The ontinuous wavelet
transform and its
disretization
1. The ontinuous transform
Supp ose that 2L 2
(R). We dene the familyof funtions a;b
by
a;b
(x)= 1
p
jaj
x b
a
; a6=0; b2R: (8.1)
Often one only onsiders the ase where a>0 and the normalization 1
p
jaj
isusedsothatk a;b
k
L 2
(R)
=k k
L 2
(R)
. Othernormalizations an,ofourse,
b eusedas well. Then wean dene
(W f)(a;b)= D
f;
a;b E
= Z
R f(x)
a;b
(x)dx:
(8.2)
It islear that jW f(a;b)jk k
L 2
(R) kfk
L 2
(R) .
Wehave thefollowing result:
Theorem8.1. Assume 2L 2
(R)nf0g issuh that
C def
= Z
R j
^
(!)j 2
j!j
d ! <1:
Then
Z
R Z
R
W f(a;b)W g(a;b) 1
a 2
dadb=C hf;gi;
for all f and g2L 2
(R).
65
This theorem says that in a weak sense, we have (provided, of ourse,
that 6=0)
f(x)= 1
C Z
R Z
R
W f(a;b) a;b
(x) 1
a 2
dadb:
Pro of. Firstwenote that
d
a;b
(!)= p
jaje i2 b!
^
(a!);
sothat byPlanherel's theorem (2.4)we have
W f(a;b)= p
jaj Z
R
^
f(!)e i2 ! b
^
(a!)d! = p
jaj
\
^
f()
^
(a)( b):
Thuswe get,again using Planherel's theorem,
(8.3) Z
R Z
R
W f(a;b)W g(a;b) 1
a 2
dadb
=jaj Z
R Z
R
\
^
f()
^
(a)( b)
\
^ g()
^
(a)( b)db
!
1
a 2
da
= Z
R Z
R
^
f(!)^g(!)j
^
(a!)j 2
d !
1
jaj da
= Z
R
^
f(!)g (!)^ Z
R j
^
(a!j 2
1
jaj da
d!:
A simple hangeof variables nowshowsthat
Z
R j
^
(a!j 2
1
jaj
da=C ;
and thenthelaim followsfrom equation (8.3).
If one do es not want to use negative as well as p ositive values for the
dilation athenone getsalmostthesameresult,provided
Z
0
1 j
^
(!)j 2
j!j
d! = Z
1
0 j
^
(!)j 2
j!j d !
def
= : (8.4)
This isofoursethe asewhen isreal-valued.
Corollary 8.2. Assume 2L 2
(R)nf0gissuh that (8.4)holdswith <
1. Then
Z
R Z
1
0
W f(a;b)W g(a;b) 1
a 2
dadb= hf;gi;
for all f and g2L 2
(R).
Thisresult saysthat in a weak sense we have
f = 1
Z
R Z
1
0
W f(a;b) a;b
(x) 1
a 2
dadb:
2. Frames of wavelets
The ontinuous wavelet transform is not neessarily a very pratial to ol
sine it is notlear towhatextent theintegralsan atually b eomputed.
If one uses disretizations, one question to ask is when the inner pro duts
hf;
m;n
i, where where
m;n
(x) = a m
2
(a
m
x nb
), really haraterize
the funtion f. If thesequene (
m;n )
m;n2Z
is an orthonormalbasis, there
arenoproblems,butif issomequitegeneralfuntion,thereisnoreasonto
exp et thattob ethease. Butitturns outtob ep ossible togiverelatively
simple onditions for this sequene to b e a frame, see Denition 3.6. We
have thefollowing result:
Theorem8.3. Assume a
and b
>0 and that 2L 2
(R) is suh that
0<essinf
! 2R X
m2Z j
^
(a m
!)j 2
esssup
! 2R X
m2Z j
^
(a m
!)j 2
<1;
(8.5)
and
X
k 2Z
k 6=0 s
k
b
k
b
<essinf
! 2R X
m2Z j
^
(a m
!)j 2
; (8.6)
where
(s) def
= esssup
! 2R X
m2Z j
^
(a m
!)jj
^
(a m
!+s )j:
Then the sequene (
m;n )
m;n2Z where
m;n
(x) = a m
2
(a
m
x nb
) is a
frame in L 2
(R) with frame bounds
A= 1
b
0
B
B
essinf
! 2R X
m2Z j
^
(a m
!)j 2
X
k 2Z
k 6=0 s
k
b
k
b
1
C
C
A
;
B = 1
b
0
B
B
esssup
! 2R X
m2Z j
^
(a m
!)j 2
+ X
k 2Z
k 6=0 s
k
b
k
b
1
C
C
A :
Pro of. A simple alulation shows that
[
m;n
(!)=a m
2
^
(a m
!)e i 2 a
m
bn!
:
Thuswegetforf 2L 2
(R),byrstusingPlanherel'stheorem forfuntions
in L 2
(R),thenwriting theintgeral overRasa sumof integrals,thenusing
Planherel'stheoremforp erio difuntions,thenexpandingthepro dutand
again writingthesum of integralsas one integral,
X
m;n2Z jhf;
m;n ij
2
= X
m;n2Z j D
^
f; [
m;n E
j 2
= X
m;n2Z
Z
R
^
f(!)a m
2
^
(a m
!)e i2 a
m
b
n!
d!
2
= X
m2Z a
m
X
n2Z
Z
1
a m
b
0 e
i2 a m
b
n!
X
j2Z
^
f !+ j
a m
b
^
a m
!+ j
b
d!
2
(byParseval'sequality applied tothe 1
a m
b
p erio di funtion
X
j2Z
^
f !+ j
a m
b
^
a m
!+ j
b
)
= 1
b
X
m2Z Z 1
a m
b
0
X
j2Z
^
f !+ j
a m
b
^
a m
!+ j
b
2
d !
= 1
b
X
m2Z Z 1
a m
b
0
X
j2Z
^
f !+ j
a m
b
^
a m
!+ j
b
X
k 2Z
^
f !+ j+k
a m
b
^
a m
!+ j+k
b
d!
= 1
b
X
m2Z X
k 2Z Z
R
^
f(!)
^
f !+ k
a m
b
^
(a m
!)
^
a m
!+ k
b
d!
= 1
b
Z
R j
^
f(!)j 2
X
m2Z j
^
(a m
!)j 2
!
d !
+ 1
b
X
m2Z X
k 2Z
k 6=0 Z
R
^
f(!)
^
f !+ k
a m
b
^
(a m
!)
^
a m
!+ k
b
d!
We have to get some estimates for the seond term, and we get by using
theCauhy-Shwarzinequality, ahangeofvariables, andthentheCauhy-
Shwarzinequality in the sumoverm:
1
b
X
m2Z X
k 2Z
k 6=0 Z
R
^
f(!)
^
f !+ k
a m
b
^
(a m
!)
^
a m
!+ k
b
d!
1
b
X
k 2Z
k 6=0 X
m2Z Z
R j
^
f(!)j 2
j
^
(a m
!)jj
^
(a m
!+ k
b
)jd!
1
2
Z
R j
^
f(!+ k
a m
b
)j 2
j
^
(a m
!)jj
^
(a m
!+ k
b )jd!
1
2
= 1
b
X
k 2Z
k 6=0 X
m2Z Z
R j
^
f(!)j 2
j
^
(a m
!)jj
^
(a m
!+ k
b
)jd!
1
2
Z
R j
^
f()j 2
j
^
(a m
k
b )jj
^
(a m
)jd
1
2
1
b
X
k 2Z
k 6=0 Z
R j
^
f(!)j 2
X
m2Z j
^
(a m
!)jj
^
(a m
!+ k
b
)j
!1
2
Z
R j
^
f()j 2
X
m2Z j
^
(a m
)jj
^
(a m
k
b
)j
! 1
2
1
b
kfk
2
L 2
(R) X
k 2Z
k 6=0 s
k
b
k
b
:
When we ombine the results we have dedued ab ove, we get the desired
onlusion.
It is lear that for (8.5)to hold we must have a
6=1, but it turns out
that one an quite easily get suÆient onditions for the assumptions of
Theorem8.3to hold.
Corollary 8.4. Assume that 2L 2
(R;R) isreal-valued,notthezerofun-
tion,
^
is ontinuous and satises
sup
! 2Rnf0g j
^
(!)j(1+j!j
)
j!j
<1;
for some onstants > +1 > 1. Then the assumptions of Theorem 8.3
hold, provided a
has beenhosensuÆientlyloseto1 andthenb
>0 has
beenhosento besuÆiently small.
Pro of. Sine isreal-valuedwehave
^
( !)=
^
(!)and sine
^
is ontin-
uousandnotidentiallyzerotherearep ositivenumb ers!
0
,Æand sothat
j
^
(!)jÆ when!
0
j!j!
0
+. Itfollows thatif 0<ja
1j<
!
0 then
essinf
! 2R X
m2Z j
^
(a m
!)j 2
Æ 2
>0;
b eause if for example 1 < a
< 1+
!
0
and a m
j!j < !
0
then a m+1
j!j
a
!
0
<!
0
+ and thereforethere isforevery !6=0 anindex m
suhthat
!
0 ja
m
!j!
0
+ and we havethe rstinequality in (8.5).
Nextwederive someuseful inequalities. Clearly
jxj
1+jxj
jx+sj
1+jx+sj
jxj
jx+sj
;
and if jsj>2jxjthen jx+sjjsj jxjjsj 1
2 jsj=j
s
2
jso thatweget
jxj
1+jxj
jx+sj
1+jx+sj
jxj
j s
2 j
; jsj2jxj:
(8.7)
In thesamewaywe getthefollowing rudeestimate
(8.8) jxj
1+jxj
jx+sj
1+jx+sj
1
jxj
1
jx+sj
1
jxj
j s
2 j
=
1
jxj 1
2
j s
2 j
+1
2
1
jxj +1
2
j s
2 j
1
2
1
jxj 1
2
j s
2 j
+1
2
;
jsj2jxj2:
b eause 1>0. On theother hand we haveforthe samereason
(8.9) jxj
1+jxj
jx+sj
1+jx+sj
1
jxj
=
1
jxj 1
2
jxj +1
2
1
jxj 1
2
j s
2 j
+1
2
; jsj2jxj:
Ifa
>0anda
6=1and we maywithoutlossof generalityassumethat
a
>1b eause inthesums involving a
we mayreplaem by mwhih is
thesameasreplaing a
by
1
a .
It followsfrom theassumptions that there isaonstant C suh that
j
^
(!)jC j!j
1+j!j
; !2R: (8.10)
Sine
^
(0) = 0 we have P
m2Z j
^
(a m
!)j 2
= 0 if ! = 0. Let ! 6= 0 and
let m
0
b e suh that a m
0
j!j 1 but a m
0 +1
j!j > 1, i.e., a m
1
j!j> 1 where
m
1
=m
0
+1. Then
X
m2Z
^
(a m
!)
2
C 2
m0
X
m= 1 ja
m
!j 2
1+ja m
!j 2
+C 2
1
X
m=m
1 ja
m
!j 2
1+ja m
!j 2
C 2
m
0
X
m= 1 ja
m
!j 2
+C 2
1
X
m=m1 ja
m
!j 2( )
=C 2
ja m
0
!j 2
1 ( 1
a
)
2 +C
2 ja
m
1
!j 2( )
1 a
2(
) C
2 1
1 a 2
+C 2
1
1 a 2( )
:
by the formula for the sum of a geometri series and b eause ja m
0
!j 1,
ja m
1
!j 1, 2 > 0 ja 2( ) < 0. This gives the seond inequality in
(8.5).
Ifnowjsj2thenwehaveby(8.7){(8.10)andthefatthata m0
j!j1
and a m
1
j!j>1that
X
m2Z
^
(a m
!)
^
(a m
!+s)
C 2
m
0
X
m= 1 ja
m
!j
s
2
+
+C 2
1
X
m=m1 ja
m
!j 1
2
s
2
+1
2
=C 2
ja m
0
!j
1 a
s
2
+
+C 2
ja m
1
!j 1
2
1 a 1
2
s
2
+1
2
:
FromthisweseethatthereisaonstantC
1
=C 2
2 +1
2
1
1 a
+
1
1 a 1
2
!
suh that
(s)C
1 jsj
+1
2
; jsj2:
Then
X
k 2Z
k 6=0 s
k
b
k
b
2C
1 b
+1
2
1
X
k =1 k
+1
2
;
when 0<b
1
2
. Thus we see that(8.6)holds provided
2C
1 b
+1
2
1
X
k =1 k
+1
2
<Æ 2
;
whih isp ossible if b
issuÆiently small.
