congruency criteria so is a wavelet set. Note that F 3 is the
Littlewood-Paley set. We have
2 ) 2 ) 2 8 ):
;
2)
;
;
F0 = ; 8
3
3
3
3
(v) The wavelet set
7
5
13 7
13
; 2 ; 4 ) 5
4 ) 4 2) 2 3) 4 2 ) 6 2 )
is the union of 6 disjoint intervals, all but one of which are positive. This
illustrates that wavelet sets can be very asymmetric in structure.
44
XINGDE DAI AND DAVID LARSON
(vi) Let 0 < < < < < < 2 : The sets of item (ii) admit further
\splitting" into multiparameter families of wavelet sets:
E = ;2 ;2 + 2) ;2 + 2 ;) ; + ; + )
+ ) + 2) 2 + 2 2 + 2)
E
= ;2 + 2 ;2 + 2) ;2 + 2 ; + ) ; + ; + )
+ + ) + 2 + 2) 2 + 2 2 + 2)
E
= ;2 ;2 + 2) ;2 + 2 ;2 + 2) ;2 + 2 ;)
; + ; + ) ; + ; + ) + )
+ + ) + 2) 2 + 2 2 + 2)
2 + 2 2 + 2):
This process can be continued. It is perhaps curious that E and E
have 6 disjoint intervals, yet E
has 10: It will be shown (see example
A.4) that these arise naturally from operator-interpolation starting with the
Littlewood-Paley set E0 and the family fE : 0 < < 2 g:
(vii) Another curious easily-checked family of wavelet sets is
8 + 2) ; 4 + ; 2 )
;
G = ; 8
3 3
3
3
2
2
4
8
3 3 + ) 3 + 2 3 )
for 0 3 : Note that these are simple perturbations of the set E which
is used in the analysis of Meyer's family Me in Proposition 5.5. (We thank
Eugen Ionascu for this example.)
(viii) Let A 32 ) be an arbitrary measurable subset. Then there is a
wavelet set W such that W \ 32 ) = A: For the construction, let
and
Let
B = 2 3) n 2A
C = ; ; 2 ) n (A ; 2)
D = 2A ; 4:
W = 3
2 2) A B C D:
We have W \ 32 ) = A: Observe that the sets 32 2) A B C D are
disjoint. Also observe that the sets
1 B 2C D
3
2)
A
2
2
4. WAVELET SETS
45
are disjoint and have union ;2 ;) 2): In addition, observe that the
sets
3
2 2) A B ; 2 C + 2 D + 2
are disjoint and have union 0 2): Hence W is a wavelet set.
(ix) Let A ( 83 3) be an arbitrary measurable subset. Then there is
a symmetric (by reection through the origin) wavelet set W such that W \
( 83 3) = A: For the construction, let
B = ; 12 A + 2 and C = 2) n (2B 21 A):
We claim that the symmetric set
W = ;(A B C) (A B C)
satises our requirements. Observe that the sets A B C are disjoint and
contained in (0 1): Then observe that the sets 21 A 2B C are disjoint and
have union 2) so W is 2-dilation congruent to ;2 ;) 2) modulo
a null set. Then note that
1 A = ;B + 2 and 2B = ;A + 4:
2
So the sets ;A + 4 ;B + 2 C are disjoint and have union 2) and
the sets A ; 4 B ; 2 ;C are disjoint and have union ;2 ;): So W
is 2-translation congruent to ;2 ;) 2) and hence to 0 2): This
shows that W is a wavelet set. By the construction we have W \ ( 83 3) = A:
(x) Wavelet sets for arbitrary (not necessarily integral) dilation factors other
then 2 exist. For instance, if d 2 is arbitrary, let
2 )
;
A = ; d2d
+1 d+1
2
B = d2 ; 1 d 2
+ 1 )
2d2 )
C = d2d
+ 1 d2 ; 1
and let G = A B C: Then G is a d-wavelet set. To see this, note that
fA + 2 B C g is a partition of an interval of length 2: So G is 2-translationcongruent to 0 2): Also, fA B d;1C g is a partition of the set ;d ;) d) for = d22; 1 and = d22; 1 so from this form it follows that
fdnG : n 2 Zg is a partition of Rnf0g: Hence if := F ;1 ( p12 G ) it follows
that fd n2 (dn t ; l) : n l 2 Zg is an orthonormal basis for L2 (R) as required.
For dilation factors 1 < d < 2 a similar type of construction yields a d-wavelet
set. (We thank Puhong You for this example.)
46
XINGDE DAI AND DAVID LARSON
(xi) There exist unbounded wavelet sets. Let fAn : n = 0 1 2 g be a
measurable partition of 2): Then the sets fAng are disjoint, and for
;n
n 1 we have 2;nAn 0 ): Let B1 = 1
n=1 2 An : Then let
;n
B = 1
n=02 An
C = ;2 ;)n(B1 ; 2)
n ;n
D = 1
n=12 (2 An ; 2):
Note that the set D is unbounded. We leave to the reader the verication that
W = B C D
satises the dilation-translation congruency criteria so is a wavelet set. (We
thank Eugen Ionascu for this example.)
The following gives an operator algebraic characterization of s-elementary
wavelets. (See also Remark 4.4 above.)
Theorem 4.6. Let 2 W (UDT ): Let V = (T) be the unique unitary
operator in C (UDT ) with V = T: Then TV = V T if and only if
for some wavelet set E:
jbj = p12 E
Proof. Suppose E is a wavelet set. Dene a unimodular function h(s) on R
by setting h(s) = e;is for s 2 E and extending 2-dilation periodically by
h(s) = h(2;ns) s 2 2n E n 2 Z
and h(0) = 1: Then Mh 2 fD T g0 C (D T) and Mh b = Tbb so by
uniqueness we must have Mh = Vb : Since Tb = Me;is the operators Tb and
Mh commute, completing the \if" part.
For the converse, assume that T V = V T: By Lemma 3.1, V commutes
with D also. So V 2 fD T g0: Thus Vb is a multiplication operator, by
Theorem 3.5. Let 2 R: It follows that V commutes with T : Let
n l 2 Z: We compute
hV T Dn T l i = hT V Dn T l i
= hT Dn T l V i
= hT Dn T l;1 i:
By Lemma 3.2, Dn T ;1 = T;2;n Dn so Dn T l;1 = T;2;n Dn T l : Thus
hV T Dn T l i = hT T;2;n Dn T l i:
4. WAVELET SETS
Also,
It follows that
47
hT V Dn T l i = hT T Dn T l i
= hTT Dn T l i
= hT T ;1 Dn T l i:
hT (T;2;n ; T ;1 )Dn T l i = 0
for all n l 2 Z and 2 R: Now take the Fourier transform of this equation.
We have
hTb b (Tb;2;n ; Tb;1)Db n Tbl bi = 0 2 R
for each xed n l 2 Z: Since fTb : 2 Rg generates
D(R) = fMg : g 2 L1 (R)g
as a von Neumann algebra, the closed linear span of
fTb b : 2 Rg
is the set of square integrable functions on R with support suppb: Hence
supp((Tb;2;n ; Tb;1 )Db n Tbl b)
must be disjoint (a.e.) from supp(b) for each n l: We have Tb;2;n = Me2;n si
and Tb;1 = Mesi : For n 6= 0 the function (e2;n si ; esi ) is nonzero (a.e.). So
for n 6= 0
supp((Tb;2;n ; Tb;1 )Db n Tbl b)
b Since Db nTbl = Tb2;n l Db n and Tb2;n l = Me;2;n lsi coincides with supp(Db n Tbl ):
b = supp(Db nb):
supp(Db n Tbl )
b ;n(
)): So
Let E = supp(b): Then Db n b = 2; n2 (2
supp(Db n b) = 2nE:
So the above argument says that 2nE \ E = n 6= 0: Thus f2nE : n 2 Zg
is a disjoint family. Since fDb n Tbl bg is a basis for L2 (R) the union of their
supports must dier from R by at most a null set. So f2nE : n 2 Zg is a
partition of R:
Since fDb n Tbl b : n l 2 Zg is an orthonormal basis for L2(R) and f2nE :
n 2 Zg is a partition of R where E = suppb it follows that fTbl b : l 2 Zg is
an o.n. basis for L2 (E): Then Lemma 4.1 implies that
E 0 2) mod 2
and
jbj = p12 E as required.
48
XINGDE DAI AND DAVID LARSON
CHAPTER 5
Operator Interpolation of Wavelets
In this section we derive a method of operator-theoretic interpolation between
certain special pairs of wavelets, and between single wavelets and special families
of them. This generalizes Proposition 1.5 When applied to s-elementary wavelets,
this yields a new construction of a class due to Meyer, and generalizes that
class. While the s-elementary wavelets do not have \good" regularity properties
because their Fourier transforms are discontinuous, they can be basic building
blocks from which certain other wavelets with better regularity properties can be
derived. It is hoped that further work here may yield synthesis of more general
wavelets in terms of interpolation families of s-elementary wavelets.
If are wavelets let V := V be the (unique) unitary operator in
C (D T ) with V = : Suppose that V normalizes fD T g0 in the sense that
V (fD T g0)V = fD T g0:
This will happen (5.3) if and are s-elementary. In this case the algebra, before
closure, generated
by fD T g0 and V is the set of all nite sums (polynomials)
P
of the form An V n with coecients An 2 fD T g0: The closure in the strong
operator topology is a von Neumann algebra. If V n = I for some n this is isomorphic to the (nite-dimensional) cross-product (10], 14]) of fD T g0 under
the automorphism group induced by V: This has a special matricial form. Now
suppose further that every power of V is contained in C (D T): This occurs only
in special cases, yet it occurs frequently enough to yield some general methods.
Then since C (D T) is closed under left multiplication by fD T g0 by Theorem
3.9(vi), this \cross-product" is contained in C (D T ) so its unitary group
parameterizes a path-connected subset of W (D T ) that contains and via
the correspondence U ! U: We say that wavelets in this set are interpolated
from f g and that f g admits operator-interpolation. More generally, if
2 W (D T) is xed and F W (D T) is a family such that each V 2 F normalizes fD T g0 and
GroupfV : 2 Fg C (D T)
then if U is a unitary in the von Neumann algebra generated by fD T g0 and
49
50
XINGDE DAI AND DAVID LARSON
fV : 2 Fg we say that the wavelet U is interpolated from f Fg and
that ( F ) admits operator-interpolation.
We follow with some basics of operator interpolation for s-elementary wavelets.
Let E and F be wavelet sets. Let : E ! F be the 1-1 onto map implementing the 2-translation congruence. Since E and F both generate partitions
of R n f0g under dilation by powers of 2 we may extend to a 1-1 map of R
onto R by dening (0) = 0 and
(s) = 2n(2;n s) for s 2 2nE n 2 Z:
We adopt the notation EF for this, and call it the interpolation map for the
ordered pair (E F ):
Lemma 5.1. In the above notation, EF is a measure-preserving transformation
from R onto R:
Proof. Let := EF : Let R be a measurable set. Let n = \ 2nE n 2
Z and let En = 2;n n E: Then f n g is a partition of and we have
m((En )) = m(En ) because the restriction of to E is measure- preserving.
So
X
X
m(( n )) = m(2n (En))
m(( )) =
=
=
n
X
n
X
n
2 m((En )) = 2nm(En )
n
X
Xn
n
m(2 En) = m( n ) = m(
n
n
):
A function f : R ! R is called 2-homogeneous if f(2s) = 2f(s) for all
s 2 R: (More generally a-homogeneous means f(as) = af(s): ) Equivalently,
f is 2-homogeneous i f(2n s) = 2nf(s) s 2 R n 2 Z: Such a function is completely determined by its values on any subset of R which generates a partition
of R n f0g by 2-dilation. So EF is the (unique) 2-homogeneous extension of
the 2-translation congruence E ! F: The set of all 2-homogeneous measurepreserving transformations of R clearly forms a group under composition. Also,
the composition of a 2-dilation-periodic function with a 2-homogeneous function
is clearly 2-dilation periodic. These facts will be useful.
If : R ! R is a measure-preserving transformation, let U denote the
unitary composition operator dened by
U f = f ;1 f 2 L2 (R):
We write UEF := UEF : Clearly (EF );1 = FE and (UEF ) = UFE : We have
UEF bE = bF since EF (E) = F:
Theorem 5.2. Let E and F be wavelet sets. Then:
5. OPERATOR-THEORETIC INTERPOLATION OF WAVELETS
51
b Tb)
(i) UEF 2 CbE (D
b Tbg0
(ii) UEF normalizes fD
F
b Tbg00:
(iii) If E 6= F then UE 2= fD
Proof. (i) Write = EF and U = UEF : We have U bE = bF since (E) = F:
We must show
We have
U Db n Tbl bE = Db n Tbl U bE n l 2 Z:
(U Db n Tbl bE )(s) = (U Db n e;ils bE )(s)
= U 2; n2 e;il2;n sbE (2;n s)
;n ;1
= 2; n2 e;il2 (s) bE (2;n;1 (s)):
Note that E (2;n;1 (s)) = 1 i 2;n;1 (s) 2 E i s 2 (2n E) = 2n(E) =
2nF i 2;ns 2 F: Also, if 2;;nn ;s 1 2 F then 2;n s ; ;1 (2;;nns) = 2;n s ;
2;n;1 (s) 2 2Z: Thus e;il2 (s) bE (2;n ;1(s)) = e;il2 s bF (2;ns):
Also, we have
(Db n Tbl U bE )(s) = (Db n Tbl bF )(s) = 2; n2 e;il2;n s bF (2;n s):
b Tbg0 has the form Mh for
(ii) By Theorem 3.5 the generic element of fD
1
some 2-dilation periodic function h 2 L (R): Since UEF = U for = EF we have UEF Mh UFE = Mh
;1 : We have
(h ;1 )(2s) = h(;1 (2s)) = h(2;1(s)) = h(;1 (s)) s 2 R
so h ;1 is 2-dilation periodic, completing the proof.
(iii) Since E 6= F EF is not (a.e.) the identity map. So for some set E0 E
of positive measure and some nonzero integer n we have
EF (s) = s + 2n s 2 E0 :
b Tbg00: Then for each 2-dilation
Suppose by way of contradiction that UEF 2 fD
1
periodic function h 2 L (R) we have
UEF Mh = Mh UEF :
So Mh = UEF Mh UFE = Mh
;1 where = EF : Since h;=1 h ;1 for each
bounded 2-dilation periodic function h it follows that s(s) is an integral
multiple of 2 for each s 6= 0 (a.e.). Hence (ss) is an integral multiple of 2
(a.e.). It follows further, then, that there is a measurable subset E1 E0 of
positive measure, and an integer k such that (s) = 2k s for all s 2 E1: Then
2k s = s + 2n for all s 2 E1: Since n 6= 0 we must have k 6= 0: But this
forces E1 to be a singleton (hence null) set, a contradiction.
Corollary 5.3. C (D T ) is nonabelian for every 2 W (D T):
52
XINGDE DAI AND DAVID LARSON
b Tb) is nonabelian, for
Proof. Part (iii) of Theorem 5.2 shows that CbE (D
instance, for E = ;2 ;) 2): The result then follows from Proposition
1.8.
The above theorem shows that a pair (E F) of wavelet sets (or, rather,
their corresponding s-elementary wavelets) admits operator-interpolation if and
b Tb) since the
only if GroupfUEF g is contained in the local commutant CbE (D
b Tbg0 is automatically satised. It is easy
requirement that UEF normalizes fD
to see that this is equivalent to the condition that for each n 2 Z n is a
2-congruence of E in the sense that (n (s) ; s)=2 2 Zfor all s 2 E which in
turn implies that n (E) is a wavelet set for all n: Here = EF : This property
holds trivially if is involutive (i.e. 2 = identity).
In cases where \torsion" is present, so (EF )k is the identity map for some
b Tbg0 and U := UEF
nite integer k the von Neumann algebra generated by fD
has the simple form
f
k
X
n=0
Mhn U n : hn 2 L1 (R) with hn (2s) = hn(s) s 2 Rg
and so each member of this \interpolated" family of wavelets has the form
p1
2
k
X
0
hn(s)n (E )
for 2-dilation periodic \coecient" functions fhn(s)g which satisfy the necessary
and sucient condition that the operator
k
X
n=0
Mhn U n
is unitary.
A standard computation shows that the map sending
k k function matrix (hij ) given by
Pk M U n to the
0 hn
hij = h(ij ) ;(i;1)
where (i j) = (i+1) modulo k is a -isomorphism. (This matricial algebra
b Tbg0 by ad(UEF ).) So, for instance, if k = 3 then is the cross-product of fD
maps
Mh1 + Mh2 UEF + Mh3 (UEF )2
to
0 h1
h2
h3 1
@ h3 ;1 h1 ;1 h2 ;1 A :
h2 ;2 h3 ;2 h1 ;2
5. OPERATOR-THEORETIC INTERPOLATION OF WAVELETS
P
53
This shows that k0 Mhn U n is a unitary operator i the scalar matrix (hij )(s)
is unitary for almost all s 2 R: This Coecient Criterion yields interpolation
formulas.
The involutive case seems to be common. See examples A.1, A.4 and A.8
in the Appendix. Example A.4 shows that uncountable commutative groups of
involutive interpolation maps exist. Example A.9 shows that the case 3 =
identity is possible. Question: Is the case n = identity attainable for each
positive integer n? Let us say that a pair of wavelet sets (E F ) is an interpolation
pair if (EF )2 = identity. In this case EF = FE : More generally, it is natural
to dene an interpolation family of wavelet sets (based at E) to be a family E of
wavelet sets, \based" at a single special xed wavlet set E 2 E with the property
that
fEF : F 2 Eg
is a group. For a nite interpolation family the matricial cross-product form for
the von Neumann algebra generated by fD T g0 together with the corresponding
group of interpolation unitaries will yield interpolation formulas. (For an innite
family the von Neumann algebra may not be -isomorphic to the cross-product.)
Problem E. Characterize those groups which are isomorphic to groups of interpolation maps for interpolation families of wavelet sets.
The Coecient Criterion for the case k = 2 yields:
Proposition 5.4. If (E F ) is an interpolation pair then
b = h1(s)bE (s) + h2(s)bF (s)
(s)
()
is the Fourier Transform of an orthogonal wavelet whenever h1 and h2 are
2-dilation-periodic functions on R with the property that the matrix
h
1
h2
h2 EF h1 EF
( )
is unitary (a.e.).
We shall show that Meyer's (family of ) wavelets have the above form. Meyer's
class is (c.f. 8],p.117):
8 p1 is2 3s
e cos 2 (; 4 ; 1)] s 2 ; 83 ; 43 )
>
2
>
is
1
>
< p12 e is2 sin 2 (;3s23s ; 1)] s 2 ;2434; 23 )
bMe (s) = p2 e 2 sin 2 ( 2 ; 1)] s 2 3 3 )
>
p12 e is2 cos 2 ( 43s ; 1)] s 2 43 83 )
>
>
:0
otherwise
for s 2 R where is a real-valued function which satises the relation
(s) + (1 ; s) = 1 s 2 R:
54
XINGDE DAI AND DAVID LARSON
Normally, one chooses so that bMe has desired regularity properties. If is
taken with (s) = 0 for s 0 and (s) = 1 for s 1 then if is continuous,
or in class C k or C 1 then the function bMe is in the same class. Any choice
of a measurable real valued function satisfying (s) + (1 ; s) = 1 yields a
(perhaps \badly behaved") wavelet, however.
Proposition 5.5. The wavelets bMe have the interpolation form ():
Proof. Decompose the support of bMe as E F where E = ; 83 ; 43 ) 23 43 ) and F = ; 43 ; 23 ) 43 83 ): These are the wavelet sets E; 3 and
E 3 respectively, from 4.5(ii). Then (EF )2 = identity, as shown in A.1. For
this degenerate case we have
s + 4 s 2 ; 8 ; 4 )
F
E (s) = s ; 2 s 2 2 3 4 ) 3
3 3
Let
4
2 4
4 2
4 8
E; = ; 8
3 ; 3 ) E+ = 3 3 ) F; = ; 3 ; 3 ) F+ = 3 3 ):
Then
bMe = f bE + gbF with f dened on E by
f(s) =
(
e isis2 cos 2 (; 43s ; 1)] s 2 E; e 2 sin 2 ( 23s ; 1)] s 2 E+ and extended to R 2-dilation periodically by setting f(0) = 0 and f(s) =
f(2;n s) for s 2 2nE n 2 Z and similarly with g dened on R by
g(s) =
(
e isis2 sin 2 (; 23s ; 1)] s 2 F;
e 2 cos 2 ( 43s ; 1)] s 2 F+
and g(0) = 1 g(s) = g(2;n s) for s 2 2nF n 2 Z: With this extension
is (; 3s ; 1)] s 2 F ;
2
2
f(s) = ee is4 cos
sin 2 ( 43s ; 1)] s 2 F+ and
e is4 sin (; 3s ; 1)] s 2 E ;
g(s) = eis cos 2 ( 3s4;
1)] s 2 E :
2
2
We must show that (a.e.) the matrix
f(s) g(s)
g((s)) f((s))
+
5. OPERATOR-THEORETIC INTERPOLATION OF WAVELETS
55
is unitary, s 2 R where = EF : It will suce to verify this for s 2 E: We
have (E; ) = F+ and (E+ ) = F; : So on E; i(s+4)
+ 4) ; 1)]
(f )(s) = f(s + 4) = e 4 sin 2 ( 3(s 4
3s + 2)]
= ;e is4 sin 2 ( 4
3s ; 1))]
= ;e is4 sin 2 (1 ; (; 4
3s ; 1)]
= ;e is4 cos 2 (; 4
where we use the fact that (x) + (1 ; x) = 1 x 2 R: And on E+ ; 2) ; 1)]
(f )(s) = f(s ; 2) = ei(s;2) cos 2 (; 3(s 2
3s + 2)]
= eis cos 2 (; 2
3s ; 1)]:
= eis sin 2 ( 2
Similarly,
( is 3s
2
(g )(s) = e is2sin 2 ( ; 43s ; 1)] s 2 E;
;e cos 2 ( 2 ; 1)] s 2 E+ :
For convenience, let 1 = 2 (; 43s ; 1) and 2 = 2 ( 23s ; 1): Then for
s 2 E; we have
f(s)
g(s)
g((s)) f((s))
=
=
!
e is2 cos 1 e is4 sin 1
e is2 sin 1 ;e is4 cos 1
cos 1 sin 1 e is2 0is
sin 1 ; cos 1
0 e4
!
a product of two unitaries, hence unitary. Similarly, for s 2 E+ we have
f(s)
g(s)
g((s)) f((s))
=
=
!
e is2issin 2 eis cos 2
;e 2 cos 2 eis sin 2
sin 2 cos 2 e is2 0 :
; cos 2 sin 2
0 eis
Again each factor is unitary. The proof is complete.
We note that we have proven that Me satises the denition of an orthogonal wavelet without carrying out sensitive integral identities. No regularity
hypothesis on are needed for this.
56
XINGDE DAI AND DAVID LARSON
Proposition 5.5 shows that Meyer's wavelet can be obtained by interpolation
between a pair of wavelet sets by showing that it satises our \coecient criterion." But it does not give algorithms, or give insight into the special form of
bMe: We now address these matters. (See especially Example 5.13.)
Let (E F ) be an interpolation pair. The reason that the matrix criterion
() does not automatically yield algorithms is that the condition involves composition with EF : We have
b Tbg0 UEF ) CbE (D
b Tb)
w (fD
so the unitary group of this interpolation von Neumann algebra (denote it by
B(E F ) ) parameterizes the interpolated wavelets. We consider a special abelian
subgroup of this group which is particularly easy to parameterize. Let
B0 (E F) = B(E F ) \ (UEF )0 :
Then
b Tbg0 \ fUEF g0 UEF ):
B0 (E F) = w (fD
Theorem 5.2 (iii) shows that B(E F) is always nonabelian if E 6= F so the
inclusion B0 (E F ) B(E F ) is always proper.
Lemma 5.6. If h 2 L1 and if is a measure-preserving transformation of
R onto R then U Mh = Mh U if and only if h = h :
If f 2 L2 (R) then (U Mh f)(s) = h(;1 (s))f(;1 (s)) and
(Mh U f)(s) = h(s)f(;1 (s)): These are equal for all f i h = h ;1 i
h = h :
Proof.
Since a matrix
is unitary i both
a b
b a a b 2 C
ja + bj = 1 and ja ; bj = 1
for the abelian case condition () reduces to the condition that both
(h1 + h2 ) and (h1 ; h2)
are unimodular. So we have an algorithm:
Proposition 5.7. Let (E F ) be an interpolation pair of wavelet sets, and
let f and g be arbitrary 2-dilation- periodic unimodular functions on
f EF = f and g EF = g: Then
b = ( f +2 g )bE + ( f ;2 g )bF
( )
is the Fourier transform of a wavelet.
R
with
5. OPERATOR-THEORETIC INTERPOLATION OF WAVELETS
57
Proof. Apply the above discussion with h1 = (f + g)=2 and h2 = (f ; g)=2:
Remarks 5.8. (i) Note that Proposition 5.7 generalizes Proposition 1.5 in the
special case of wavelets. The form ( ) is clearly equivalent to the form
b(s) = ei(s)cos (s)bE + i sin (s)bF ]
for arbitrary real-valued measurable 2-dilation-periodic functions with
EF = and EF = :
(ii) If we drop the requirement in part (i) that EF = and only
require that be 2-dilation-periodic, then b is still the Fourier transform of a
wavelet. This is because arbitrary 2-dilation-periodic unimodular functions are
wavelet multipliers in the sense of Remark 3.7. The requirement on cannot be
dropped, however.
By construction, the coecient criterion () is invariant under 2-dilationperiodic wavelet multipliers. We next show it is also invariant (in the only
appropriate sense possible) under 2-translation-periodic wavelet multipliers.
Proposition 5.9. Let (E1 E2 ) be an interpolation pair of wavelet sets, and
let h1 h2 be 2-dilation-periodic functions on R such that
h (s)
1
h2(s)
h2 ((s)) h1((s))
is unitary (a.e.) on R where := EF : Let f be an arbitrary 2-translation-
periodic unimodular function.
Dene 2-dilation-periodic functions f1 f2 on R by setting fi = f jEi
and extending 2-dilation-periodically to R n f0g and setting fi (0) = 1: Let
ehi(s) = hi(s)fi (s) s 2 R i = 1 2: Then
e
eh2(s) !
h1 (s)
eh2((s)) eh1((s))
is unitary (a.e.) on R:
Proof. Since eh1 eh2 are 2-dilation periodic it is only necessary to verify that
the last matrix is unitary (a.e.) on E1: For s 2 E1 let k(s) be the unique
integer such that 2k(s)s 2 E2: Then 2k(s)(s) = (2k(s)s) 2 E1: We have
f1 (s) = f(s) and f2 (s) = f2 (2k(s)s) = f(2k(s) s): Also,
f1 ((s)) = f1 (2k(s)(s)) = f1 ((2k(s)s)) = f((2k(s) s)) = f(2k(s)s)
where we used dilation-periodicity of f1 translation-periodicity of f 2-homogeneity
of and the fact that f1 jE1 = f jE1 : In addition,
58
XINGDE DAI AND DAVID LARSON
f2((s)) = f((s)) = f(s)
since f2jE2 = f jE2 and acts as a 2-congruence on E2 : Hence for (a.e.)
s 2 E1
e
h (s)f(s) h (s)f(2k(s)s) eh2(s) !
h1 (s)
1
2
=
eh2((s)) eh1((s))
h2 (s)f(s) h1 (s)f(2k(s) s)
h (s) h (s) f(s)
0
1
2
=
h2 ((s)) h1 ((s))
0 f(2k(s) s) a product of two unitaries, hence unitary.
Sometimes parts of two wavelet sets can be combined to form a third wavelet
set.
Proposition 5.10. Let (E F) be an interpolation pair, and let G R be a
measurable set which is 2-dilation stable in the sense that 2G = G and which
is invariant under EF : Then
(G \ E) (Gc \ F)
is also a wavelet set.
Proof. Let
h1 = G and h2 = Gc = 1 ; G :
The hypotheses imply that hi is 2-dilation periodic and hi EF = hi i = 1 2:
Since (h1 + h2) and (h1 ; h2 ) are unimodular, an application of () or
Proposition 5.7 implies that
b := p1 h1 E + p1 h2F
2
2
is the Fourier transform of a wavelet. Since b = p12 K where K = (G \
E) (Gc \ F) K must be a wavelet set.
We next show that there are always nontrivial f and g satisfying the hypothesis of Proposition 5.7. The proof is in fact a construction.
Theorem 5.11. Let (E F) be an interpolation pair. Let
!(E F ) = fs 2 E : EF (s) 2 2n E for some nonnegative integer n:g:
Then !(E F ) has positive measure. If h(s) is an arbitrary bounded measurable
function on !(E F) then h extends uniquely to a 2-dilation periodic function
(which we still denote by h ) on R which satises the condition that h EF = h:
Proof. For each n 2 Z let
5. OPERATOR-THEORETIC INTERPOLATION OF WAVELETS
59
En = fs 2 E : EF (s) 2 2nE g:
Then fEn : n 2 Zg is a partition of E: We have
!(E F) = fEn : n 0g:
If s 2 En let se = 2;n EF (s) 2 E: Then
EF (se) = EF (2;n EF (s)) = 2;ns
so es 2 E;n: This argument is reversible. So
2;n EF (En ) = E;n n 2 Z:
Suppose !(E F ) is null. Then En is a null set for n 0: So EF (En) is
null since EF is measure-preserving. Thus
E;n = 2;n EF (En)
is null. But n2ZEn = E and m(E) = 2 a contradiction. Hence !(E F)
has positive measure.
Now let h(s) be an arbitrary bounded measurable function on !(E F):
First extend to E by setting, for s 2 E;n with n > 0
h(s) = h(2nEF (s)):
Then extend to R 2-dilation periodically by setting h(s) = h(2;l s) for s 2 2l E:
We claim h = h EF : If s 2 En with n > 0 then 2;n EF (s) 2 E;n: So by
2-dilation periodicity of h and the denition of the extension to E;n
h(EF (s)) = h(2;nEF (s)) = h(2n EF (2;n EF (s))) = h(s)
where we use
EF (2;n EF (s)) = 2;nEF (EF (s)) = 2;ns:
For s 2 E;n with n > 0 we have
So
2n EF (s) 2 En:
h(EF (s)) = h(2nEF (s)) = h(EF (2n EF (s)))
= h(2nEF (EF (s))) = h(2ns) = h(s):
Note that E0 = E \ F: So for s 2 E0 EF (s) = s: Thus the result for this
case is trivial.
Corollary 5.12. Let (E F ) be an interpolation pair of wavelet sets, and let be a wavelet obtained by the method of operator-interpolation () between E
60
XINGDE DAI AND DAVID LARSON
b is not a wavelet set, then f D T Pg0 lies strictly between
and F : If supp()
0
C I and fD T g :
Proof. We shall use the characterization of f D T P g0 given by Corollary
b E F: Let !(E F) be as in Theorem 5.11, let h 2
3.15. We have supp()
L1 (!(E F )) be arbitrary, with the 2-dilation extension given by the proposition
satisfying h EF = h: We claim that h coincides on E F with a 2-periodic
function. It will be sucient to show that if s 2 E F and if s+2n 2 E F for
some n 2 Z then h(s+2n) = h(s): If s 2 E then s+2n 2 E F implies
s + 2n 2 F and EF = s + 2n: Hence h(s + 2n) = h(EF (s)) = h(s): The
argument for s 2 F is similar. Since !(E F ) has positive measure, this shows
that fD T P g0 is nontrivial. If fD T P g0 = fD T g0 then by Corollary 3.15,
every 2-dilation-periodic function coincides on supp(b) with a2-translationb is contained in a wavelet set, so since
periodic function. This implies supp()
b is not a wavelet
is a wavelet, it must ll out the wavelet set. Thus if supp()
set, then fD T P g0 6= fD T g0:
Example 5.13. We give an example which demonstrates how wavelets with
special properties, such as bMe can be constructed algebraically from basic
elements of the interpolation theory without resorting to integral identities. Let
4 2 4
E = ; 8
3 ; 3 ) 3 3 ) = E; E+
and
2 4 8
F = ; 4
3 ; 3 ) 3 3 ) = F; F+
as in the proof of Proposition 5.5. Then (E F ) is an interpolation pair, and
EF (E; ) = E; + 4 = F+ EF (E+ ) = E+ ; 2 = F;:
We have !(E F) = ; 83 ; 43 ): Then Theorem 5.11 and Proposition 5.7 imply
that if f and g are arbitrary unimodular functions dened on ; 83 ; 43 ) then
bfg is (the Fourier transform of) a wavelet, where
8 p1 ( f (s)+g(s) )
>
2
2
>
1 ( f (2s);g(2s) )
>
p
< 2
2
bfg (s) = > p12 ( f (2s;4)+2 g(2s;4) )
>
p12 ( f (s;4);2 g(s;4) )
>
: 0
s 2 ; 83 ; 43 )
s 2 ; 43 ; 23 )
s 2 23 43 )
s 2 43 83 )
otherwise
So suppose (s) is an arbitrary measurable real-valued function on ; 83 ; 43 ):
5. OPERATOR-THEORETIC INTERPOLATION OF WAVELETS
61
Let f = ei g = e;i : Write b := bfg : Then
8 p1
>
cos (s)
2
>
i sin (2s)
>
p
< 2
b (s) = > p12 cos (2s ; 4)
>
pi sin (s ; 4)
>
: 02
s 2 ; 83 ; 43 )
s 2 ; 43 ; 23 )
s 2 23 43 )
s 2 43 83 )
otherwise.
Note that b cannot be continuous, for continuity would imply
8 ):
)
=
i
sin
(
;
lim4 ; cos (; 4
3
3
s!(; 3 )
Since (s) is real this implies sin(; 83 ) = 0 but also b (; 83 ) = 0 implies
cos (; 83 ) = 0: Hence cos (; 83 ) = 0 = sin (; 83 ) an impossibility.
Now dene k(s) := i for s 2 E; and k(s) := 1 for s 2 E+ and extend
2-periodically to R: Dene h(s) := ;i for s < 0 and h(s) := ;1 for s 0
a 2-dilation periodic unimodular function. Since h k are wavelet multipliers,
hkb is the Fourier transform of a wavelet. On E F
8 1
>
< i
k(s)h(s) = > ;
: ;;1i
So
s 2 E;
s 2 F;
s 2 E+
s 2 F+
8 p1
>
cos (s)
2
>
1 sin (2s)
>
p
< 2
(khb )(s) = > ; p12 cos (2s ; 4)
>
p1 sin (s ; 4)
>
: 02
s 2 E;
s 2 F;
s 2 E+
s 2 F+
otherwise.
This shows that Fourier transforms of wavelets in our class can be real-valued.
For this to be continuous, computations at s = ; 83 ; 43 43 would imply
4
8
cos (; 8
3 ) = 0 cos(; 3 ) = sin (; 3 )
and
; cos (; 43 ) = sin (; 83 )
an impossibility. So no (Fourier transform of a) wavelet hkb is can be continuous.
Next, dene a wavelet multiplier q by setting q(s) := e 2 for
s 2 F and
i(s;2)
2
extending 2-periodically to R: Then on E+ we have q(s) = e
= ;e is2 62
XINGDE DAI AND DAVID LARSON
and on E; we have q(s) = e
wavelet
i(s+4)
2
= e is2 : Then the (Fourier transform of a)
8 p1 is
>
e cos (s)
2
>
is
1
>
p
< 12 e is sin (2s)
b (s) := (qhkb )(s) = > p2 e cos (2s ; 4)
>
p12 e is sin (s ; 4)
>
: 0
s 2 E;
s 2 F;
2
s 2 E+
2
s 2 F+
otherwise.
can be continuous. If is any continuous real-valued function on ; 83 ; 43 ]
with (; 83 ) = 2 and (; 43 ) = 0 then b is continuous on R and will be
in class C k if (s) 2 C k and if the rst k right/left derivatives of (s) vanish
at these two points. Wavelets of Meyer's class bMe are of this form. Indeed, if
we let
3s ; 1)
(s) = 2 (; 4
then
3s ; 1)] and sin (s ; 4) = cos ( 3s ; 1)]
cos (2s ; 4) = sin 2 ( 2
2 4
2
2
where we use the property (s) + (1 ; s) = 1 hence
b = bMe:
Example 5.14. Meyer's class Me eectively demonstrates Corollary 5.12.
We have
2 ) 2 8 )
supp(Me ) = E F = ; 8
;
3 3
3 3
with E F as in Example 5.13. We have
4 ):
;
!(E F) = ; 8
3 3
If h is an arbitrary bounded function on ; 83 ; 43 ) note that ; 83 ; 43 ) +
4 = 43 83 ) E F and extend h by dening h(s) = h(s ; 4) for
s 2 43 83 ): Then note that K := ; 83 ; 43 ) 43 83 ) is a 2-dilation generator
of a partition of Rnf0g and extend h to R by h(s) = h(2;ns) s 2 2nK
n 2 Z and h(0) = 0: Then h is 2-dilation-periodic and if s 2 ; 43 ; 23 )
then s+2 2 23 43 ) hence 2s+4 2 43 83 ): So h(s+2) = h(2s+ 4) =
h(2s) = h(s) as required. The reasoning is similar for s 2 23 43 ): So if
k(s) is dened on ; 83 ; 23 ) a 2-translation generator of a partition of R
by k(s) = h(s) and extended 2-periodically to R then k(s) = h(s) also
on 23 83 ): That is, h agrees with k on E F: Thus by Corollary 3.15,
b Tb Pbg0: Moreover, by the uniqueness part of this construction, every
Mh 2 fD
b Tb Pbg0 has this form.
operator in fD
The following is simple but useful.
5. OPERATOR-THEORETIC INTERPOLATION OF WAVELETS
63
Proposition 5.15. Let (E F ) be an interpolation pair of wavelet sets. Suppose
h1 h2 are as in () . Then the wavelet b in () satises (a.e.)
8 1 s2E nF
>
< 221 s 2 F n E
2
F
2
b
b
j(s)j + j(E (s))j = > 1 s 2 E \ F
>
: 0
otherwise
Note: this does not imply that b is discontinuous, since the discontinuity of
jbj2 + jb EF j2 can be due entirely to the discontinuity of EF : This is the case
for Me in particular.]
Proof. We have
8
>
<
b(s) = >
:
p12 h1 (s)
s2E nF
p12 h2 (s)
s2F nE
p12 (h1 (s) + h2 (s)) s 2 E \ F:
If s 2 E n F then EF (s) 2 F n E: So
b(EF (s)) = p1 h2 (EF (s)):
2
Since
h
h2
1
h2 EF h1 EF
is a unitary matrix a.e., we must have
jh1(s)j2 + jh2 (EF (s))j2 = 1 a.e.
Thus
jb(s)j2 + jb(EF (s))j2 = p12 a.e.
Similarly, if x 2 F n E the same equality holds. If s 2 E \ F then EF (s) = s:
The matrix () is then
h (s) h (s) 1
2
h (s) h (s) :
2
1
The condition that this is unitary implies that jh1(s) + h2(s)j = 1 hence
jb(s)j2 = 21 = jb(EF (s))j2 :
Let @(K) = K \ K c denote the boundary of a set K:
Corollary 5.16. Let (E F) be an interpolation pair. Suppose E F are nite
unions of intervals. A necessary condition for there to exist h1 h2 satisfying
() such that b of () is continuous on R is that
@(E F ) \ EF (@(E F)) = :
64
XINGDE DAI AND DAVID LARSON
Proof. If b is continuous then b must vanish on @(E F): If s 2 E F
then Proposition 5.15 shows that b cannot vanish at both s and EF (s): So
either
s 2= @(E F)
or
EF (s) 2= @(E F ):
Now suppose s0 2 @(E F) is arbitrary. Since E and F are wavelet sets which
are nite unions of intervals, s0 must be a limit of points sn 2 E F on which
(EF (s) ; s) is constant. Then
EF (sn ) ! EF (s0 )
so
b(EF (sn )) ! b(EF (s0 )):
Also
b(sn ) ! b(s0 ):
Thus
jb(EF (sn ))j2 + jb(sn )j2 ! jb(EF (s0 ))j2 + jb(s0 )j2:
By Proposition 5.15,
jb(EF (sn ))j2 + jb(sn )j2 21
b 0) = 0 this implies that (
b EF (s0)) 6= 0: Thus
for all n: Since (s
EF (s0 ) 2= @(E F ):
The above Proposition raises some questions: (1) Is the necessary condition
in Corollary 5.16 sucient? (2) Can the hypothesis that E F are nite unions
of intervals be removed?
Example 5.17. The class of interpolation pairs given in A.1 of the Appendix
gives a good demonstration of Corollary 5.16 and yields a natural generalization
of the class Me: For 0 < < 3 follow the scheme of Example 5.13, letting
E = E; = ;2 ; 2 ; ; ) ; 2 ; 2)
and
F = E = ;2 + 2 ; + ) + 2 + 2):
The description of b (s) is analogous, with the exception that on E \ F =
;2 + 2 ; ; ) + 2 ; 2) b (s) = cos (s) + i sin (s): Then
jb (s)j = 1 on E \ F: Dene k(s) analogously, with the exception that on
5. OPERATOR-THEORETIC INTERPOLATION OF WAVELETS
65
E \ F let k(s) = e is2 (b (s));1 : Let h(s) and q(s) be as in Example 5.13.
Then b := qhkb has the form
8 p1 is2
> 2 e cos (s)
s 2 ;2 ; 2 ;2 + 2)
is
>
1
p2 e 2 >
s 2 ;2 + 2 ; ; )
>
1 e is
>
p
2 sin (2s)
s 2 ; ; ; + )
< 2
b (s) = > p12 e is2 cos (2s ; 4) s 2 ; + )
>
p12 e is2 s 2 + 2 ; 2)
>
>
is
1
p 2
>
2 2 ; 2 2 + 2)
>
: 02 e sin (s ; 4) sotherwise.
As in Example 5.13, if is any continuous real-valued function on ;2 ;
2 ;2 + 2) with (;2 ; 2) = 2 and (;2 + 2) = 0 it is easily
checked that b (s) is continuous on R: For small > 0 b (s) can be thought
of as a wavelet obtained from the Littlewood-Paley wavelet multiplied by e is2
(a wavelet by Remark 4.4) by \rounding down the corners" appropriately. Also
note that by modifying k(s) we can replace p12 eis on E \ F in the description
of b (s) by any function (s) of uniform modulus p12 that is continuous on
the closure of E \ F and which takes the same values at the endpoints, and still
achieve continuity of b : We can achieve regularity properties by appropriate
choices of the parameters, as with bMe :
We conclude with a problem.
Problem F. If (E F ) is an interpolation pair of wavelet sets, suppose f g
are functions in L1 (R) such that
f bE + gbF
is the Fourier transform of a wavelet. Are there 2-dilation-periodic functions
h1 h2 2 L1 (R) with
b = h1 bE + h2bF
such that fh1 h2g satisfy the Coecient Criterion ()?
66
XINGDE DAI AND DAVID LARSON
CHAPTER 6
Concluding Remarks
6.1. Unitary Equivalence
Unitary representations 1 2 of a group G on Hilbert spaces H1 H2
are called equivalent if there is a unitary operator W 2 B(H1 H2) such that
W1 (g)x = 2 (g)Wx g 2 G x 2 H1 :
Let G0 be an allowable subset of G in the sense of the remark after Example
1.11, and suppose 1 2 are wandering vector representations of (G G0) with
complete wandering vectors 1 2 respectively for 1(G0) 2(G0 ): There is a
natural equivalence relation on pairs ( ) which extends the usual equivalence
relation for representations, namely, require that the unitary W implementing
the equivalence of 1 and 2 satisfy W1 = 2: If is xed, then the
corresponding induced equivalence relation on W ((G0 )) is that 1 2
i there exists V 2 U((G0 )0 ) with V 1 = 2 : (This is analogous to the
equivalence of MRA's introduced in 15], with scaling functions i in place of
the i : ) This is equivalent to the condition that the vector states !1 and !2
agree on C ((G )): If U(C1 ((G0))) 6= U((G0)0 ) as is most often the case
by Proposition 1.7, then there exist inequivalent complete wandering vectors, as
happens for hD T i: An application of Proposition 1.3 then shows that the set
of equivalence classes of W ((G0 )) can be parametrized by the coset space
U(C ((G0 )))=U((G0 )0 )
for any single choice of 2 W ((G0 )): This raises a natural Question: If
U B(H) is a unitary system, when is U(U 0 ) normalized by every element of U(C (U ))? Equivalently, when is U(U 0 ) a normal subgroup of
Group(U(C (U )))? Is this true for the wavelet case U = UDT ? Theorem
5.3 sheds some light in that it shows that if = E is s-elementary, then the
interpolation unitaries in C (D T) all normalize U 0 = fD T g0:
In a related direction, if and are inequivalent elements of W (U ) then
there is an element V 2 Group(U ) such that hV i 6= hV i: So the equiv67
68
XINGDE DAI AND DAVID LARSON
alence classes of W (U ) are completely parameterized by the vector functions
! : Group(U ) ! C
given by ! (V ) = hV i: As noted in the introduction, we have
Group(UDT ) = fDn T : n 2 Z 2 D g:
So the function : Z D ! C given by (n ) = hDn T i is a complete
unitary equivalence invariant for wavelets. Two rather immediate properties are:
(1) jbj = jbj if and only if (0 ) = (0 ) 2 D and (2) jbj = p12 E
for some wavelet set E if and only if (n ) = 0 whenever n 6= 0: It may be
worthwhile to explore this invariant further.
6.2. Higher Dimensional Systems
Let m be a positive integer, and let Tk be the unitary operator of translation
by 1 in the kth coordinate direction on L2 (Rm ): Let A be an m m invertible
matrix with real entries. Then the operator D : L2 (Rm ) ! L2 (Rm ) dened by
(Df)(x) = jdetAj 21 f(Ax) x 2 Rm f 2 L2 (Rm )
is easily shown to be unitary. If A is a strict dilation in the sense that A;1
is a strict contraction (i.e. kA;1 k < 1), or more generally if the eigenvalues of
A all have modulii strictly greater than 1 (so D is similar to a strict dilation),
then A dilates the unit ball B1 of Rm in the sense that
m
n
1
n=1A B1 = R :
In this case it can be shown that the unitary system
UDT1T2 Tm := fDn T1l1 T2l2 Tmlm : n li 2 Z 1 i mg
has complete wandering vectors. There are also called orthogonal wavelets, as
in the 1-dimensional case. ( For all choices of A orthogonal wavelets exist. For
some choices of A but not for all, it can be shown that such wavelets exist
which have \good" regularity properties, so are useful in applications.) Using
the n-dimensional Fourier transform (which is the tensor product of n copies
of the 1-dimensional Fourier transform) essentially all of Chapter 3 generalizes
to this setting. Indeed, since the unitaries Ti commute the unitary system
UDT1T2 Tm factors as the product of two abelian groups, and so the structural
results of Chapter 2 all apply, yielding analogs of Lemma 3.1 and Theorem 3.9,
exactly as in Chapter 3. We leave proofs of the details and extensions of the
other results, to the reader. The denition of wavelet set, and of s-elementary
wavelet, makes sense for these dilation-translation systems in Rm and it can
be shown that they always exist. (A proof of existence of wavelet sets for general
A has been obtained by the present authors together with D. Speegle, who is at
present a graduate student at Texas A&M University. In addition, Speegle has
obtained a proof that the family of s-elementary wavelets is a connected subset
6.3. MULTIRESOLUTION ANALYSIS
69
of the unit ball of L2 (Rm ): ) Much of the interpolation theory of Chapter 5
(formally) adapts to this setting, and we leave details to the reader.
6.3. Multiresolution Analysis
The method of multiresolution analysis is a very important technique for
deriving wavelets. In fact, in many respects the formulation of MRA (c.f. 8, 19,
21]) captures the \essence" of wavelet theory. The MRA technique is a method
of starting with a suitably chosen unit vector for which the set fT l : l 2 Zg
is a Riesz basis for its closed span in L2(R) and using it to derive a complete
wandering vector for UDT : Suitably \chosen" means that the 2-scale relation
D;1 2 spanfT l : l 2 Zg is satised and that spanfDn T l : n 2 Z+ l 2
Zg = L2 (R) where Z+ = f0 1 2 g: This method can be given an operatortheoretic formulation. (c.f. 6, 15]). However, its connection with the local
commutant, which is the main new tool used in this paper, is at most indirect.
The reason for this is that since fDnT l : n 2 Z+ l 2 Zg is a semigroup (apply
Lemma 3.2) we have C (UDT ) = fD T g0 (see the remark after Lemma 1.1),
so two inequivalent scaling functions cannot be related via a unitary in the local
commutant of fD T g at one of them.
6.4. A Connection With Some Work of Guido Weiss
Several months after this manuscript was submitted we discovered that there
is a connection between some of the work we have presented in this article and
some of the work of Guido Weiss and his group who have been working on a
program of a unied approach to wavelet theory via the Fourier transform. We
learned of this connection in late spring 1995, when our colleagues Charles Chui,
Bill Johnson, and John Mc Carthy saw talks each of us gave separately, and
alerted both Larson and Weiss. We thank our colleagues very much for pointing
this out. In the past year we have had some fruitful interaction between our
respective groups concerning this.
It turned out that the class we call s-elementary wavelets in Chapter 4 were
also discovered completely independently, and in about the same time period, by
Professor Weiss, his colleague E. Hernandez (U. Madrid), and Weiss' students X.
Fang and X. Wang. They called them MSF wavelets, and they were introduced in
a series of three papers, the rst of which is due to Fang and Wang (Construction
of minimally supported frequency wavelets, J. Fourier Anal. Appl. 2 (1996)
315-327) and the second and third to Hernandez, Wang and Weiss (Smoothing
minimally supported frequency (MSF) wavelets: Parts I and II, J. Fourier Anal.
Appl. (1996)). The rst of these was submitted within two days of submission of
this memoir. As we noted in Remark 4.4 , any function supported on a wavelet
set which is unimodular on the wavelet set is the Fourier transform of a wavelet.
The class of MSF wavelets were dened to include this more general type also.
So our wavelet sets are the just the support sets of Fourier transforms of MSF
70
XINGDE DAI AND DAVID LARSON
wavelets.
The reasons that the two groups were led to consider this special class of
wavelets were dierent. In Professor Weiss' program techniques were developed
to \smooth" appropriate MSF wavelets to produce new wavelets with continuous
Fourier transform, and this led to new results concerning multiresolution analysis
(MRA) wavelets. In our case, we greatly needed concrete examples of pairs of
wavelets with simple enough structure to enable us to experiment by hand with
operator computations, and we found that we could modify the Littlewood-Paley
set.
However, there is some denite overlap in the results we obtained. The characterization in this paper (Lemma 4.3) of wavelet sets as measurable sets which generate measurable partitions of the line under both 2-dilation and 2-translation
is also a result of Fang and Wang. They also obtained an example of an unbounded wavelet set. It is dierent from Example 4.5(xi) in this paper. There
are some overlaps in the classes of examples of wavelet sets constructed, but there
are also big dierences. The smoothing technique of Weiss and our method of
interpolation have apparently very little in common, yet they lead to some of the
same classes of wavelets. Both give new ways of obtaining Meyer's family: ours
by interpolating between two wavelet sets and their's by smoothing Shannon's
wavelet. At the same time, each seems to yield classes that apparently cannot
be obtained using the others' techniques. Most notably, we seem to have been
giving our research students some of the same wavelet problems to work on. This
fact became apparent during a conference last summer (see 6.7.3). We note that
the connectedness problem (i.e. Problem A in this paper) was also raised by
Professor Weiss.
We remark that much of the above mentioned work of Professor Weiss and his
group (but not the overlap with ours) are detailed in the recent excellent book
(E. Hernandez and G. Weiss, A First Course on Wavelets, CRC Press, Boca
Raton). Professor Weiss mentioned to us last summer that, because of the short
timing involved in our realization of the extent of overlap between our groups,
it was not possible to insert appropriate comments concerning this in their book
before it appeared. He asked us to instead insert a description of this matter in
our memoir before publication. We thank him for his support of our work.
6.5. Status of Problems
In this manuscript we had posed a number of problems, with the main ones
designated by the letters A - F. In the past year, problems B, C, D, the nite
group case of E, and F have been solved, although they seemed hopelessly out
of reach at the time this paper was written. This reveals something about how
rapidly the theory is evolving. Problem A is the connectedness problem, and at
the time of this writing it is still open, although much positive progress has been
made. This problem was also raised independently by Professor Weiss. D. Han
6.6. EXAMPLES
71
and V. Kamat in their thesis work at Texas A&M University, and independently
W.S. Li, J. Mc Carthy and D. Timotin, have proven that Theorem 2.16 is valid
when U0 is nonabelian. Both groups used this result to anwer Problem B negatively. Problem C was answered negatively by D. Han, and independently by Li,
McCarthy and Timotin the counterexamples are dierent. Problem C0 and D
were answered negatively by Dai, Gu, Larson and Liang. The nite group case
of Problem E was answered positively by Q. Gu in his thesis work at A&M he
showed that every nite group is attainable. Problem F was answered positively
by Gu and Larson.
6.6. Examples
The referee of this manuscript kindly suggested that we include some additional material, and in particular it was strongly suggested that we include
some concrete examples of wavelet sets in the plane. The existance proof, mentioned above, is contained in the preprint (\Wavelet sets in Rn ", by X. Dai,
D. Larson and D. Speegle, which is to appear in J. Fourier Anal. Appl.). It
is basically constructive, but the constructive technique does not directly yield
any examples of \elegance". So we follow the referee's advice and include two
examples that we worked out (except for the graph) shortly after this manuscript
was accepted for publication in October 1995. We remark that, motivated by
our existance result, other examples have been worked out by P. Soardi and D.
Weiland, and Q. Gu and D. Speegle, and perhaps others. In fact, the graph
we inserted in Example 6.6.1 (but not the example itself) was inspired by the
beautiful graphs in the recent preprint by Soardi and Weiland entitled \Single
wavelets in n-dimensions" in which precise methods are detailed for constructing a family of MSF wavelets (and hence wavelet sets) with fractal-like nature
in the plane and in higher dimensions for dilation factor 2I: In addition, we note
that Gu and Speegle have shown (not yet published) that interpolation pairs of
wavelet sets exist in the plane for certain matricial dilation factors, and hence
that single-function wavelets exist in L2 (R2 ) which are not MSF wavelets.
Example 6.6.1 Consider the dilation matrix A = 2I, where I is the identity
matrix on R2 : For n 2 N dene 2-dimensional vectors ~ n ~n by:
~ n := 4n1;1 ( 2 2 ) 2 R2
n
X
~n :=
~ k k=1
~0 := 0:
72
XINGDE DAI AND DAVID LARSON
Dene
G0 = 0 2 ] 0 2 ]
Gn = 41n G0 + ~n E =
1
k=1
Gk 2G0 nG0
C = G0 E + (2 2)
B = 2G0n(G0 E):
Dene
A1 = B C
A2 = f(;x y) : (x y) 2 A1g
A3 = f(;x ;y) : (x y) 2 A1 g
A4 = f(x ;y) : (x y) 2 A1g
W1 = A1 A2 A3 A4 :
Then the set W1 is a 2-dimensional wavelet set for the system UDT1 T2 in
section 6.2, where (Df)(x) = jdetAj1=2f(Ax) = 2f(2x) x 2 R2 f 2 L2 (R2 ):
Proof. We have
1
4E = (4G1) (
k=2
4Gk )
= (G0 + (2 2)) (
= (2 2) + G0 E
= C:
Since
1 1
~
4k;1 G0 + k;1 + (2 2))
k=2
E B = 2G0nG0
is a 2-dilation generator for the rst quadrant, and 4E = C so A1 is a 2-dilation
generator for this. So W1 is a 2-dilation generator for the 2-dimensional plane
R2:
Also, it is clear that W1 is congruent modulo translations along the coordinate
axes by integral multiples of 2 to the set ; ] ; ]:
The following graph was inspired by the graphs of Soardi and Weiland, as
noted above.
6.6. EXAMPLES
73
6
-
Figure 1: Wavelet set W1 in R2
Example 6.6.2 Let A and E be as in Example 6.6.1. For n 2 N dene
2-dimensional vectors ~ n ~n by:
~ n := 4n1;1 ( 2 0) 2 R2 n
X
~ k ~n :=
k=1
~0 := 0:
Dene
G0 = 0 2 ] ; 2 2 ]
Gn = 41n G0 + ~n E =
1
k=1
Gk 2G0nG0
C = G0 E + (2 0)
B = 2G0n(G0 E):
Dene
A1 = B C
A2 = f(;x y) : (x y) 2 A1 g
W2 = A1 A2:
Then the set W2 is a 2-dimensional wavelet set.
74
XINGDE DAI AND DAVID LARSON
Proof. We have
1
4E = (4G1) (
k=2
4Gk)
= (G0 + (2 0)) (
= (2 0) + G0 E
= C:
1 1
~
4k;1 G0 + k;1 + (2 0))
k=2
Since
E B = 2G0nG0
is a 2-dilation generator for the right half plane, and 4E = C so A1 is a 2-dilation
generator for this. So W2 is a 2-dilation generator for the 2-dimensional plane
R2:
Also, it is clear that W2 is congruent modulo translations along the coordinate
axes by integral multiples of 2 to the set ; ] ; ]:
6.7. Acknowledgments
6.7.1. We wish to thank the referee for some excellent suggestions for additional further directions of research that we were either unaware of or only
partially aware of. We take the opportunity to note that some of these ideas
have signicantly inuenced the subsequent projects we have undertaken.
6.7.2. The work in this manuscript was begun in 1992 when the rst author was a summer participant in the NSF MSRG] funded Workshop in Linear
Analysis and Probability at Texas A&M University.
6.7.3. The work in this manuscript was presented, in various stages of development, in a number of conference talks including AMS Special Sessions in
October 1992 (Dayton), January 1993 (San Antonio), October 1993 (College Station), January 1994 (Cincinnati), and hour talks at GPOTS-94 (U. Nebraska)
and SEAM-95 (Georgia Tech). Two talks by the second author on this and
subsequent work were given at a NATO Advanced Study Institute on \Operator Algebras and Applications", August 1996 (Samos, Greece). (An expository
conference proceedings article is available entitled \Von Neumann algebras and
wavelets".) Special credit is due to the AFOSR, NSA and UNCC for funding a
conference in April, 1993 (UNC-Charlotte) which featured early stages of this
work among work of others. Credit is also due to the AFOSR, NSF and UNCC
for sponsoring a similar conference on operator theory and wavelet theory in
July, 1996, in Charlotte which promoted interaction between our research group
and the group of Professor Weiss.
Appendix: Examples of Interpolation Maps
The class of interpolation maps between wavelet sets exhibits many dierent
types of algebraic structural properties that are relevant to this article. We give
some examples illustrating these properties. Many are referenced as counterexamples in this article. These examples also serve to illustrate the manner in
which one can compute with interpolation maps.
Consider the path of wavelet sets of Example 4.5 (ii) given by
E = ;2 + 2 ; + ) + 2 + 2)
for ; < < : The interpolation maps 12 := EE12 have some useful
properties. First consider the special case where 1 = ; and 2 = : Here
the union E1 E2 is the symmetric set
;2 ; 2 ; + ) ; 2 + 2):
Example A.1. We will show that ; is involutive for in the range 0 3 : Write := ; : To compute we can partition
E; = A1 B1 C1 D1 and E = A2 B2 C2 D2
where
A1 = ;2 ; 2 ;2 + 2) B1 = ;2 + 2 ; ; )
C1 = ; + ) D1 = + 2 ; 2)
and
A2 = ;2 + 2 ; ; ) B2 = ; ; ; + )
C2 = + 2 ; 2) D2 = + 2 ; 2):
Then
A1 + 4 = D2 B1 = A2 C1 ; 2 = B2 D1 = C2
and we also have the arithmetic relations 2B2 = A1 and D2 = 2C1: Thus on
A1 we have (s) = s + 4 2 D2 on C1 we have (s) = s ; 2 2 B2 and
75
76
XINGDE DAI AND DAVID LARSON
(s) = s for s 2 B1 D1 : So for s 2 A1 since (s) 2 D2 and D2 = 2C1
we have 21 (s) 2 C1 so using the 2-homogeneity of we have
2(s) = 2( 21 (s)) = 2( 21 (s) ; 2) = (s) ; 4 = s + 4 ; 4 = s:
Similarly, for s 2 C1 we have 2(s) 2 A1 so
2(s) = 12 (2(s)) = 21 (2(s) + 4) = (s) + 2 = s ; 2 + 2 = s:
On B1 D1 we have (s) = s so 2 (s) = s: Thus 2(s) = s for all s 2 E; and hence 2(s) = s for all s 2 R since is determined by its restriction to
E;:
Example A.2. Next consider the example above, but for in the range
< : We will show that in this case is not involutive,and moreover, the
3
2
restriction of 2 to E; is not a 2-congruence. In addition the orbit of a point
in E; can be unbounded. Here a dierent (from that of A.1) partitioning is
required to compute ; : Let
A1 = ;2 ; 2 ;3 + ) B1 = ;3 + ; ; )
C1 = ; 2) D1 = 2 2 ; 2)
and
A2 = ;2 + 2 ;2) B2 = ;2 ; + )
C2 = + 3 ; ) D2 = 3 ; 2 + 2):
We have
A1 + 2 = B2 B1 + 4 = C2 C1 + 2 = D2 D1 ; 2 = A2 :
So
8 s + 2 s 2 A1
>
< 4 s 2 B1
(s) = > ss +
2 s 2 C1
: s ;+ 2
s 2 D1 :
The additional arithmetic relationships available in A.1 to compute 2 are not
available here without further partitioning. The form of 2 is complicated. For
instance, for s = ;2 ; 2 we have (s) = s+2 = ;2: Since 3 < 2 we have ;4 2 B1 : So
2 (s) = 21 (;4) = ; 21 (;4 + 4) = 2 ; 2 2 2C1:
So 2 (s) 6= s at this point (and at nearby points to the right of s = ;2 ; 2 ).
Continuing, we have
3 (s) = (2 ; 2) = 2( ; ) = 2( ; + 2) = 6 ; 2:
APPENDIX: EXAMPLES OF INTERPOLATION MAPS
77
Since 3 < 2 we have ; < (6;8 2) < 2 so 81 3(s) 2 C1 so
4(s) = 8( 18 3 (s)) = 8( 81 3 (s) + 2) = 3 (s) + 16 = 22 ; 2:
A simple induction argument shows that the orbit of s = ;2 ; 2 under is
unbounded. Now, observe that since 3 < 2 we have ; 52 ; 2 2 A1 : Let
s1 = ; 52 ; 2 : Then (s1 ) = s1 +2 = ; 2 ; 2 : So 4(s1 ) = ;2 ; 2 2 A1 :
So
2 (s1 ) = 41 (4(s1 )) = 41 (4(s1 ) + 2) = (s1 ) + 2 = s1 + 5
2:
Thus (2 (s1 ) ; s1 )=2 is not an integer. Hence 2 is not a 2-congruence on
E;: So the corresponding composition operator U is an element of the local
b Tbg at bE; whose square is not in the local commutant.
commutant of fD
Example A.3. Next, x the initial wavelet set
E0 = ;2 ;) 2)
and let the nal wavelet set be E for 0 < : Write
:= EE0 :
It is easy to compute that for 0 2 we have, on E0 8 s + 4 s 2 ;2 ;2 + 2)
>
<
s 2 ;2 + 2 ;)
(s) = > s
s
;
2
s 2 + )
: s
s 2 + 2)
and for 2 < < we have, on E0 8 s + 4 s 2 ;2 ;)
>
< 2 s 2 2)
(s) = > ss +
+ )
: s; 2 ss 22 2
+ 2)
For 0 2 one may compute that is involutive. Indeed, if
s 2 + ) then
(s) = s ; 2 2 ; ; + ):
So 2(s) 2 E0: We have
2(s) = 21 (2(s)) = 21 (2(s) + 4) = (s) + 2 = s:
The case s 2 ;2 ;2 + 2) is computed similarly. On the other hand, for
< < is not involutive. Indeed, for s = we have (s) = s+2 = 3
2
so 21 (s) 2 2 + ) and hence
2 (s) = 2( 12 (s)) = 2( 3
2 ; 2) = ;:
78
XINGDE DAI AND DAVID LARSON
Example A.4. For 0 2 as above, one may compute that and commute as maps from R onto R so their corresponding composition operators
U := U and U := U commute. Moreover, here the compositions are all 2-congruences of E0 so
f( )(E0) : 0 2 g
is a 2-parameter family of wavelet sets. This is how the family E if Example
4.5 (vi) was derived Further compositions lead to multiparameter families. It
follows that the group (under composition) generated by f : 0 2 g is an
abelian group of measure-preserving involutions of R such that for each element
b Tb): Since
of the group the composition operator U is contained in Cb0 (D
b Tbg0 so does U : Letting
each U normalizes fD
F = f : 0 2 g
this means that f0 Fg admits interpolation. It is, in a sense, prototypical of
involutive interpolating families.
For 0 < 2 one can compute, on E0
8
>
s
s 2 ;2 ;2 + 2)
>
>
+ 2 ;2 + 2)
>
< ss+ 4 ss 22 ;;2
2
+ 2 ;)
( )(s) = > s
s
2
+
)
>
>
s
;
2
s
2
+
+ )
> s
:
s 2 + 2):
For instance, for s 2 ;2 + 2 ;2 + 2) we have (s) = s + 4 2
2 + 2 2 + 2) so 12 (s) 2 + + ) + 2): So
( )(s) = 2( 12 (s)) = 2( 12 (s)) = s + 4:
Verifying commutativity on this interval, we have (s) = s so
( )(s) = (s) = s + 4
as required. The computations for the other subintervals of E0 are similar.
Example A.5. For 0 < if either or is greater then
2 then and can fail to commute, and the composition can fail to act 2-congruently
on E0: For instance, let = 4 and = 34 : Then, on E0 8 s + 4
>
<
(s) = > ss; 2
: s
4
s 2 ;2 ; 32 )
s 2 ; 32 ;)
s 2 54 )
s 2 54 2)
APPENDIX: EXAMPLES OF INTERPOLATION MAPS
and
8 s + 4
>
< + 2
(s) = > ss ;
: s 2
3
4
A routine computation yields
s 2 ;2 ;)
s 2 32 )
s 2 32 74 )
s 2 74 2)
8
>
s
>
>
4
>
< ss ++ 2
( )(s) = > s ; >
>
>
: ss; 2
4
and
79
3
4
s 2 ;2 ; 32 )
s 2 ; 32 ;)
s 2 32 )
s 2 32 138 )
s 2 138 74 )
s 2 74 2):
8
>
s + 8
>
>
s
>
< s+ 4
( )(s) = s + 2
>
>
>
: ss; 2
3
4
4
The map = 4 34
s 2 32 138 )
s 2 ;2 ; 32 )
s 2 ; 32 ;)
s 2 54 )
s 2 54 32 ) :
s 2 32 74 )
s 2 74 2):
fails to act 2-congruenctly on E0 because for
((s) ; s) = ; 1 2= Z:
2
2
The image of E0 under 4 34 is not a wavelet set because it is not
2-translation congruent to 0 2): So
b Tb):
U 4 U 34 2= Cb0 (D
On the other hand, 34 4 is a 2-congruence of E0: The image ( 34 4 )(E0 ) is the wavelet set of Example 4.5 (v). We have
b Tb):
U 34 U 4 2 Cb0 (D
b Tb) can contain unitaries A B
This shows that the local commutant Cb0 (D
b Tb) but
with AB 2 Cb0 (D
b Tb):
BA 2= Cb0 (D
b Tb) is non-abelian.
In particular it shows that Cb0 (D
80
XINGDE DAI AND DAVID LARSON
Example A.6. Consider the map 34 above. We can compute 234 on E0 as
8 s + 8
>
< ; 2
2
(s) = > ss ;
: s s 2 ;2 ;)
s 2 32 )
3
4
s 2 32 74 )
s 2 74 2):
It follows that 334 (s) = s for all 2 E0 and hence for all s 2 R: For instance,
letting := 34 for s 2 ;2 ;) we have 2 (s) = s + 8 2 6 7): So
1 2 (s) 2 3 7 ) E : Thus
0
4
2 4
3(s) = 4( 14 2 (s)) = 4( 41 2 (s) ; 2) = 2 (s) ; 8 = s:
Computations for the other intervals are analogous. However, 2 is not a
2-congruence of E0 because 2(s) ; s is not an integral multiple of 2
for s 2 32 74 ): So 34 does not admit operator-interpolation. We have
b Tb) but
U 2 Cb0 (D
b Tb)
U = U2 2= Cb0 (D
b Tb) is not self-adjoint.
showing that Cb0 (D
Example A.7. Consider the set of Journe in Example 4.5 (i). Let
4 4
32
J = ; 32
7 ;4) ; ; 7 ) 7 ) 4 7 )
and let J denote the interpolation map between E0 and J: On E0 we have
8 s + 6
>
< 2
J (s) = > ss +
2
: s ;; 6
s 2 ;2 ; 107 )
s 2 ; 107 ;)
s 2 107 )
s 2 107 2):
Using 2-homogeneity, we may compute the inverse map, on E0 as
8 s + 4
>
< 32 ;
1
J (s) = > ss +
3 2
: s ;; 4
s 2 ;2 ; 87 )
s 2 ; 87 ;)
s 2 87 )
s 2 87 2):
Then J;1 is not a 2-congruence of E0 so again, we have
b Tb):
UJ 2= Cb0 (D
APPENDIX: EXAMPLES OF INTERPOLATION MAPS
81
The compositions with 7 are not dicult to compute. On E0 we have
8 s ; 2 s 2 ;2 ; 10 )
>
7
>
< s + 2 s 2 ; 1087 ;)
( 7 J )(s) = > s
s 2 7 )
8 10
>
s
;
2
s
7 )
: s ; 6 s 22 107 2)
7
and
8 s
s 2 ;2 ; 127 )
>
>
< s + 6 s 2 ; 12107 ; 107 )
(J 7 )(s) = s + 2 s 2 ; 7 ;)
>
s 2 87 )
>
: ss +; 6
s 2 87 2):
So 7 J acts 2-congrently on E0 but J 7 does not. Thus
and
b Tb)
U 7 UJ 2 Cb0 (D
b Tb):
UJ U 7 2= Cb0 (D
Example A.8. Consider Example 4.5 (viii). Here, A 32 ) is a prescribed
measurable subset, and B C D are derived from A so that f 32 2) A B C Dg
partitions a wavelet set W: Let W be the interpolation map between E0 =
;2 ;) 2) and W: The sets
f 3
2 2) A B ; 2 C + 2 D + 2g
partition 0 2): So the sets
f 3
2 2) A B ; 4 C + 2 Dg
partition E0: Hence
8 s
s 2 32 2)
>
>
s2A
< s
W (s) = s + 4 s 2 B ; 4
>
>
: ss; 2 ss 22 CD +: 2
From the construction we have B = 2C + 4: So
1 B = C + 2 and 2C = B ; 4:
2
So for s 2 B ; 4 W (s) = s + 4 2 B hence
1
2 (s) = 2 ( 1
W
W 2 W (s)) = 2( 2 W (s) ; 2) = s:
82
XINGDE DAI AND DAVID LARSON
Similarly, for s 2 C + 2 W (s) = s ; 2 2 C so
2 (s) = 1 (2 (s)) = 1 (2 (s) + 4) = s:
W
2 W W
2 W
Thus the map W is involutive.
Example A.9. We construct an example of an interpolation map between
two wavelet sets E and F for which 3 =identity, and 2 is a 2-congruence
on the initial wavelet set E: Hence
b Tb)
GroupfU g CbE (D
so admits operator-interpolation.
Let
31
31
A = ; ; 2 ) 7
2 8 ) B = ; 4 ; 8 ) 7 2 ):
Let
15 ) A = 15 31 )
A1 = ; ; 2 ) A2 = 7
3
2 4
4 8
15
15
B1 = ; 4 ; 8 ) B2 = 7 2 ) B3 = 2 31
4 ):
Then
A1 + 8 = B2 A2 + 4 = B3 A3 ; 4 = B1 A1 = 4B1 2A2 = B2 2A3 = B3 :
It follows that A and B are both 2-translation congruent and 2-dilation congruent. If fact, both are translation congruent to 158 ) and both are dilation
congruent to ;2 ;) 74 3116 ): It follows, using an exhaustive induction
technique, that there exists a (nonunique) measurable set C disjoint from A and
B such that
E := A C and F := B C
are wavelet sets. Let
:= EF :
We have (s) = s for s 2 C (s) = s + 8 2 B2 for s 2 A1 (s) =
s + 4 2 B3 for s 2 A2 and (s) = s ; 4 2 B1 for s 2 A3 : So on A1 since 2A2 = B2 we have 12 (s) 2 A2 hence
2 (s) = 2( 21 (s)) = 2( 21 (s) + 4) = s + 16:
Similar computations yield 2(s) = s ; 4 for s 2 A2 2 (s) = s ; 2 for
s 2 A3 and of course (s) = s for s 2 C: So 2 is a 2-congruence on E as
required. To compute 3 note that for s 2 A1 we have (s) 2 B2 = 2A2 so 2(s) 2 (2A2 ) = 2(A2 ) = 2B3 = 4A3: Thus
3 (s) = 4( 14 2 (s)) = 4( 14 2 (s) ; 4) = 2 (s) ; 16 = s
APPENDIX: EXAMPLES OF INTERPOLATION MAPS
83
as required. For s 2 A2 we have (s) 2 B3 = 2A3 so 2 (s) 2 (2A3 ) =
2B1 = 21 A1 so
3(s) = 12 (22 (s)) = 21 (22 (s) + 8) = 2 (s) + 4 = s:
For s 2 A3 we have (s) 2 B1 = 14 A1 so 2 (s) 2 14 (A1 ) = 14 B2 = 12 A2 so
3(s) = 12 (22 (s)) = 21 (22 (s) + 4) = 2 (s) + 2 = s:
Thus 3 =identity on E Hence on R as claimed.
84
XINGDE DAI AND DAVID LARSON
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