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Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 2
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Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 3
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Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 5
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Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 6
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Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 7
X
k
flfl
fl^g(! +2…k)
flfl
fl
2
=
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flfl
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k
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k
flfl
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flfl
fl
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X
k
flfl
fl^g(! +2…k)
flfl
fl
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k
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k
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X
k
hk’(2t?k) fhkg2 l2 (6)
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Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 8
(6)
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HTFourierMD¤
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1
P
k
hk’(2t?k)e?it!dt
= 12 P
k
hke?ik!=2
+1R
1
’(2t?k)e?i(2t?k)!=2d(2t?k)
= m0(!=2) ^’(!=2)
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k
hke?ik!
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rof
b
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ìμ
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X
k
hk = 2,(7)
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k
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k
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p.183).
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nf
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k
’(t?k)b?
R1
0 j
P
k
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k
R1
0 j’(t?k)jdt
= R+1?1 j’(t)jdt
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Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 9
f(t) =
X
k
ckei2k…t
?Fourier"
ck = R10 P
l
’(t?l)e?i2k…tdt
= P
l
R1
0 ’(t?l)e
i2k…tdt
= R+1?1 ’(t)e?i2k…tdt
=?’(2k…):
?(8)
Tw¤ck = 0?k 6= 0'?’(2k…) = 0 ?k 6= 0b
?
T?’(!) = m0(!=2)?’(!=2)
H |
i
7! = 2k… p¤
j?’(0)j2
= P
k
j?’(2k…)j2
= P
k
jm0(k…)j2j?’(k…)j2
= P
k
(jm0(2k…)j2j?’(2k…)j2 +jm0((2k +1)…)j2j?’((2k +1)…)j2)
= jm0(0)j2P
k
j?’(2k…)j2 +jm0(…)j2P
k
j?’(… +2k…)j2
= j?’(0)j2 +jm0(…)j2P
k
j?’(… +2k…)j2:
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m0(!)￥?
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[P
k
h2k = P
k
h2k+1b
Q-,?
1",μ
f(t) = Pk ’(t?k)
= PkPl hl’(2t?2k?l)
= PkPl hl?2k’(2t?l)
= Pl(Pk hl?2k)’(2t?l)
= Pl ’(2t?l)
= f(2t):
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Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 10
X
k
’(t?k) =?’(0):
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^
Z?(6)￥3b ?f’(t?k)gk2Z1S??"5
jm0(!)j2 +jm0(! +…)j2 = 1;8! 2 [0;2…],(10)
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1 = P
k
j?’(2! +2k…)j2
= P
k
jm0(! +k…)j2j?’(! +k…)j2
= P
k
jm
0(! +2k…)j
2j?’(! +2k…)j2 +jm
0(! +… +2k…)j
2j?’(! +… +2k…)j2￠
= P
k
jm
0(!)j
2j?’(! +2k…)j2 +jm
0(! +…)j
2j?’(! +… +2k…)j2￠
= jm0(!)j2 +jm0(! +…)j2,
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n
hnhn?2k = 2–k;0;8k 2 Z,(11)
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1
2
P
k
hke?ik!
1
2
P
k
hneik!
= 14 P
k
P
n
hnhn?k
e?ik!:

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n
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n
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Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 11
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j=1
m0(2?j!)￠?’(0)
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1Q
j=1
m0(2?j!)b
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k
hke?ik! ?? Om0(0) = 1 ?
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k
jhjjkj" < +1(0 < "? 1):
5
m0(!) = 1+O(j!j"):
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ìμ
jm0(!)?1j
=
flfl
flflP
k
hk(e?ik!?1)
flfl
flfl
2P
k
jhkj￠min(1;jk!j)
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k
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j=1
m0(2?j!)
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l ?# ??B? ??f
b
￡ ?iJ2 > J1?J1 sv
Hμ
log
J2Q
j=J1
m0(2?j!) =
J2P
j=J1
logm0(2?j!)
=
J2P
j=J1
O(2?j"j!j")
= o(1):
 ?i%??"K

T?￥o(1)! 2 KBáb#μ
J2X
j=J1
m0(2?j!) = 1+o(1)
! 2 KBáb?￡
ü
1Q
j=1
m0(2?j!) ?i?"
Bá
l ? Kf
)) ??.
? ?9y ?
í ke￥ ???bá
ì÷?
μ?ú￥?5?[# V?b?
?6?Xü￡
üm0(…) = 0bá
ì5óíB??2Tb
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 12
??? ? ? ?x 2Rnf0gá
ìμ
1Y
j=1
cos(2?jx) = sinxx,
??? ? ? ?10
!m0(!) V[s31
m0(!) =
1+e?i!
2
L
M0(!) (12)
?M0(…) 6= 0b
!f
M0(!) ?? O
@M0(!) = 1+O(j!j–)
– > 0b ?ik 2 Z+
P¤
Bk = max
!
flflM
0(!)￠￠￠￠￠￠M0(2k?1!)
flfl1=k < 2L?1=2,(13)
:? = L? 12?logBk > 0b5?’(!) =
1Q
j=1
m0(2?j!)μ
j?’(!)j? C (1+j!j)?12?" ;8! 2R (14)
￡á
ìμjm0(!)j? 1+Cj!j–0b?j!j? 1
H
j?’(!)j? e
1P
j=1
logjm0(2?j!)j
eC
P1
j=1(2?jj!j)
–0
C0
#?j!j? 1
H(14)? ?.C￡j!j > 1￥ f?bá
ìμ
1Q
j=1
(1+e?i2?j!2 )
= Q
j
e?i2?j?1! cos(2?j?1!)
= Q
j
e?i2?j?1!Q
j
cos(2?j?1!)
= e?i!=2(2=!)sin(!=2)
[
^’(!) = e?iL!=2
2
! sin(
!
2)
L
:
1Y
l=0
f(2?kl!)
?
f(!) = M0(!=2)￠￠￠M0(2?k!)
yj!j > 1il0 2 Z+
P¤2kl0? j!j < 2k(l0+1)b5· = 2?(l0+1)k!
@j·j? 1bV7
y1M09
@M0(!) = 1+O(j!j–)?
B?￥2Tμ
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 13
1Q
l=l0+1
flflf(2?kl!)flfl = 1Q
j=0
flflf(2?(j+l
0+1)k!)
flfl
=
1Q
j=0
flflf(2?jk2?(l
0+1)k!)
flfl
=
1Q
j=0
flflf(2?jk·)flfl
=
1Q
j=0
jM0(2?j·)j
C
]
H:fl = log2 Bkμjf(!)j? Bkk8! 2R
l0Q
l=0
flflf(2?kl!)flfl? Bk(l0+1)
k
Bk+logj!jk
= Bkk ￠j!jlogBk = Cj!jfl
[?j!j > 1
Hμ
flfl
fl^’(!)
flfl
fl? Cj!j?L ￠j!jfl = C0j!j?L+log2 Bk < Cj!j?12?"
??(14)
T Vw¤^’ 2 L2b?v￥LD?l￥Bk|?á’ μ?ú￥?5?
èHarrf
á
ìμm0(!) = 12(1 + e?i!)b#L = 1,Bk = 1?
H(14)
T?￥" = 12b
V7
flfl
fl^’(!)
flfl
fl? C(1+j!j)?1
á
ì |
m0(!) =
1+e?i!
2
2?
a+be?i!￠
'h0 = a=2h1 = (2a+b)=2h2 = (a+2b)=2h3 = b=2.
?Hq(7)m0(0) = 1¤
a+b = 1:
????Hq(11)?k = 1
Hw¤
h2h0 +h3h1 = 0:
'
(a+2b)a+b(2a+b) = 0:
#μa+b = 1ab =?12.Ka¤a = 1+
p3
2b =
1?p3
2
6?B?
^a =
1?p3
2.
á
�4"
￥DaubechiesloD4
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 14
h0 = 1+
p3
4 ;h1 =
3+p3
4 ;h2 =
3?p3
4 ;h3 =
1?p3
4,
V
M0(!) = 1+
p3
2 +
1?p3
2 e
i!;
¤
B1 = max!
flfl
flfl
fl
1+p3
2 +
1?p3
2 e
i!
flfl
flfl
fl =
1+p3
2 +
p3?1
2 =
p3 < 22?1:
?(14)¤
flfl
fl^’(!)
flfl
fl? C(1+j!j)?(2?log 32)
?2?log 32 > 1#D41Haarf
;áb
5.SSS??????￥￥￥   111HHHqqq
?B??á
ìXü￡
üf’(t?k)g1S??"￥ 1Hq
^
1
T? ?b? ?7X
￡
ü?’
^YV
￥FourierMD^’(!) =
1Q
j=1
m0(2?j!)
?l
H?
1
T Vw¤1?
ro
m0(!)￥Hq(10)b?μHq(10)i?
 ￡???b? V?/
￥ è0 Ab
f
’(t) = 13′[0;3](t)á
ìμ
^’(!) = 1
3i!(1?e
i3!)
?N¤
m0 =
^’(2!)
^’(!) =
1
2(1+e
i3!)
^2…?
ùf
bA ?μm0(0) = 1jm0(!)j2 +jm0(! +…)j2 = 1b
?
^f
h^
’; ^’
i
(!) = P
k
flfl
fl^’(! +2k…)
flfl
fl
2
1 ??f
 O
h^
’; ^’
i
(23…) = 0b?? ?3,
f’(k?k)gk2Z?
^Rieszb
á
ì5óB? sHqb
??? ? ? ?11
!1?m0(!)￥Hq(10)? ?5? ?10￥Hq/f’(t?l)gl2Z1S??
"b
￡
7
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 15
’n(!) =
nY
j=1
m0(2?j!)′[?2n…;2n…](!)
A ’n(!) !?’(!)a:e:b

1f’n(t?l)g1S??"b
1R
1
’n(t)’n(t?l)dt
= 12… R11?’n(!)?’n(!)eil!d!
= 12…
2n…R
2n…
nQ
j=1
jm0(2?j!)jeil!d!
= 12…
2n+1…R
0
nQ
j=1
jm0(2?j!)j
2
eil!d!
= 12… R2n…0 (
n?1Q
j=1
jm0(2?j!)j2)￠jm0(2?n!)j2 eil!d!
+ 12… R2n+1…2n… (
n?1Q
j=1
jm0(2?j!)j2)￠jm0(2?n!)j2 eil!d!
= 12… R2n…0 (
n?1Q
j=1
jm0(2?j!)j2)￠jm0(2?n!)j2 eil!d!
+ 12… R2n…0 (
n?1Q
j=1
jm0(2?j(! +2n…))j2 ￠jm0(2?n! +…)j2 eil(!+2n…)d!
= 12… R2n…0 (
n?1Q
j=1
jm0(2?j!)j2)￠(jm0(2?n!)j2 +jm0(2?n! +…)j2)eil!d!
= 12… R2n…0 (
n?1Q
j=1
jm0(2?j!)j2)eil!d!
= ￠￠￠ = 12… R4…0 jm0(!=2)j2eil!d!
= 12… R2…0 jm0(!=2)j2eil!d! + 12… R4…2… jm0(!=2+…)j2eil!d!
= 12… R2…0 eil!d!
= –l;0:
?N ?^w¤R ’n(t?m)’n(t?l)dt = –m;lb

2á
ì￡
ü
j?’n(!)j? C(1+j!j)?12;? > 0 (15)
?è
C?G ??n;!b
_? ?10￥w?á
ìμ
j?’n(!)j =
flfl
fl2?n sin !2=sin(2?n?1!)
flfl
fl
L ￠ nY
j=0
flflM
0(2?j!)
flfl′
[?2n…;2n…](!)
?¨?j·j? …2
Hμjsin·j? …2 j·já
ì¤
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 16
flfl
flfl 1
sin(2?n?1!)′[?…;…](2
n!)
flfl
flfl? …
22
n+1j!j?1 ;8!
[
flfl
flfl 2?n sin !2
sin(2?n?1!)′[?2n…;2n…](!)
flfl
flfl? 2?n…
22
n+1j!j?1? C(1+j!j)?1:
%?k
7
n = km+q;0? q < k:
??M0(!) ??[2…1?
ù#μ?b?
Hμ
nQ
j=0
jM0(2?j!)j
=
kmQ
j=0
jM0(2?j!)j
km+qQ
j=km+1
jM0(2?j!)j
=
mQ
j=0
flflf(2?jk!)flfl qQ
j=1
flflM
0(2?j?km!)
flfl
C
mQ
j=1
flflf(2?jk!)flfl
!l01?

P¤2kl0?j!j < 2k(l0+1)b
?m > l0
H?
mY
j=0
flflf(2?jk!)flfl = l0Y
j=0
flflf(2?jk!)flfl mY
j=l0+1
flflf(2?jk!)flfl
?? ?10￥￡
ü -By0??VCj!jfl ;fl = logBkaBy0μ?b
?m < l0
H?jf(!)j? Bkk(?^
!Bk > 1)á
ìμ
mY
j=0
flflf(2?jk!)flfl? Bkm
k? B
k(l0+1) = Cj!jfl
?"á
�
j?’n(!)j? C(1+j!j)?L(1+j!j)fl? C(1+j!j)?12?"
'(15)? ?b
?Bk? 1
Hμj?’n(!)j? C(1+j!j)?L.

3f’(t?l)g1S??"b
?(15)j’n(!)j2μ V￥ e?f
C(1 +j!j)?1?2"#?Lebesgue e?
l ?? ??’n !
’L2?b

1?X￡
ü
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 17
1
2…
Z +1
1
j?’n(!)j2eil!d! = –l;0
#μ
1
2…
Z +1
1
j?’(!)j2eil!d! = –l;0
?Nw¤f’(t?l)g
^S??"b
?1(13)?k"1
H￥Hq
n5?Daubechies
19884'
m0(!) =
1+e?i!
2
L
M0(!)
?sup
!
jM0(!)j? 2L?12
2Mallat
1989ó
 ?/￥ sHq
inf
j!j?…=2
jm0(!)j > 0
3Cohen
1990·MallatHq?￥…=2 V?1…=3b'm0(!)[?…=3;…=3]?
àμ
,
?b
4 -
￥ è0m0(!) = 1+e?i3!2á
ìμm0(§…3) = 0b
? ?11?á
ì ?¨
1?m0￥Hq(10)[#(12)(13)bá
ìóB?Ff
'
&
￥ sHqb
??? ? ? ?12:
!m0
^ ??￥2…?
ùf

@Hq(10)'
jm0(!)j2 +jm0(! +…)j2 = 1;m0(0) = 1:
?
1Q
j=1
m0(2?j!)
l ?’ OμRiesz/?Hq
[?’;?’](!) =
X
k
j?’(! +2k…)j2? C > 0;a:e,(16)
5f’(t?k)gk2Z
^S??"b
￡á
ì V[`? ?11￥￡
ü?
*"?l￡ù?
f?’ngb ?a1ù?(16)
T
a
ü V[
pK%? Vn
C ?p.p.7980b
?(1)￡? ?(16)￥Hq V
1￡S??￥ 1Hq
X
k
j?’(! +2k…)j2 = 1
1 ?^¤b
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 18
(2)? ?12￥Hq/,?
àμHq(16)
H,á
ì ˉ V￡
ü’ 2 L2(R) Ok’k? 1.
Y
L
,_
? ?11?￥￡
ü,?l?’n(!).5μ?’n(!) !?’(!) Ok?’nk = 2….y1j?’n(!)j? 0,?Fatou?
?Z +1
1
j?’(!)j2d!? limn!1
Z 1
1
j?’n(!)j2d! = 2…:
1?S???Cohen(1990)DLawton(1990)sY? ?1¤
 sA1Hqbá
ì?
?
CohenHq %? V? IDaubechiesp.p,182—194b
?l?"K?1
2…D[?…;…]]?
Congruent ?

1jKj=2…('K￥?D[…;…]M])

2 ?! 2 [?…;…]il 2 Z
P¤! +2l… 2 Kb
è
!
K = [?27…=8;?13…=8][[?…;?…=2][[?…=4;5…=8][[3…=4;…][[3…=2;7…=4]:
5K
^
2…D[?…;…]]?￥bK V[ A?5| uW[?…;…]?￥0 uW￡?…2; …2?￡5…8 ; 3…4?g M
?
cut ?a| -?_·
üM2…a?_P
üM4…aTL
pasteí ?b
??? ? ? ?13:
Cohen
!m0 (!)
^ ??￥2…?
ùf

@Hq(10) O
1Q
j=1
m0 (2?j!)??
l ?’(!)b5f’(t?k)gk2Z
^S??"? O??/
Hq? ?
i?"K
2…]?? uW[?…;…], Ocμe?O￥
#×
P¤
inf
j>0
inf
!2K
flflm
0
2?j!￠flfl > 0:
LawtonHq[
ro m0 (!)￥"
fhkg￥?
Tó
DCohenHq?Nb
6.111???\Vj = f0g[Vj = L2(R)
'??á
ì I
nMRA?l?￥Hq
2'
\Vj = f0g[Vj = L2(R):
á
ìL?f’(t?k)gk2Z
^V0￥Riesz'
2
T? ?bN
H ?^￡f’j;kgk2Z
^Vj￥Rieszb
? ?
!f 2 C10 (R)5
X
k
jhf;’j;kij2 ! 0?j !?1,(17)
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 19
￡
ü |?
r
P¤suppf ‰ [?r;r].:Ij;k = [?2jr?k;2jr?k]?j < 0; sl
H,Ij;p \
Ij;q = ;(p 6= q)b5μ
P
k
jhf;’j;kij2? 2j
R
jtj?r
jf(t)j￠j’(2jt?k)jdt
!2
2j kfk21 ￠2r￠P
k
R
jtj?r
j’(2jt?k)j2 dt
= 2rkfk21P
k
R
Ij;k
j’(t)j2 dt
= 2rkfk21P
k
R+1
1 ′Ij;k (t)j’(t)j
2 dt
= 2rkfk21
+1R
1
′[kIj;k (t)j’(t)j2 dt:
f
′[
k
Ij;k (t)"?
?[?
l ??0?j !?1
Hb
[+))
l ??0b?
?’ 2 L2? e?
l ?? ?¤
lim
j!?1
+1Z
1
′[kIj;k (t)j’(t)j2 dt =
+1Z
1
lim
j!?1
′[
k
Ij;k (t)j’(t)j
2 dt = 0:
? ?
!f’(t?k)gk2Z
^V0￥Riesz:Pj1L2 (R)?Vj￥g?
05 ?f 2 C10 (R)
μ
lim
j!?1
kPjfk = 0 (18)
￡?V0 ‰ Vj Vw¤f’j;kgk2Z
^Vj￥Riesz'μ
Akfk2?
X
k
jhf;’j;kij2? Bkfk2 8f 2 Vj:
?? ?4VjiS??#Pj
^??g?
0y7
^1
a￥b
[ ?
f 2 C10 (Rn)μ
kPjfk2? 1A P
k
jhPjf;’j;kij2
= 1A P
k
jhf;Pj’j;kij2
= 1A P
k
jhf;’j;kij2
?(17)'￡¤(18)b
??? ? ? ?14\jVj = f0gb
￡
!f 2 \jVj ?ó￥? > 0??C10 (R)L2 (R)?if0 2 C1o (R)
P
¤kf?f0k < "2byf 2 Vj;8j#Pjf = fb
[μ
kPjf0?fk = kPj (f0?f)k?kf0?fk <?2:
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 20
?(18)
Tij0 < 0
P¤?j < j0
Hμ
kPjf0k <?2:
[
kfk?kPjf0k+kPjf0?fk <?
?N¤f = 0b
Cá
ì_?￡
ü
[Vj = L2 (R):
á
ì? ?L2 (R)￥B?
á0 bW
C10

R
·
=
n
f,f 2 L2;?f 2 C10 (R)
o
:
??? ? ? ?C10

R
·
L2 (R)?
áb
￡f 2 L2 (R)μ?f 2 L2 (R)b??C10 (R)L2 (R)?
ái?
ffng‰ C10

R
·

P¤
ffn
! 0
[kf?fnk! 0?n !1b
??? ? ? ? ?f 2 C10

R
·
ij0
P¤?j > j0
Hμ
X
k
jhf;’j;kij2 = 12…
Z +1
1
flfl?’?2?j!￠flfl2flfl
fl?f (!)
flfl
fl
2 d! (19)
￡?Plancherel
T[#
n
e?ik2?j!
o
k2Z
^L2 [0;2j+1…]
￥S??á
ìμ
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 21
P
k
flfl
flR+1?1 f (t)2j2’(2jt?k)dt
flfl
fl
2
= P
k
flfl
fl 12… R+1?1?f (!)2?j2?’(2?j!)eik2?j!d!
flfl
fl
2
= 12… P
k
flfl
flflR2j+1…0 P
n
f (! +2j+1…n)?’(2?j! +2…n)eik2?j(!+2j+1…n).p2j+1…d!
flfl
flfl
2
= 12… R2j+1…0
flfl
flflP
n
f (! +2j+1…n)?’(2?j! +2…n)
flfl
flfl
2
d!
= 12… R2j+1…0 P
n;l
f (! +2j+1…n)?f (! +2j+1…l)?’(2?j! +2…n)?’(2?j! +2…l)d!
= 12… P
n;l
R2j+1…(n+1)
2j+1…n
f (!)?f (! +2j+1…(l?n))?’(2?j!)?’(2?j! +2…(l?n))d!
= 12… P
n;l
R2j+1…(n+1)
2j+1…n
f (!)?f (! +2j+1…l)?’(2j!)?’(2?j! +2…l)d!
= 12… P
l
R+1
1
f (!)?f (! +2j+1…l)?’(2?j!)?’(2?j! +2…l)d!
= 12… R+1?1
flfl
fl?f(!)
flfl
fl
2j?’(2?j!)j2 d! + 1
2…
P
l6=0
R+1
1
f(!)?f(! +2j+1…l)?’(2?j!)?’(2?j! +2…l)d!:
?f￥|"1?"#?j sv
H ?l 6=0,μ
supp?f(!)\supp?f(! +2j+1…l) = ;:
V7

T?=[1
,b?N￡¤
19b
?
19? ?f 2 C10 (R),5·
3F
B?[RjRj ! 0?j !1.
??? ? ? ?15
!f’j;kg1Vj￥Riesz?’(!)μ? O?’(!)0? ?? O?’(0) 6= 0.5
[j2ZVj = L2(R):
￡
!f? [j Vjbá
ì￡
üf = 0b > 0??C10 (?R)L2(R)?
áif0 2
C10 (?R)
P¤
kf?f0k <?:

1Pjf = 0;8j 2 Z.
!f = f1 +f2;f1 2 Vj;f2 2 V?j5?f?[j Vjw¤< f;f1 >= 0 O
< Pjf;Pjf >=< f;P?j Pjf >=< f;Pjf >=< f;f1 >= 0:
#Pjf = 0b

2kPjf0k <?.
?
^y1
kPjf0k = kPj(f0?f)k?kf0?fk <?:

3? OHq ?g 2 Vj,μ
k
X
k
< g;’j;k > k2? Bkgk2:
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 22
+YPjf0μ
BkPjf0k2? Pk j < Pjf0;’j;k > j2
= Pk j < f0;Pj’j;k > j2
= Pk j < f0;’j;k > j2
= 1=2…R1?1j?’(2?j!)j2j?f0(!)j2d!:
'μZ
+1
1
flfl?’(2?j!)flfl2flfl
fl?f0(!)
flfl
fl
2d!? 2…B?2:
?’(!)μ? O0) ???Lebesgue e?
l ?? ??j !1
Hμ
j?’(0)j2
Z 1
1
j?f0(!)j2d!? 2…B?2kf0k? C?:
[
kfk?kf?f0k+kf0k < (1+C)?:
V7f = 0b
? ?15??F
Hq?’(0) 6= 0?
^A1￥b
L=
?? ?6￥￡
ü,á
ìμ
??? ? ? ?16
!m0(!)1 ??￥2…?
ùf
 OQ1j=1 m0(2?j!)
l ?? ??f
’(!)b Of’(t?
k)gk2Z
3?V05μ

1?’(0) 6= 0;?’(2k…) = 0;8k 6= 0,

2m0(0) = 1,

3?’ 2 L1(R)
HX
k
’(t?k) =?’(0):
??’(0)D[jVj￥1"￥)
 Vn
C ?,p.67.
7.111???fff
￥￥￥|||"""
?
rof
m0(!) = 1=2Pk hke?ik!á
ì?fhkg1
(mask)bB?×1￥ f?
^
fhkg?oμμK??1
,b'μ
hk = 0;?N > N2N < N1:
?
H?￥f
￥|"μz￥?éb
??? ? ? ?17:
!m0(!) = 1=2PN2k=N1 hke?ik! Oμ
m0(0) = 1;'
N2X
k=N1
hk = 2:
5?í ke
’(!) =
1Y
j=1
m0(2?j!)
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 23
?l￥f
’(t)￥|"c?[N1;N2].
￡?31¨?1?ˉ×
￥FourierMD￥Paley-Wiener? ?%? VnDaubechies,pp
176-177.
31·￥
^í k
9 V
á
3?|"￥f
b è ?
’(t) =
8
><
>:
1; 0? t < 1;
1=2; 1? t < 2;
0; 
:
￥FourierMD?’(!) = (i!)?1=(2?e?i!?e?2i!)#
m0(!) =?’(2!)?’(!) = (1+e
i!)(2+e?2i!)
2(2+e?i!),
1 ??￥2…?
ùf
??
^ ??[
T#
 μíKéb
8.
ZZZ???￥￥￥333
?
Z?(6)
’(t) =
X
k
hk’(2t?k);
-
á
ì¨í ke
q×
=ó
 p3ZEb/
á
ì)
?°¤3Eb
ZEBá
ì I
n ?/￥Y}V?
·0(t) = ′[?1=2;1=2](t);
·n+1(t) = Pk hk·n(2t?k);n = 0;1;￠￠￠,
?

Y}V?f·ngμK5á
ì?
Z?(6)Y} V3b? ?6￥Hq/ V
[￡
ü?·n(!) !?’(!),·n(t) ! ’(t)L2?b ￡
ü Vn
C ?,pp 107-108.
??
E?1Subdivision
ElossC[ -üX$ |
ì
ù?b
?fhkgμμK￥

H V[°¤ú ?￥ p3b
ZE=
!
’(t) =
N2X
k=N1
hk’(2t?k); (20)
?? ?16á
ì V[’(0) = 1,'
X
k
’(t?k) =?’(0) = 1,(21)
?? ?17supp’ ‰ [N1;N2].#?
?oμ’(N1 +1);’(N1 +2);￠￠￠ ;’(N2?1) V
d
,b? V?(20)#(21)·B p3b
? ?m=2￥???N1 < m=2 < N2,?(20) V
¤
’(m2 ) =
N2X
k=N1
hk’(m?k):
×ˉ

V?á
ì V[ú ?19
’(t)t = m=2j?"￥=é?
￥
′b
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 24
9.lllooofff

I
n??MRA'V0iS??b??Vj ‰ Vj+1 ?j# V?lWj1VjVj+1?
￥???'
Vj+1 = Vj 'Wj:
¨Qj:L2(R)?Wj￥g?
0
Qj,L2(R) ! Wj:
1?0 bWWj,á
ìμ
??? ? ? ?: ?j,μ
(1) Wj?Wj+1,
(2) f(t) 2 Wj? O??f(2t) 2 Wj+1.
￡
1 ?f 2 Wj;g 2 Wj+1??Wj ‰ Vj+1;Vj+1?Wj+1¤< f;g >= 0,#
1?
?b

2
!f(t) 2 Wj,5μ< f(t);g(t) >= 0;8g(t) 2 VjV7μ< f(2t);g(2t) >= 0;8g(2t) 2
Vj+1.?f(t) 2 Vj+1,
[f(2t) 2 Vj+2,#f(2t) 2 Wj+1b
Q- V ?
￥￡
üb
??? ? ? ?18
!fVjg
^L2￥B?MRAf?(t?k)gk2Z
^W0￥S??5f?j;kgj;k2Z
^L2(R)￥
S??.
￡y1f?0;kgk2Z
^W0￥S??#f?j;kg
^Wj￥S??".?? -B? ?￥

2?f (t) 2 Wj
Hμf (2?jt) 2 W0
[
f?2?jt￠ =
X
k
ck?0;k(t);ck =
Z
f?2?jt￠?(t?k)dt;
V7
f(t) =
X
k
ck2?j2?j;k(t)
Vμf?j;kgL?V?
a
üf?j;kgk2Z
^Wj￥b
? -B? ?￥
1w¤Wj?Wj0;j 6= j0b?Vj+1 = Vj 'Wj¤VJ+1 = V?J J'
j=?J
Wj#
8f 2 L2(R)μ
PJ+1f?P?Jf =
JX
j=?J
X
k
< f;?j;k >?j;k:
7J !1i?MRA￥?é
2¤
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 25
f =
X
j;k
< f;?j;k >?j;k:
#f?j;kgj;k2Z
^L2(R)￥S??b
???lllW0￥
3?í??1(
)lo
mother wavelet.
/
á
ì)
8"/lof
b
??? ? ? ?
!’ 2 L2(R)1f
5g 2 L2(R);g?V0? O??
[?g;?’](!) =
X
k
g(! +2k…)?’(! +2k…) = 0 ;a:e,(22)
￡?g;’ 2 L2(R)
H[?g;?’] 2 L1[0;2…] O(22)?￥í k)
^ '
l ?￥?
^y1
Z 2…
0
X
k
j?g(! +2k…)j￠j’(! +2k…)jd! =
Z +1
1
j?g(!)’(!)jd!?jjgjj￠jj’jj:
 ?k 2 Zá
ìμ
hg;’0;ki = 12… R+1?1?g(!)?’(!)eik!d!
= 12… R2…0 P
n
g(! +2n…)?’(! +2n…)eik(!+2n…)d!
= 12… R2…0 [?g;?’](!)eik!d!
[hg;’0;ki
^L1[0;2…]
 V￥f
[?g;?’]￥Fourier"
.
?g?V0
Hhg;’0;ki = 0;8kyN[?g;?’](!) = 0; a:e:
Q- ?[?g;?’] = 0; a:e:5hg;’0;ki = 0V7 ?f = P
k
ck’0;k 2 V0μhg;fi =
0'g?V0b
á
ìL?f
’
@S(’) = V0 O
’(!) = m0(!2)?’(!2)jm0(!)j2 +jm0(! +…)j2 = 1; m0(0) = 1:
n5ùsf 2 W0?
Há
ìμ
f 2 V1f?V0:
??B?Hq
üM?MV
U? ?1i2…?
ùf
m(!)
P¤
f(!) = m(!
2)?’(
!
2):
??=?HqD -B? ?w¤
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 26
0 = [?f;?’](!)
= P
k
f(! +2k…)?’(! +2k…)
= P
k
m(!2 +k…)?’(!2 +k…)m0(!2 +k…)’(!2 +k…)
= P
k
[m(!2 +2k…)m0(!2 +2k…)j?’(!2 +2k…))j2
+m(!2 +(2k +1)…)m0(!2 +(2k +1)…)j?’(!2 +(2k +1)…)j2]
= m(!2)m0(!2)P
k
j?’(!2 +2k…)j2 +m(!2 +…)m0(!2 +…)P
k
j?’(!2 +… +2k…)j2
= m(!2)m0(!2)+m(!2 +…)m0(!2 +…):
[
m(!)m0(!)+m(! +…)m0(! +…) = 0;a:e,(23)
?V
ü?ê_
(m0(!);m0(! + …))D_
(m(!);m(! + …))
^??￥.??m0(!)Dm0(! +
…) a:e:?]
H1
,i?
ùf
(!)
P¤
m(!) =?(!)m0(! +…); (24)
V?l1
(!) =
(?m(!+…)
m0(!) ; m0(!) 6= 0;
m(!)
m0(!+…); m0(! +…) 6= 0;m0(!) = 0:
} ?(23)
T¤
(!)+?(! +…) = 0;a:e:
[?(!) = P
k
ke?ik!2k = 0;8k.# V
(!) = e?i!”(2!);”(!)12…?
ùf
,(25)
Q-A ??(24)(25) ??￥m(!)
@(23)b
á
ì+Y |”(!) · 15μ
m1(!) = e?i!m0(! +…)
= e?i! 12 P
k
ˉhke?ik(!+…)
= 12 P
k
(?1)kˉhkei(k?1)!
= 12 P
k
(?1)k?1ˉh1?ke?ik!:
#á
ì:
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 27
m1(!) = 12
X
k
gke?ik!;
?
gk = (?1)k?1h1?k:
?¨FourierMD?lf

(!) = m1(!
2)?’(
!
2)
= e?i!2 m0(!2 +…)?’(!2):
/
á
ì￡f?(t?k)k2Zg
^S??"b1No3￡
ü
h?;
i
= 1;a:e:
7á
ìμ
P
k
flfl
fl(! +2…)
flfl
fl
2
= P
k
flflm
0(!2 +k… +…)
flfl2flfl?’(!
2 +k…)
flfl2
= P
k
hflfl
m0(!2 +2k… +…)flfl2flfl?’(!2 +2k…)flfl2 +flflm0(!2 +2k…)flfl2flfl?’(!2 +(2k?1)…)flfl2
i
= flflm0(!2 +…)flfl2P
k
flfl?’(!
2 +2k…)
flfl2 +flflm
0(!2)
flfl2P
k
flfl?’(!
2?… +2k…)
flfl2
= flflm0(!2 +…)flfl2 +flflm0(!2)flfl2
= 1
1￡f?0;kg
^W0￥á
ì3￡ ?if 2 W0iflkg2 l2
P¤
f(t) =
X
k
dk?0;k(t),(26)
?(24)(25)f VV
U1
f(!) =,(!)(!)
?
2…R
0
j”(!)j2d! = 2
…R
0
j?(!)j2d!b
??f 2 V1;f = P
k
fk’1;k OμP
k
jfkj2 = kfk2b?
H?f(!) = m(!=2)?’(!=2)?￥?]
?
ùf
m,μ
2…Z
0
jm(!)j2 d! = 2…
X
k
jfkj2 = 2…kfk2:
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 28
?(24)μ
2…R
0
jm(!)j2 d!
=
2…R
0
j?(!)j2jm0(! +…)j2 d!
=
…R
0
j?(!)j2jm0(! +…)j2 d! +
2…R
…
j?(!)j2jm0(! +…)j2 d!
=
…R
0
j?(!)j2jm0(! +…)j2 d! +
…R
0
j?(! +…)j2jm0(! +2…)j2 d!
=
…R
0
j?(!)j2 d!
#μ
2…R
0
j”(!)j2 d!
= 2
…R
0
j?(!)j2 d!
= 2
2…R
0
jm(!)j2 d!
= 4…kfk2 < +1:
[f VV
U1(26)￥?
T ?fdkg1”(!)￥Fourier"
b
8[
á
ì￡
ü

??? ? ? ?19
!fVjgj2Z1L2(R)￥s’1S??f
 O2…?
ùf
m0(!)
@
m0(0) = 1jm0(!)j2 +jm0(! +…)j2 = 1;
[#?’(!) = m0(!2)?’(!2)b?l?(t)1
(!) = e?i!2 m0(!
2 +…)?’(
!
2) (27)
5f’j;k(t)gj;k2Z
^L2(R)￥S??b
? μ

?é￥f

^?·B￥?]￥?D? VMμB?
11￥2…?
ùy
0‰(!),(!) = ‰(!)(!)bnDaubechies,p,135b
á
ì V[ |‰(!) = ‰0eim!;m 2 Z;j‰0j = 1 ?
üM?(t)b
10??????lllooo￥￥￥ è è è000

1Haar
’(t) = ′[0;1](t);?’(!) = 1i!(1?e?i!)
[
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 29
m0(!) =?’(2!)?’(!) = 12(1+e?i!);
'h0 = h1 = 1lof

-
F
￥B?μ|
(!)
=?e?i!2 (1+e?i(!2 +…)2 )?’(!2)
= 1i!(1?e?i!2 )2
V7
(t) = ′[0;1
2]
(t)?′[1
2;1]
(t):
M?￥1"1
’(t) = ’(2t)+’(2t?1);?(t) =?(2t)+?(2t?1):

2Battle-Lemarie.
1?"Hf
Tf
?a??
¤￥bBQ
Hμ
’(t) =
(
1?jtj; jtj? 1;
0; ;
?
H
’(!) = 1i!(ei!2?e?i!2 )2 = (2! sin !2)2:
#μ
m0(!) = cos2 !2:
[?’;?’](!)
= P
k
j?’(! +2k…)j2
= P
k
sin2 !
2
(!2 +k…)2
·2
= sin4 !2 ￠
1
sin4 !2?
2
3
1
sin2 !2
·
= 1? 23 sin2 !2
= 13(1+2cos2 !2):
[’D
￥?
üM????"b
7
’#(!) =?’(!)p[?’;?’](!) = p3 4sin
2 !
2
!2(1+2cos2 !2)12
5f’#(t?k)gk2Z
3?
??MRAb?’#X?
^?|"￥b
N
H
rof
1
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 30
m#0 (!) = cos2 !2
1+2cos2 !
2
1+2cos2 !
12
[
(!) = e?i!2 m#0 (!
2 +…)?’
#(!
2)
= p3e?i!2 sin2 !4
1+2sin2 !
4
1+2cos2 !2
·1
2
1
1+2cos2 !4
·1
2,
B?NQ
H V¨FourierMD?lf
’(t)
’(!) = e?i?!2
2
! sin
!
2
N+1
?? = 0?N= 
? = 1?N=
}
b
L=
?N1 
H
’(t) = ′[?1
2;
1
2]
￠￠￠￠￠￠?′[?1
2;
1
2]
(t)(N +1Q )
?N1
}
H
’(t) = ′[?1
2;
1
2]
￠￠￠￠￠￠?′[?1
2;
1
2]
(t? 12)
^?|"s
NQ[
T?8
?CN?1￥f
'NQ"Hf
b
??m0(!) = ei[N+12 ]!(1+e?i!2 )N+11 ??[
T#’(t)￥
Z?1
’(t) =
8
>><
>>:
2?2M
2M+1P
j=0
Cj2M+1’(2t+M?j); N = 2M;
2?2M?1
2M+2P
j=0
Cj2M+2’(2t+M +1?j); N = 2M +1:
N?[?’;?’]￥Vr
T9 V[A
Uby7 V¨??V?f
’#(t)b V[
￡
ü’#(t)μ·
hb

3Meyer
!”(t)
^C1f

@
”(t) =
(
0; t? 0;
1; t? 1;
”(t) +,(1?t) = 1b μμK;á?￥ è01”1(t) = t;0 < t < 1,”2(t) = sin2(…t2 );0 <
t < 1,[
!”3(t) = t4(35?84t+70t2?20t3)?b
?l
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 31
’(t) =
8
><
>:
1; j!j? 23…;
cos[…2”( 32…j!j?1)]; 23…?j!j? 43…;
0; ;
5μ[?’;?’](!) = 1:b
Y
L
?|"[#?
ù?éo3￡
j?’(!)j2 +j?’(2…?!)j2 = 1;8! 2 [0;2…]:
?0? !? 2…3
H4…3? 2…?3? 2…

TA ?? ?b] ? V￡4…3? !? 2…￥ f?b
?2…3? !? 43…
Hμ
j?’(!)j2 +j?’(2…?!)j2
= cos2[…2”( 32…!?1)]+cos2[…2”( 32…(2…?3)?1)]
= cos2[…2”( 32…!?1)]+sin2[…2”( 32…!?1)]
= 1:
[’
3?B???MRAb
?
rof
m0(!)á
ìμ
m0(!) =
X
k
’(2! +4k…):
Kaá
�
(!) =
8>
<
>:
e?i!2 sin[…2”( 32…j!j?1)]; 2…3?j!j? 4…3 ;
e?i!2 cos[…2”( 34…j!j?1)]; 4…3?j!j? 8…3 ;
0; 
:
Meyer
^C1￥ O y
/???
^·
/?b%? V? I
C ?p.p,118-119b
11Daubechieslllooo
-
￥ è0?Haar?;áBattle-LemarieDMeyer
àμ?|"b'??|o
;á
?|lo￥/b?
^Daubechies1988
M
n5ó￥b
V
ro m0(!) 7
Sbá
ìXμ
m0(0) = 1; jm0(!)j2 +jm0(! +…)j2 = 1; (10)

m0(!) =
1+e?i!
2
N
M0(!),(12)
??Haarlo??N = 1á
ì?? |N? 2bN| e?’￥?5?.
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 32
á
ì?? ??|"￥f
loy7L?fhkg?oμμK?d
,i O1
L
?
'
!μ
Z?
’(t) =
N2X
k=N1
hk’(2t?k):
üV
üM’1(t) = ’(t+N1)a

T?1
’1(t) =
N2?N2X
k=0
hk+N1’1(2t?k):
[á
ì?^
!hk = 0?k < 0?
Hμ[
T
H(z) = 12
X
k
hkzk:
5á
ìμ
m0(!) = H(e?i!)
O
H(z) =
1+z
2
N
Q(z) ;
Q(z)1
L"
[
T OQ(1) = 1;Q(?1) 6= 0:
C)
8"?(10)
T¤[
TQ(z)b
??Q(e?i!) = Q(ei!)
[jQ(e?i!)j2 = Q(e?i!)Q(ei!)1!￥
}f
#
^cos!￥[
Tb ?¨1?cos!2 = sin2 !2
?
^sin2 !2￥[
T:-1P(y);y = sin2 !2b
y1
1+e?i!
2
·N
￠
1+ei!
2
·N
=?1+cos!2 ￠N
=?cos2 !2￠N
= (1?y)N,
?(10)¤
(1?y)NP(y)+yNP(1?y) = 1; P(y)? 0 8y 2 [0;1],(28)
1 p3}
Z?(28)5óB?￡ù2Tb
??? ? ? ?(Bezout):
!P1P2
^Q
1n1n2￥o
í[
T5i·B￥Q
sY??Vn2?
1n1?1￥[
Tq1q2
P¤
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 33
p1(t)q1(t)+p2(t)q2(t) = 1:
￡nDaubechies p.p.169-170.
?P(y)￥Q
??VN?1
H?Bezout? ?3
^·B￥|(28)?1
P(y) = (1?y)?N?yN(1?y)?NP(1?y);
?Taylor? ??=[￥Q
9ú?N?1#N?1Q[
T31
P0(y) =
N?1X
k=0
CkN+k?1yk:
B?3P(y) V1
P(y) = P0(y)+yN (y):
} ?(28)¤
(y)+ (1?y) = 0:
7 (y) = R(12?y)5μR(y) =?R(?y)b
[B?3
P(y) =
N?1X
k=0
CkN+k?1yk +yNR(12?y):
?á
ì?R(y) · 05μ
flflQ(e?i!)flfl2 = N?1X
k=0
CkN+k?1
sin2 !2
·k
:
á
ì31 p[
TQ(z)bRiesz;óB?/?ZEb
Riesz??? ? ? ?
!a0;a1;￠￠￠ ;aM1
L
aM 6= 0
P¤
A(!) = a02 +
MX
k=1
ak cos(k!)? 0;! 2R:
5i
L"
[
T
B(z) =
MX
k=0
bkzk
P¤
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 34
flflB?e?i!￠flfl2 = A(!):
￡ I
n2MQ[
T
PA(z) = 12
MX
k=?M
ajkjzM+k;
A ?μ
PA(z) = zMA(!);z = e?i!

PA(z) = z2MPA(1z) z 2 C:
y1aM 6= 0#PA(0) 6= 0'
,?
^PA(z)￥?b
?

?
Tá
ìw¤ ?z
^?51z9
^? Oy1fakg
^
L￥z9
^?b#á
ì¤
?
L?
r1;r2;￠￠￠ ;rk; 1r
1; 1r
2;￠￠￠ ; 1r
k;
ˉ?
z1;z2;￠￠￠ ;zj;z1;z2;￠￠￠ ;zJ;￠￠￠ ; 1z
1; 1z
2; 1z
J;￠￠￠ ; 1z
1; 1z
2;￠￠￠ ; 1z
J
:
'PA(z) V[s31
PA(z) = 12aM
KY
k=1
(z?rk)(z? 1r
k
)
!? JY
j=1
(z?zj)(z?zj)(z? 1z
j
)(z? 1z
j
)
!
K +2J = M.
?z = e?i!
H
flfl
flfl(z?zj)(z? 1
zj)
flfl
flfl = jzjj?1jz?zjj2
#á
ìμ
A(!) = jA(!)j = jPA(z)j = 12 jaMj￠
KY
k=1
1
jrkj
!
￠
JY
j=1
1
jzjj2
!
￠
flfl
flfl
fl
KY
k=1
(z?rk)
flfl
flfl
fl
2
￠
flfl
flfl
fl
JY
j=1
(z?zj)(z?zj)
flfl
flfl
fl
2
[B(z) V |1
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 35
B(z) =
1
2
flfl
flfl
flaM
KY
k=1
1
rk
JY
j=1
1
z2j
flfl
flfl
fl
!12
￠
KY
k=1
(z?rk)￠
JY
j=1
(z?zj)(z?zj):
B￥Q
1K +2J = M,A ?B?
^·B￥b|ê |
a?￥Sy0
P¤Q(1) = 1b
è?N = 2
H
flflQ(e?i!)flfl2 = 1X
k=0
Ck2+k?1(sin2 !2)k = 1+2sin2 !2 = 42?cos!
#a0 = 4;a1 =?1b?Riesz? ?PA(z)1
PA(z) = 12(?1+4z?z2) =?12(z?(2?p3))(z?(2+p3)):
ê |ê??ê??￥?2+p3T1r15μ
B(z) = (12 12+p3)12(z?(2+p3)) = 11+p3(z?(2+p3))
?"
Q(z) =?B(z) = 1?
p3
2 z +
1+p3
2,
V7
H(z) =
1+z
2
2
Q(z) = 12
1+p3
4 +
3+p3
4 z +
3?p3
4 z
2 + 1?
p3
4 z
3
!;
Dá
ì[ -¤?￥M]b
0 0.5 1 1.5 2 2.5 3?0.4
0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1?0.5 0 0.5 1 1.5 2?1.5
1
0.5
0
0.5
1
1.5
2
Figure 1,Daubechies 4-"
￥f
’(Pm)lo?(·m).
?
1??Q￥Q
1N? 1#H(z)￥Q
12N? 1'
ro ￥"
￥é
12Nb
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 36

2Riesz￥
E ?

^/?￥?
#?}
Z?￥ p?b?B?o
¨í
ZE p¤b

3 ?:N
H￥f
1’N(t)5
￥|"1[0;2N +1]b?f
’N(t)YV(27) V/
f
N(t)
￥|"
^
I
1$

4??(10)?
^S??￥A1Hqf’N(t? k)gk2Z
^?1S??"
.à
^Riesz?
31éB?￥￡
Lb
Lawton￡
ü?m0
^ ??[
T
Hf?j;kg
^? O'μ
??? ? ? ?20
!m0(!)1 ??[
T
@?
T(10)'
m0(0) = 1; jm0(!)j2 +jm0(! +…)j2 = 1:
?í ke
1Q
j=1
m0(2?j!)?l?|"f
’(t)?(27)?lf
(t)'
(t) =
X
k
(?1)kˉh?k+1’(2t?k)
5 ?f 2 L2(R)μ
X
j;k
j < f;?j;k > j2 = kfk2 (29)
'f?j;kg(j;k)2Z2
^L2(R)￥? Ob
￡￡￡(i)?? ?8￥￡
ü?Hq(10)w¤(11)? ?'
X
n
hnˉhn?2k = 2–k;0
(ii) ?mn 2 Zμ
X
k
(hm?2kˉhn?2k +(?1)m+nˉh?m+2k+1h?k+2k+1) = 2–m;n (30)
Y
L
?m;n]1
}
H'm = 2p;n = 2qμ
P
k
(h2p?2kˉh2q?2k +(?1)2p+2qˉh?2p+2k+1h?2q+2k+1)
= P
k
(h2kˉh2(q?p)+2k + ˉh2(q?p)+2k+1h2k+1)
= P
k
hkˉhk+2(q?p)
= 2–2(p?k);0 = 2–m;n
? f? V ?
￥￡
ü.
(iii)
!f 2 C10 (R)b5 ?jμfhf;?j;kigk2Zfhf;?j;kigk2Z 2 l2b
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 37
??fμ?|"ir > 0
P¤suppf ‰ [?r;r]b:Ik = [?2jr?k;2jr?k]b5i
k > 0
P¤?jk1?k2j > K
HIk1\Ik2 = `b ?k 2 Zs3k = Km+q(0? q < K)b
á
ìμ
P
k jhf;’j;kij
2
2j P
k
R+1
1 jf(t)’(2
jt?k)jdt
·2
2j P
k
Rr
r jf(t)j
2 dt
·?Rr
r j’(2
jt?k)j2dt
·
kfk2P
k
R
Ik j’(t)j
2dt
kfk2
+1P
1
K?1P
q=0
R
IKm+q j’(t)j
2dt
kfk2
K?1P
q=0
R
[mIKm+q j’(t)j
2dt
kfk2 K ￠k’k2,
#fhf;’j;kigk2Z 2 l2b ?
V￡fhf;?j;kigk2Z 2 l2b
(iv)?’(t) = P
k
hk’(2t?k)?(t) = P
k
(?1)k h?k+1’(2t?k)w¤’0;k (t)= 1p2 P
m
hm?2k’1;m (t)
0;k (t) = 1p2 P
m
(?1)m h?m+2k+1’1;m (t)b
[,?(30)¤
P
k
jhf;’
0;kij
2 +jhf;?
0;kij
2￠
= 12 P
k
P
m;n
h
m?2khn?2k +(?1)
m+n h
m+2k+1h?n+2k+1
￠hf;’
1;mih’1;n;fi
= 12 P
m;n
2–m;nhf;’1;mih’1;n;fi
= P
m
jhf;’1;mij2
?=??
T? p?DQ?
^ ??￥y1%?￥k1?mn￥?
^μKb?"á
ìμ
X
k
jhf;’0;kij2 =
X
k
jhf;’1;kij2?
X
k
jhf;’0;kij2:
?Nw¤ ?jμ
X
k
jhf;?j;kij2 =
X
k
jhf;’j+1;kij2?
X
k
jhf;’j;kij2
 ?J > 0, pa¤
J?1X
j=?J
jhf;?j;kij2 =
X
k
jhf;’J;kij2?
X
k
jhf;’?J;kij2 (31)
?i?,?f 2 C10 (R)?(17)
T?J !1
Hμ
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 38
X
k
jhf;’?J;kij2 ! 0:
?(19)
T¤
X
k
jhf;’J;kij2 = 12…
+1Z
1
flfl?’(2?J!)flfl2flfl
fl?f(!)
flfl
fl
2 d! +R
J
?RJ ! 0?J !1b
y1jm0(!)j? 1#j?’(!)j? 1b?
H9μ?’(!)! = 0) ??b?Lebesgue e?
l ??
?μ
lim
J!1
X
k
jhf;’J;kij2 = 12…
+1Z
1
flfl
fl?f(!)
flfl
fl
2d! = kfk2,
(31)?
7J !1'￡¤(29)b
Ka, ?f 2 L2(R)¨
á? Vw¤
1￡b
??Bc￥? ?5k?j;kk = 18j;k 2 Z
H? Of?j;kg(j;k) 2 Z2?1S
??b?
HVk?j;kk = k?k¤k?k = 1b
[? ?20￥Hq/ ?k?k = 1
5f?j;kg
^L2(R)￥S??b
Cá
ì)
Daubechieslo￥;á?b
m0(!) = (1+e
i!
2 )
NM0(!);
?M0(!)
@
jM0(!)j2 = PN(sin2 !2) =
N?1X
k=0
CkN+k?1(sin2 !2)k,(32)
???lll:
!fi? 0;fi = n+fl;0? fl < 1;n 2 Z+b?f 2 Lipfi ?
flflf(n)(t+h)?f(n)(t)flfl? Cjhjfl ;8jhj? 1:
??? ? ? ?21:
!?’(!)
@
j?’(!)j? C(1+j!j)?1?fi;8! 2R:
5’ 2 Lipfi.
￡￡￡A ’ 2 L1#μ
’(t) = 12…
Z
’(!)ei!td!;
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 39
’(n)(t+h)?’(n)(t) = 12…
Z
’(!)ei!t(i!)n(eih!?1)d!
V
flfleih!?1flfl?jh!jfl
w¤
flfl’(n)(t+h)?’(n)(t)flfl? Cjhjfl Z flflfl^’(!)flflflj!jfi d!? Cjhjfl,
1?[
TPN(y)á
ìμ
??? ? ? ?[
TPN(x)
@
(1)?0? x? y
Hμ
x?N+1PN(x)? y?N+1PN(y);
(2)?0? x? 1
Hμ
PN(x)? 2N?1(max(1;2x))N?1,(33)
￡
1
x?(N?1)PN(x) =
N?1X
k=0
CkN?1+kx?(N?1?k)
A ?
^x￥hf

2??PN(x)
@
xNPN(1?x)+(1?x)NPN(x) = 1
7x = 1=2¤
PN(12) = 2N?1:
?x? 12
Hμ
PN(x)? PN(12) = 2N?1:
7?x > 1=2
Hμ
x?N+1PN(x)? 2N?1PN(12)
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 40
[
PN(x)? (2x)N?1 ￠2N?1:
??? ? ? ?:Bk = sup
!
flflM
0 (!)￠￠￠M0
2k?1!￠flfl1=k ?M
0 (!)
@(32)b5
B1 < 2N?1;B2 < 2N?1N
 0:18b
￡￡￡?k = 1
Hμ
B1 = sup!
flfl
flflN?1P
n=0
CnN+n?1?sin2 !2￠n
flfl
flfl
12
N?1P
n=0
CnN+n?1
12
<
2N?1
N?1P
n=0
CnN?1+n2?n
12
=?2N?1PN?12￠￠12 = 2N?1:
?k = 2
Hμ
jM0 (2!)j2 = PN?sin2 !￠;sin2 ! = 4sin2 !2
1?sin2 !2
·;
[
B2 = sup
!
jM0 (!)M0 (2!)j12 = sup
0?y?1
jPN (y)PN (4y(1?y))j14,(34)
?0? y? 1212 +
p2
4? y? 1
Hμ
y4y(1?y) 2
0; 12
:
#(34)??(33)μB?y0??V2N?17
:/￥B?y0??V2N?1 ￠2N?1
[
?
Hμ
PN (y)PN (4y(1?y))? 23(N?1):
7?12? y? 12 +
p2
4
H?(33)¤
PN (y)PN (4y(1?y))? 2N?1 (2y)N?1 ￠2N?1 (8y(1?y))N?1
26(N?1)(y2(1?y))N?1
26(N?1)(4=27)N?1(y2(1?y)? 4=27?0? y? 1)
Lecture Notes on Wavelets,Chapter 4,by D.Q,Dai,2003 41
Ka¤
B2? (26 ￠ 427)N?14
= 2(2?34 log3)(N?1)
< 2N?1N

@
N < (34 log3?1)(N?1)
734 log3?1 … 0:1887;# V |? = 0:18.
??? ? ? ?22Daubechieslo’N(t)á
ìμ
flfl
flc’N(!)
flfl
fl? C(1+j!j)?1N
? > 0
V7’N(t) 2 Lip(?N).