The Journal of Fourier Analysis and Applications
Volume 4, Issue 6, 1998
On W a v e l e t S e t s
Eugen J. Ionascu, David R. Larson, and Carl M. Pearcy
Communicated by Guido Weiss
ABSTRACT
It is proved that associated with every wavelet set is a closely related "regularized"
wavelet set which has very nice properties. Then it is shown that for many (and perhaps all) pairs E, F
of wavelet sets, the corresponding MSF wavelets can be connected by a continuous path in L2(R) of
MSF wavelets for which the Fourier transform has support contained in E U F. Our technique applies,
in particular, to the Shannon and Journe wavelet sets.
1.
Introduction
In [2], the notion of a wavelet (measurable) set was introduced and the question was raised
whether the set of all one-dimensional wavelets is connected (in the norm topology on L2(R)).
Wavelet sets were also introduced independently and simultaneously as the support sets of minimally
supported frequency (MSF) wavelets in the sequence of papers [3, 5] and [6], in which the connectivity
problem was raised. (See also the recent excellent book [4].) The problem has been treated by several
additional authors (cf. [7, 9, 10]) but is not solved in complete generality. The purpose of this note
is to introduce a different approach to the connectivity problem which seems to be very natural and
simple and proves somewhat more in several particular cases.
We begin by introducing some preliminary terminology and notation. The measurable space
under consideration will always be R together with the a-ring L of Lebesgue measurable sets, and
Lebesgue measure on this space will be denoted by #. The LZ-space with respect to /z will be
written, as above, simply as L2(R). Throughout the article, we shall need the equivalence relation
on L defined by F -,~ G if F V G is a null set I/z]. Following roughly the ideas in [2], we redefine
some notions and establish the connections between them. Recall (cf. [2]) that a function w 6 L2(R)
is a wavelet if the family of functions {tOj,k}j,k~ x defined by
wj,i(s) = 2J/2w (2Js + k) , s 6 R, j, k 6 Z ,
is an orthonormal basis for L2(R).
We say that a subset G of N with positive measure is a wavelet set if ~ X1 C
(1.1)
= .7-(w),
where w is a wavelet in L2(R) and .Y"is the Fourier-Plancherel transform on L2(N). A measurable
Math Subject Classifications. 42C 15, 46H25.
Keywords and Phrases. Wavelet, regularized, wavelet set, wandering measurable set, conti~mous path.
9 1998Birkh~userBoston.Allrights reserved
ISSN 1069-5869
Eugen J. Ionascu,DavidR. Larson,and CarlM. Pearcy
712
subset G of ]R is called a 2-dilation generator of a partition of ]R if the sets
2~:G := {2ks : s E G ] , k E Z ,
are disjoint and
Uk~z2k G
"-" R, and G is a 2zr-translation generator of a partition of ]R if the sets
G + 2ksr := {s + 2krr : s E G}, k E Z ,
are disjoint and Uk~z(G + 2kzr) ~ ]R. A measurable subset G of ]R is translation congruent modulo
2rr to a (measurable) set H if there exists a measurable bijection ~o : G --~ tp(G) such that ~0(s) - s
is an integral multiple of 2zr for every s in G and r
--~ H. Analogously, G is said to be dilation
congruent modulo 2 to a (measurable) set H if there exists a measurable bijection ~ : G ~ ~ ( G )
such that ~t(s)/s is an integral power of 2 for every s in G and Or(G) ,~ H.
L e m m a 1.
The following conditions are equivalent for any measurable subset G of]R:
(a) G is a wavelet set,
(b) there exists a set G' E L such that G "~ G' and G ~ is a 2-dilation generator of a partition of]R
and a 2zr-translation generator of another partition of]R,
(c) the Littlewood-Patey wavelet set E = [-2~r, - z r ) O [rr, 2rr) is translation congruent modulo 2Jr
and dilation congruent modulo 2 to G.
Proof.
Assume that G is a wavelet set. Then, by definition, there exists a wavelet w such that
l
VTTb-~Xo = ~-(w). Thus, the family of vectors {Wj.k }j,kEZ defined in (1.1) is an orthonormal basis
of L2(]R). Since ~" is a unitary transformation and .~(wj,o) = ~ X 2 1- J G
for all j in Z, it
follows that all of the sets 2JG fl 2~G, j ~ k (j, k E Z), are null sets. We can clearly find a
measurable subset G1 of G such that Gl -- G and the family {2JGI}j~z is disjoint. Since for all
j, k 6 Z w e h a v e
eiks
(wj.,,))(s)
=
]R ,
s
and {~(wj,l:)}j,k~z is an orthonormal basis of L2(]R), we get that Uk~z 2kG "~ Uk~z 2kGl "~ ]R,
which proves that Gl is a 2-dilation generator of a partition of ]R.
We assert that for e v e r y j , k ~ Z, j r k, the set ( G + 2 j r r ) M ( G + 2 k s r ) i s a n u l l s e t .
If not, there exists s ~ Z\{0} such that/z(G A (G + 2esr)) > 0. Set F = G f3 (G + 2err), so
F u ( F - 2s
C G. Since/z(F) > 0, there exists a subset F1 of F of positive measure such that
F1 f'l (F1 - 2e~r) = O. Thus,
IIx,=, -
11 =
(1.3)
(F,) > o.
On the other hand, using (1.2) and the fact that the family {2JG}j~z is almost disjoint, we have
fR ] .
< ~ ( w y . k ) , xe~ - X ( F , - 2 t , , ) >
__
eiks
(s)
-
•j,O f eiks ( x F , ( S ) _ X(F,_21zr)(s))ds
,/V(N Ja
aj,o
(~
eikSds
eikSds)
O, y,k
Z,
On WaveletSets
713
which contradicts (1.3). Thus, (G + 2jzr) N (G + 2krr) ~ (GI + 2jrr) f'l (G1 + 2kzr) is a null
set for all j --r k (j, k ~ Z). Thus, we can find a subset G' of GI such that G ~ ~ G1 and the
family {G' + 2jzr}j~z is disjoint. Since II~ = U k c z ( E ' + 2ksr) (with E' = [0, 2~r)), if we define
tp : G' --~ E ' by r
= s - 2kzr ifs e G~ : = G' Cl (E' + 2kzr), r becomes an injective measurable
map. Then from (1.2) we obtain that
8k,o~//z(G) = fReiksXG(S)dS:fG eeikSdS=k~Z
i k s d S ' fG'k
= k~zf~
eikSds=f~
(G'k)
eikSds
(G')
,/-2-~fe,(I/,,/2-~)eikSx,(G,)(s)ds.
=
Since the family of functions {s --+ (1/,/-2-~)eikSXE,(S)}keZis an orthonormal basis for L2(E'),
we conclude from the above computation that X~(G')is the constant function (s//x(G)/(2sr))XE, in
L2(E'). Hence,/z(G) = 2zr and r
~ E ~, so
U (G' + 2kn')
k~Z
k~Z
=
u
j~Z
(~:. (,0~;> + ~,~ + ,>~))
jeZ
=
= u
(~,~ (,0~;> + ~,,~))
j~Z
u (~,,z (,0<~;> + ~ - ) ) ~ u : ' + ~-,> = ~.
e~Z
(1.4)
~Z
This proves that G' is a 2zr-translation generator of a partition of R. Since G' C G1 and G' ~ G,
G' is also a 2-dilation generator of a partition of ]R, so (a) implies (b).
We prove now that (b) implies (c). Thus, suppose that the set G' is as in part (b). We consider
the equivalence relation p on the null set R \ ( U k s z 2kG ' U {0}) defined by sl p s2 if sl/s2 is an
integral power of 2. By the axiom of choice there exists a subset G of R ~ U k s Z 2 t G ' such that
contains exactly one point from each equivalence class (rood p). Clearly, G is a null set and the set
= G' U G has the property that the family {2kG}/~sz is a measurable partition of R\{0}. Hence,
if we define ~ : E --~ G by ~ ( s ) = 2-ks for s E E A (2kG), if/is a measurable bijection, and so
since G "-" G, E is dilation congruent modulo 2rr to G. Similarly one shows that E is translation
congruent modulo 2zr to G.
To establish the implication (c) =~ (a), let ~ : E --+ ~ ( E ) "~ G be the measurable bijection
which satisfies ~(s)/s E {2k : k E Z} for every s ~ E. If we consider the measurable sets
Ek : = {s ~ E : ~(s)/s = 2k}, k ~ Z, then 2JEt A 2~Em C 2JE N 2kE = 0 for all s m ~ Z and
all j, k ~ Z, j -~ k. Thus, for j ~ Z\{0} fixed we have
/
2J~(E) f'l ~(E)
\k~Z
=
\k~Z
/
U 2~+~e~2'E,=U2'(E'-~E,) = ~
(1.5)
This implies that the family {2JG}jeZ is almost disjoint, and a computation like (1.4) shows that
U j ~ z 2j G ~ R. Therefore, by (1.2), it suffices to show that the family of functions
I'-" (I/7~) e/"~">l,:.
<"~>
EugenJ. lonascu,DavidR. Larson, and CarlM. Pearcy
714
is an orthonormal basis for L2(G). For this purpose let ~o : E --* ~o(E) -,~ G be a measurable
bijection which satisfies ~o(s) - s ~ {2kzr : k ~ Z} for every s ~ E. We consider E k := {s ~ E :
~o(s) - s = 2kzr }, k 6 Z, and an arbitrary function f ~ L2(G). Then
fG f(s)e-ikSds
=
f~(E) f(s)e-ikSds
=
E
e~Z
(Et)
f.
--
- fe:(
f(s)e-ik'ds = Z
eEZ
fe
'+2e~r f (s)e-ikSds
f (t + 2&r)e-ikt-2iken dt
~o(t))e lktdt, k ~ Z .
(1.7)
-
First, letting f ( s ) = e ijs in (1.7), we get that the family (1.6) is orthonormaI (since g ( G ) =
/z(cp(E)) = ~-]~k~Zlz( Ek + 2kzr) = /z(E) = 2zr). Second, (1,7) tells us that the only element of
L2(G) that is orthogonal to the family (1.6) is f = 0. Thus, the family (1.6) is an orthonormal basis
for L2(G), and the lemma is proved.
[]
Now let G q L be an arbitrary wavelet set. Then we know from Lemma l(b) that there exists
a wavelet set G ' ~ G such that the families {G' + 2k~r}k~Z and {2kG'}kzZ are disjoint and satisfy
U(G'+2k ) ~R,
LI2 G' ~R
k~Z
k~Z
It turns out to be very convenient for the constructions to follow to know something stronger, namely,
that G' may be chosen to also satisfy
U (o' + 2k.) = R,
U 2k ' =
kcZ
k~Z
(1.8)
Strangely enough, this is always possible, as is established by the following result.
Theorem 1.
Let G be a wavelet set. Then there exists a (wavelet) set G ~ ~ L such that G ~ ,- G, G ~ is a
2-dilation generator of a partition of R and a 2It-translation generator of another partition of •,
and, furthermore, the equalities in (1.8) hold. Moreover, there exist unique measurable bijections
and ~t of E onto G' satisfying the conditions
~(s)/sE {2k:kEZ},
r
sEE.
(1.9)
Proof.
According to part (c) in Lemma 1 there exist measurable bijections ~o : E --+ ~o(E) ~ G
and ~ : E --+ ~ ( E ) "~ G such that ~o(s) - s ~ {2kzr : k q Z} and ~ ( s ) / s ~ {2k : k fi Z} for every
s q E. Let G1 = ~o(E) M ~p(E) and inductively define the sequences {Gn}nr and {En}n~n of sets
by
En = ~o-1 (Gn) n q/-1 (Gn), n E N,
Gn+l = ~o(En) n ~ (En), n E N\{1}.
(1.10)
It is easy to see that both sequences {Gn}, {En} are decreasing and that En "" E, Gn "" G for every
n in N. Let J~ := AnEN En, G := Anon Gn, and observe that
/z (/~) = lim/z (En) = 2zr ----lim/z ( G . ) = / z (G) ,
n
?1
(1.11)
On WaveletSets
715
from which we get that E ,,~ E and G -,~ G. We assert that ~p(/~) = G and ~p(/~) = G. We first
show that ~0(E) C G. Consider s ~ E and fix n ~ N\{I}. Since s ~ En, there exists t ~ Gn such
that s = r
and there exists ~ E En-l such that t = ~(~). Hence, ~p(s) = ~ ( ~ ) ~ ~P(En-l),
and since ~o(s) ~ ~o(En) C ~o(En-l), it follows from (1.10) that ~o(s) ~ Gn, and thus that ~o(s) ~
and ~ ( E ) C G. Analogously one establishes that ~(J~) C G, ~o-l(G) C E, and ~ - l ( ~ ) C /~,
which shows that ~p(J~) = G = ~(J~).
We define now G' = G U (EkJ~). From (1.11) we get G ~ "-~ G, and if we define ~ [resp.
~] : E ~ G' to be ~o [resp. ~ ] on E and the identity function on E \ E , we obtain that r and
are measurable bijections (with range G') satisfying (1.9). (To verify the injectivity of ~b and ~, let
Sl ~ E\J~, s2 ~ E, and r
= r
[resp. ff(Sl) = if(s2)]. Then ~P(S2) [resp.q/(s2)] = sl E E.
Since s2, ~(s2) ~ E and s2 - ~/(s2) is an integral multiple of 2re, we must have ~p(s2) = s2 = Sl and
similarly for ~.) Two computations similar to (1.4) show that both equalities in (1.8) hold, and two
computations similar to (1.5) show that the sequences {G' + 2k:r }i~Z and {2tG'}k~Z are disjoint.
Finally, the uniqueness of ~ and ~ follows immediately from (1.9) and the fact that G' is both a
2-dilation generator and a 2zr-translation generator of partitions of IR.
[]
Definition 1.
If G C ~ is any wavelet set and G' is related to G as in Theorem 1, we say that G ~ is a
r e g u l a r i z a t i o n of G and that G ~ is a r e g u l a r i z e d wavelet set. (Easy examples show that such
a G ~need not be unique.)
2.
T h e Main Result
The following concept will be very useful. If (X, M) is any measurable space and g : X -+ X
is a measurable bijection with measurable inverse, a measurable set W C X is called a wandering
set for g if the family {g(k)(W)}k~Z is disjoint and X = [.Jk~Zg(k)(W), where g(k) is the identity
function on X when k = 0 and the composition o f g [resp. g - l ] with itself Ikl times when k > 0
[resp. k < 0]. (In particular, if X = 0, then W = 0 is a wandering set for g.)
With these preliminaries disposed of, we now turn to our principal construction. Let F be any
wavelet set, let F ~ be a regularization of F, and let ~b : E ~ F', ff : E ~ F ' be the measurable
bijections given by Theorem 1. Since ~ - l and ~ - l are measurable as well [via (1.9)], the function
h : E --+ E defined by h = ~b-1 o ~ and its inverse are measurable bijections. We consider a
measurable partition E = E1 t.) E2 U E3 U . . . t.) Eoo of E, where
(2.1)
k=l
For each n ~ N U {oo}, h maps each En onto itself and h Ie, is a measurable bijection with measurable
inverse. The following is our principal theorem.
T h e o r e m 2.
Let E be the Littlewood-Paley wavelet set as above, F any wavelet set, and F ~a regularization
of F. Suppose there exists a wandering set W for the function h leoo defined above. Then there exists
a family of(regularized) wavelet sets {Gt}t~[O, ll such that Go = E, GI V F~ is at most countable,
and Gt ( E U F for all t ~ [0, 1]. Moreover, the function t ~ .Tr-l(--~XGt) defined on [0, 1] is
continuous (with respect to the norm topology on L2(R)) and consists entirely of wavelets.
Proof.
Let ~ and q/as above. For t E [0, 1] and n ~ N we define the sets
pt
=
[-2rC-(2-t)~r)U[(2-t):r,
2~r),
Wt = W f q p t ,
Ptn= E n N P t ,
EugenJ. lonascu, DavidR. Larson, and CarlM. Pearcy
716
n--I
f2t
=
Uh(k)(wt),
oo
f2tn=Uh(k)(Pnt), f 2 t = f 2 t u U a
k~Z
k=0
~,
(2.2)
k=2
where the En are as in (2.1). Then the desired family of wavelet sets is given by
Gt := ( E \ a , ) t.; ~b (f2t),
t 6 [0, 1].
(2.3)
Clearly, the sets Wt, P n,
t aoo,
t
t f2t (and consequently Gt) are measurable for all t ~ [0, 1] and
a n,
n 6 N. It is easy to see that Go = E (since p0 = 0). To show that GI V F ' is at most countable, we
note that ~b(s) = s for all s in El except possibly a countable set. (Indeed, i f s 6 El, then h(s) = s
and this leads to the equation ~b(s) = s + 2 l r c = 2ms = ~ ( s ) for some l, m ~ Z. I f m = 0 then I = 0,
which shows that ~b(s) = s. I f m # 0 then s = - 2 l J r / ( 2 m - 1), and the set of such points is countable.)
A calculation using (2.2) shows that G1 = ( E \ a l ) u ~ b ( a l ) = E1 U~b(E\E1) = E l U ( F ' \ ~ b ( E l ) ) ,
so G1 V FI is countable. From (2.3) we see that Gt C E U F ' for all t ~ [0, 1].
To check that each Gt is a wavelet set, by Lemma 1 it suffices to show that E is translation
congruent modulo 2Jr and dilation congruent modulo 2 to Gt. For this purpose let us observe that
h ( a t ) = f2t and thus ~b(at) = ~ ( a t ) for all t 6 [0, I]. Moreover, we have ~b(f2t) fq E = 0 for
all t 6 [0, 1]. (Indeed, if ~b(s) 6 E for some s in ~t, then ~b(s) = s by (1.9) and the fact that
{E + 2kzr }k~Z is a disjoint family. Hence, s 6 E :1 F', and since ~ ( s ) 6 F', we must have r
= s
[since {2kFI}~Z is a disjoint family]. This implies that h ( s ) = ( r
o (O)(s) = s and hence s 6 El.
But this contradicts the fact that f2t N E t = 0 by the definition of a t . ) We consider the measurable
maps ~ot, ~tt : E --+ Gt defined to be the identity function on E \ a t and by ~ot = ~, ~t = ~/
on f2t for all t E [0, 1]. Since Gt = ( E \ a t ) t.J ~ ( a t ) = ( E \ a t ) U ~ ( a t ) for all t E [0, 1], and
( E \ a t ) fq ~(at) = ( E \ a t ) f') Ct(at) = ~, ~ot and ~/t are injective and map E onto Gt. It is clear
that for all t 6 [0, 1] and all s 6 E, ~ot (S) - - S is an integral multiple of 2rr and ~t ( s ) / s is an integral
power of 2. As in the proof of Theorem 1, one concludes easily that all the sets Gt are regularized
wavelets sets.
Since 5r is a unitary operator, to check that the function t --+ .Y"-I ( ~ 'tX G , ) is continuous, it
suffices to show that the function t --+ XG, is continuous on [0, I]. For this purpose let q, t2 6 [0, 1],
and, without loss of generality, suppose that tl < t2. Then Pq Q pt2 and consequently a q c f2t2.
THUS,
XGq - XG,2 2L2 =
XGq ~TGt2 2L2 = / Z ( f 2 t 2 \ a t , ) + / z (~o ( a t 2 \ a q ) )
.
(2.4)
Since ~b is/z-measure preserving, (2.4) becomes
XGq -- XG, 2 2L2 = 2(/z (at2) - / z (f2tl)) 9
(2.5)
The following lemma completes the proof of Theorem 2.
L e r a m a 2.
The function co defined by co(t) = / z ( a t ) ,
Proof.
t ~ [0, 1], is continuous on [0, 1].
We begin by defining the functions con on [0, 1] by
coo(t)=/z (at),
con(t)= U (at), n ~ N.
F r o m ( 2 . 2 ) w e see that
oo
co(t) = coo(t) + Z
cok(t), t 6 [0, 1].
k=2
Since h ( E n ) = En for all n ~ N, we obtain
Y coW) <_Y .u (ek) <_ U(E) = 2 r,
k=2
k=2
t ~[0,1],
(2.6)
On WaveletSets
717
so the series in (2.6) converges uniformly on [0, 1]. Thus, it suffices to show that for n = 0, 1. . . . .
the function wn is continuous on [0, 1]. For n = 1, 2 . . . . . we note that if 0 < tl < t2 < 1, then
n-1
f2~ \ S2t~ C Uk=0
h(k)(P~2\Ptt), and consequently we have
n-I
0 ~<~O.)n( t 2 ) - O)n (tl)_~<
Zl-l,(h(k)(Pnt2 \ pntl)) ,
n EN.
k=O
Since/z(P~t2 \ p t l ) < 2st }tl - t21 [cf. (2.2)], to establish the continuity of the wn, n ~ N, it is sufficient
to show that for k ~ Z the measure vk defined on (E, L 71 E) by vk(L) = lz(h(k)(L)), L ~ L 71 E,
k ~ Z, is absolutely continuous with respect to/Zl(Lne ). Moreover, using (2.2) and the facts that
Wt C W and W is a wandering set for hlEoo, we get
o)o(t) = ~ Iz (h(k) (Wt))
(2.7)
keTZ
and
t ~[0,1].
kEZ
kEZ
This shows that the convergence of the series in (2.7) is uniform on [0, 1], and since
lz (Wt2\Wq) < 2rr It1 - tzl, t l , t2 E [0, 11,
the continuity of w0 will also follow from the absolute continuity [/z] of the vk, k ~ Z.
In order to check the relations vk < < /ZILf'IE, we observe that the measure Vl is mutually
absolutely continuous with respect to (i.e., equivalent to) /ZlLAE. Indeed, let L be a measurable
subset of E such that/.t (L) = 0. It is clear from (1.9) that for every s in E we get h(s) = 2ms -t- 2kTr
for some integers m and k. Thus, we consider the measurable partition E = U i , j ~ z Hij o f E , where
H ij = {s E E : h(s) = 2is -F 2jrr}. Since h is a measurable bijective map, h(L) is the disjoint
union
u,, (,.
i,j~Z
=u
(L n,-,,,)+
i,jEZ
This implies that
v I ( L ) ~---- ~'~
i,jEZ
11~(2i (L71HiJ) nt-2jTr) ~ Y ~ 2il.s
i,jEZ
=0.
(2.8)
Reciprocally, if L is a measurable subset of E such that vl (L) = 0, we obtain from (2.8) that
lz(L N H ij) = 0 for every i, j 6 Z. Hence,/z(L) = Y~i,j~z lz(L N H ij) ~- O. T h u s , Vl and/z vanish
simultaneously on L 71 E, and by virtue of the relationships between vl, vk, and/~, the same is true
for all Wo k 6 Z. Since the v/r and/z are all finite measures on (E, L 71 E), for each fixed k ~ Z and
for each ~ > 0, there exists 8k > 0 such that vk(L) < e whenever/z(L) < 8k (see, e.g., [1, p. 169]).
This is what was needed, so the proof is complete.
[]
[]
R e m a r k 1.
In Theorem 2, E is the Littlewood-Paley wavelet set, but the content of the theorem is not
altered if we replace E by any regularized wavelet set.
Corollary 1.
Let F be a wavelet set for which the set Eoo defined in the preamble to Theorem 2 is a null set
(equivalently, for which almost every point of E has a finite orbit under the corresponding function
h). Then there exists a wandering set W for hlEo~, and thus the conclusions of Theorem 2 hold.
Eugen J. lonascu, David R. Larson, and Carl M. Pearcy
718
Proof. IfE~ isvoid, then Oisawanderingsetforhle~. IfE~ ~ 0, we consider the equivalence
relation 8 on E ~ given by s ~ s ~ if h (k) (s) = s' for some integer k. By the Axiom of Choice there
exists a subset W of E ~ such that W contains exactly one point from each equivalence class (mod S).
By hypothesis it follows that W is a measurable null set. Clearly, by construction, h (k)(W) N W = 0
for all k E N and E ~ = [-Jk~Z h(k)(W) 9 Hence, W is a wandering set for hie ~, and this proves the
corollary.
There are several examples of wavelet sets to which we can apply Corollary 1.
E x a m p l e 1.
First we consider the class of all wavelet sets F such that (E, F) is an interpolation
pair (see [2, p. 56]). (Indeed, in the notation of [2], if (trff) 2 is the identity function (a.e.) on JR,
one may observe that (~rff)lE, = ~bo ~ - l (a.e.) in our notation, and thus h (2) is the identity function
(a.e.) on E. This implies that E --. E1 tJ E2 and hence that Eoo is a null set.) Consequently, we
obtain the following result first mentioned in [2]: Any two wavelets associated with an interpolation
pair of wavelet sets can be connected by an arc of wavelets.
E x a m p l e 2.
There are cases (see [2, p. 83]) when (trF)
E 3 is the identity function on JR. Clearly,
these cases are also covered by Corollary 1. Indeed, Corollary 1 applies whenever there exists an
n ~ N such that h (n) is the identity function (a.e.) on E, since in this case E "-- U~=l Ej. In fact,
we can prove somewhat more.
Corollary 2.
Let F be a wavelet set such that the function h defined in the preamble to Theorem 2 has the
property that h (k) is continuous on E f o r some k E Z\{0}. Then there exists a wandering set W f o r
hie ~, and thus the conclusions of Theorem 2 hold.
Proof.
We write g for h (k), and recall that g and g - l are measurable and injective maps of E onto
itself. Since g maps connected sets to connected sets, we must have g([Tr, 2rr)) = [-2rr, - J r ) or
g([Jr, 2~r)) = [Jr, 2:r). In any case, by squaring if necessary, we may suppose that g maps [Jr, 2zr)
onto itself and thus that g (rr) = 7r and g is strictly increasing on [Jr, 2zr). Hence, lims ~ 2n g (s) = 2zr.
In other words, g can be extended to be a continuous bijection on [Jr, 2rr ], and thus a homeomorphism.
Consequently g is a homeomorphism of [zr, 2rr) onto itself, and similarly for g on [ - 2 r r , - J r ) . Let
C + be the set of fixed points ofgt[rr,2~r ) (which includes Jr), and let U + = [zr, 2Jr) \ C +. If U + = 0,
then 12t is a wandering set for glu+. Otherwise U + is a nonempty open subset of R which can
be written uniquely as a countable union U + = U j ~ j ( a j , bj) of disjoint open intervals. Clearly,
aj, bj ~ C + U {2zr} and g((ay, by)) = (aj, bj) for all j. Consequently, U + consists exactly of those
points s ~ [rr, 2zr) such that the orbit of s under g (and therefore under h) is infinite. We now exhibit
a wandering set W + for gltJ+. For each j ~ J we choose a point (say, the midpoint) sj ~ (aj, bj)
and define W + = [..Jj Iy, where lj = [sj, g(sj)) if sj < g(sj) and Ij = [g(sj), sj) if g(sj) < sj. Itis
easy to see that the family of sets {g(k)(W+)}k~Z is disjoint and Uk~Z g(~)(W+) = U+. In a similar
way we construct a wandering set W - for glu-, where U - C [-27r, - r r ) is defined analogously to
U + , and we write r~ = W + t3 W - . Clearly I~' is a wandering set for glu+UtJ- (and (g-1)lu+uu-).
The following lemma applied to X = U + U U - , M = L Iq X, and v = / z l x completes the proof of
the corollary.
/,emma 3.
Let (X, M, v) be a complete, finite measure space, and let h : X --+ X be a measurable
bijection with measurable inverse which preserves sets of measure zero and has the property that f o r
some n E N, h (n) has a wandering set. Then h also has a wandering set.
Proof.
Suppose .W is a wandering set for h (ko) (ko > 1), and consider the collection S of subsets
On WaveletSets
719
of W defined by
S:={SCW:S~I~and
{h(J)(s)}j~zisadisjointfamily } .
Obviously 0 ~ S, and we consider the partial ordering defined on S by $1 -~ $2 if S1 C $2
and v(S2\S1) > 0. Let C = {S,,}~A be a chain in S. Then A is at most countable since W
cannot contain uncountably many disjoint sets of positive measure, and an easy calculation shows
that U = Uct~A Sct E S. T h U S , U is an upper bound for the chain C, and by Zorn's Lemma, S
contains a maximal element M. By definition, the family {h(J)(M)}j~Z is disjoint, and we assert
that v ( X \ Uj~z h(J)(M)) = 0. Suppose, for the moment, that this has been established. Then, if
\
/
X \ Uj az h (j) (M) ~ ~, we put an equivalence relation on this set of measure zero as in the proof of
Corollary 1, and using the Axiom of Choice as in that corollary and the completeness of (X, M, v),
we obtain a set T of measure zero such that M U T is a wandering set for h.
Thus it suffices to verify that X \ U j e z h ( J ) ( M ) has measure zero. Suppose not. Then
R = W\(Ujezh(J)(M)) must also have positive measure. (Otherwise, by hypothesis, X =
Ue~z h(ek~
is almost contained in Uj~z h(J)(M) .) Using the fact that
U h(k~
= X, j = 1 . . . . . k o - 1 ,
e~Z
we may choose integers n l . . . . . n~o_l by induction so that the (measurable) set
Q = R n h(k~
n h(k~
N ... n h(k~176
has positive measure, and since Q c R and R n (Uj~zh(J)(M)) = O, we have
h(m)(Q)n ( U h ( J ) ( M ) I = 0 ,
\j~z
/
mEZ.
Thus M O Q ~ S, which contradicts the fact that M is a maximal element of S, so the proof is
complete.
[]
[]
R e m a r k 2.
One can prove that, in fact, the hypothesis in Lemma 3 can be weakened to the assumption
that (X, 1~) is simply a measurable space. The proof is very computational (so we do not include it
here), and is based on checking that the set
,
= wn
U
N h(JD(w)
'
(Jl ..... jko_l)~A g----I
where .4 is a subset of the product (koZ + 1) x ... x (koZ + ko - 1) with certain properties, is a
wandering set for h.
E x a m p l e 3.
An interesting wavelet set to which Corollary 2 applies is Journe's (regularized)
wavelet set (el. [2, p. 45]):
J=[-32;,-4zr)
U[-rr,-~)U[~,zr)U[4rr,
3-~-) 9
One can easily compute the corresponding function h (constructed from E and J) and obtain that
h(s) = s/4 + 3rr/2 for s 6 [ - 2 z r , - 1 0 r r / 7 ) , h(s) = 2s + 4rr if s 6 [-10zr/7), h(s) = 2 s - 4zr
720
Eugen J. lonascu, David R. Larson, and Carl M. Pearcy
for s 6 [zr, 10zr/7), and h ( s ) = s / 4 - 3zr/2 i f s ~ [10rr/7, 2~r). Obviously h is continuous, but
calculations show that E ~ is not a null set, so Corollary 1 does not apply. But the set
[ - 9 z r / 7 , - 8 r r / 7 ) U [8zr/7, 9rr/7)
is easily seen to be a wandering set for hie ~ , and thus Theorem 2 applies.
E x a m p l e 4.
The following (regularized) wavelet set G (due to Gu [8]), is an interval wavelet set
(i.e., a wavelet set which is a union of intervals), but even so, for every k 6 Z the function h (k) is not
continuous:
G : = [ - 6 4 0 r r / 2 1 , -448~r/15) U [ - 5 6 r r / 1 5 , - S z r / 3 ) U [ - 2 z r / 3 , -10~r/21) U [2zr/15, 4rr/15) .
Indeed, an analysis of the functions ~b and ~ allows one to check that h is given by
s / 4 + 2zr, s ~ [-2Jr, - 4 0 r r / 2 1 ) ,
h(s) =
16s + 32Jr, s E [ - 4 0 r r / 2 1 , - 1 5 r r / 8 ) ,
16s + 28zr, s ~ [-15zr/8, - 2 8 z r / 1 5 ) ,
2s + 2rr, s E [-28zr/15, - 3 r r / 2 ) ,
2s + 4zr, s E [ - 3 z r / 2 , - 4 ~ r / 3 ) ,
s / 2 + 2zr, s E [ - 4 z r / 3 , - J r ) ,
s / 4 - 2zr, s ~ [Jr, 16zr/15),
s / 8 - 2zr, s ~ [16zr/15, 2rr) .
(2.9)
One can trace the orbit {h(n)(--27r)}n~N and observe that it is an infinite set. But from the proof of
Corollary 2, one knows that if h (k) were continuous for some k ~ Z, then - 2 z r would be a fixed
point for h (2k). Still, it turns out that h i e ~ has a wandering set, so Theorem 2 applies again. In fact,
one can check that E = E3 !.3 E4 U E ~ , where
E3
=
[ - 7 z r / 4 , - 2 6 r r / 1 5 ) U [ - 3 r r / 2 , - 2 2 z r / 1 5 ) U [Jr, 16zr/15) ,
E4
=
[ - 2 8 r r / 1 5 , - l l r r / 6 ) U [ - 2 6 J r / 1 5 , - 5 r r / 3 ) U [ - 2 2 r r / 1 5 , - 4 z r / 3 ) U [16zr/15, 4zr/3) ,
Eoo
=
[-2zr, - 2 8 r r / 1 5 ) U [ - I lzr/6, - 7 r r / 4 ) U [ - 5 r r / 3 , - 3 r r / 2 ) U [ - 4 z r / 3 , - z r ) U [4zr/3, 2zr)
and the interval [37r/2, 2rr) is a wandering set for hie ~ .
We have reason to believe that the following conjecture, which we are presently unable to
prove, is true.
Conjecture 1.
Given an arbitrary wavelet set F, there exists a wandering set f o r the corresponding function
hie ~ associated with E and F.
References
[1] Brown, A. and Pearcy, C. (1977). Introduction to Operator Theory I, Elements of Functional Analysis, SpringerVedag.
[2] Dai, X. and Larson, D. (1998). Wandering vectors for unitary systems and orthogonal wavelets, Memoirs Amer.
Math. Soc., 134, no. 640.
[3] Fang, X. and Wang, X. (1996). Construction of minimally-supported-frequencywavelets, J. Fourier Anal. Appl.,
2, 315-327.
[4] Hernandez,E. and Weiss, G.L. (1996). A First Course on Wavelets, CRC Press, Boca Raton, FL.
[5] Hernandez,E., Wang, X., and Weiss, G.L. (1996). Smoothing minimally supported frequency (MSF) wavelets:
Part I, J. Fourier Anal. Appl., 2, 329-340.
On Wavelet Sets
[6]
[7]
[8]
[9]
[10]
721
Hernandez, E., Wang, X., and Weiss, G.L. (to appear). Smoothing minimally supported frequency (MSF)
wavelets: Part II, J. Fourier Anal. Appl.
Dai, X. and Liang, R. Path connectivity and wavelets, preprint.
Gu, Q. The construction of 4-interval wavelet sets, Chapter 4 of Ph.D. Thesis, Texas A&M Univ., 1998.
Han, D. and Lu, S. Smoothing approximations and pathwise connectivity of MRA wavelets, preprint.
Speegle, D. (to appear). The s-elementary wavelets are connected, Proc. Amer. Math. Soc.
Received June 3, 1997
Department of Mathematics, Texas A&M University, College Station, Texas, 77843
e-mail: ionascu @math.tamu.edu
Department of Mathematics, Texas A&M University, College Station, Texas, 77843
e-mail: larson@math.tamu.edu
Department of Mathematics, Texas A&M University, College Station, Texas, 77843
e-mail: pearcy@math.tamu.edu