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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 2
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 3
al = R ’j;‘ (t)’j+1;k (t)dt
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 4
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 5
0 =
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1
f(t)?f(2jt?2j+j0k0)dt
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n=0
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 6
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 7
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 8
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1
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flfl
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 9
jf(t)?f(s)j? C
X
j;k
2?(12+fi)j j?j;k(t)j;k(s)j (12)
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P
j?j0
P
k
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j?j0
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k
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j?j0
2(1?fi)j jt?sjP
k
j?0(2j?k)j( ê?tDs-W)
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j?j0
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= C2(1?fi)j0 jt?sj
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7
P
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P
k
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P
j>j0
2?fij P
k
(j?(2jt?k)j+j?(2js?k)j)
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j>j0
2?fij
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 10
jf(t)?f(t0)j? cjt?t0jfi ln2=jt?t0j
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flfl
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 11
f’0;k(t);?j;k(t)gk2Z;j?0 (15)
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1
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 12
lo V[T1
f
bW￥iYVZ 7"
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 2
?gk = (?1)kh1?k b
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l
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l
hlR f(t)2j2’(2j+1t?2k?l)dt
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l
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l
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Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 3
al = R ’j;‘ (t)’j+1;k (t)dt
= 2j+12 R ’(2jt?‘)’(2j+1t?k)dt
= 2j+12 P
m
hmR ’(2j+1t?2‘?m)’(2j+1t?k)dt
= 1p2 P
m
hmR ’(t?2l?m)’(t?k)dt
= 1p2 P
m
hm–2‘+m;k
= 1p2hk?2‘
?
1μ
bl = 1p2gk?2l
(6)
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f?
T(1)(2)
n5¤?s3"
fcj;kgfdj;kg ?aפ?f
f b ?T??é?s3, ?a ×A ??
^á
ì￥
"￥ bá
ì?YVs3"
fcj;kgfdj;kg￥s¤1?f
f￥
t?é￥ a
M b
Bcá
ì? ?￥lof
@(0) = 0'
Z +1
1
(t)dt = 0:
÷B?1á
ì? ?h
> ￥à
Q
???lll ?f
fμm(m 2Z+)¨h
> ?
Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 4
Z +1
1
tnf(t)dt = 0;n = 0;1;￠￠￠ ;m:
á
ìóB??B?￥￡ù2T b
??? ? ? ?
!f
f 2 Cm(R)?1è
Ol(0? l? m)¨?
f(l)(t)μ? bf
f(t)μ?
Oμ ?/￥
h
flfl
flfl?f(t)
flfl
flfl? C(1+jtj)?fi;fi > m+1;
? ?i￥?
j;j0k;k0μ
??1"
D
fj;k;?fj0;k0
E
= –j;j0;–k;k0,(7)
5?fμm¨h
> '
+1Z
1
tn?f(t)dt = 0;80? n? m:
￡
ü ¨B
,E.l = 0
H￥￡
ü]/
￥￡
üB"í3??￡ b
!0 < l? mi O
?n;n < ltn?f(t)￥s1
, b1￡n = lg ? b
? |j0;k0 2Z 2j0k0)f(t)TTaylorZ 7¤
f(t) =
lX
n=0
f(n)(2j0k0)
n! (t?2
j0k0)n +o(flflt?2j0k0flfll)
[ ?ó￥? > 0;i – > 0?jt?2j0k0jl < –
Hμo(jt?2j0k0jl)? "jt?2j0k0jl
|j > max(j0;0)5?(7)μ
Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 5
0 =
+1R
1
f(t)?f(2jt?2j+j0k0)dt
=
lP
n=0
f(n)(2j0 k0)
n!
+1R
1
(t?2j0k0)n?f(2jt?2j+j0k0)dt
+
+1R
1
o(jt?2j0k0jl)?f(2jt?2j+j0k0)dt
= f(l)(2j0k0)l!
+1R
1
(t?2j0k0)n?f(2jt?2j+j0k0)dt+J
= f(l)(2j0k0)l! 2?(l+1)j R+1?1 tl?f (t)dt+J:
71?Jμ
jJj? R+1?1 o
jt?2j0k0jl
·flfl
fl?f (2jt?2j+j0k0)
flfl
fldt
R
jyj?–
jyjl
flfl
fl?f (2jy)
flfl
fldy +C R
jyj>–
(1+jyj)l
flfl
fl?f (2jy)
flfl
fldy
￠2?(l+1) R
jyj?2j–
jyjl
flfl
fl?f (y)
flfl
fldy +C R
jyj>–
(1+jyj)l (1+2j jyj)?fi dy
C?￠2?(l+1)j R+1?1 jyjl (1+jyj)?fidy +C2?jfi R
jyj?–
(1+jyj)ljyj?fi dy
C?￠2?(l+1)j +C2?jfi
?
T
0 = f
(l) (2j0k0)
l!
Z +1
1
tl?f (t)dt+J2(l+1)j
?
7j ! +1¤
flfl
flflf(l) (2j0t0)
l!
Z +1
1
tl?f (t)dt
flfl
flfl? C?
#μ
f(l) (2j0t0)
l!
Z +1
1
tl?f (t)dt = 0
??f?1è
á
ìw￡i j0;k0
P¤f(l) (2j0k0) 6= 0.?l = 0;1
HA ? b7
l? 2
H ?f(l) (2j0k0) = 0;8j0;k0 2Z;? ???f(l) (t) = 0;8t 2Ry7f (t)
^l?1Q
[
T b?
Hf? V
μ? b
Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 6
Kaá
ìμR+1?1 tl?f (t)dt = 0:
|? ??¨???loá
ì¤
??? ? ? ?3
!f’j;kgj;k?L2(R)￥S??’ 2 Cm; O’(l)μ?l? m; O
? > 0;μ
j’(t)j? C (1+jtj)?m?1?" ;8t 2R
5
Z +1
1
tl’(t)dt = 0;0;? l? m:
www
!f’j;kgj;k
^L2(R)￥S??lo b5/
?é?
]
H? ?
(1)?(t)·
h
(2)? 2 C1(R) O ?l;?(l)(t)μ? b
￡￡￡
!
1
2]
H? ? Oj?(t)j? Ce?ajtj: I
n?￥Fourier-LaplaceMDz 2
C;
(z) =
Z +1
1
(t)eitzdt:
A ?(z) u×fz,jImzj < ag
=3 b
?? ?3
2¤
Z +1
1
tl?(t)dt = 0;8l 2Z+;
#(l) (!) = (i!)lR+1?1?(t)tldt;! 2 R,¤(l) (0) = 0;8l 2 Z+.
[3f
(z) z = 0￥
#×
=1
,V71
, b?w?(t) = 0 b?Df?j;kgj;k
3
?L2(R)
± b
?ó?￥
rof
m0(!)[#f
’(t)á
ì/
lo?(t)
ì￥ ó"
^
(!) = e?i!=2m0(!=2+…)?’(!=2);?’(0) = 1,(8)
??m0(…) = 0;m0(!) ! = …)μ
,? b?1 p?μ÷ú￥;á?
Hμ
Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 7
??? ? ? ?4 L 2Z+. ?m0 2 CL(R), O
2 CL;?(l)(t)μ?;8l 2 L:
? > 0,
j?(t)j? C(1+jtj)?L?1
5m0(!) ! = …μL+1×
,? b
￡￡￡ ?? ?3,R+1?1 tl?(t)dt = 0;??N?dl(0)d!l = 0;l = 0;1;:::;L.
??
T(8) VB
,1w¤
dlm0(…)
d!l = 0;l = 0;1;:::;L:
V? ?4w¤1/ μB?
h?;á?￥lo? ?
m0(!) = (1+e
i!
2 )
L+1M0(!)
￥Hq
^A?1F￥ b
3llloooDDDfff
￥￥￥???555???
á
ì
n5)
QH¨olderf
? O? I
n0 < fi < 1￥ f? bB? f?n
C ?
pp.309-317.
á
ìL
!
jf(t)?f(s)j? cjt?sjfi ;8t;s 2R,(9)
á
ì
!lof
(t)
@
j?(t)j;j?0(t)j? C(1+jtj)?2 (10)
??? ? ? ?
!l = 0;1. ?t 2Rí k)
Báμ?'
Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 8
X
k
flfl?(l)(t?k)flfl? C;8t 2R:
￡￡￡ o￡l = 0
H￥ f? b
:g(t) = P
k
j?(t?k)jA ?g[11?
ù b#
sup
t2R
g(t) = sup
0?t?1
g(t)
?
H?
10μ
sup
0?t?1
g(t)? sup
0?t?1
flfl
flfl
fl
X
k
(1+jt?kj)?2
flfl
flfl
fl?
X
k
(1+jjkj?1j)?2 < 1:
?? ??loMD￥ f?á
ìμ
??? ? ? ?5
!μZ 7
T
f(t) =
X
j;k
dj;k?j;k(t);dj;k =< f;?j;k(t) >,
5f1fi-H¨older ??￥? O??
jdj;hj? C2?(12+fi)j (11)
￡￡￡
!f
^fi-H¨older ??￥5
jdj;kj =
flfl
flfl+1R
1
f(t)?j;k(t)dt
flfl
flfl
=
flfl
flfl+1R
1
(f(t)?f(k2?j))?j;k(t)dt
flfl
flfl
C
+1R
1
jt?k ￠2?jjfi 2j2(1+j2jt?kj)?2dt
= C ￠2?(12+fi)j
+1R
1
jtjfi (1+jtj)?2dt
C02?(12+fi)j
Q- ?
11? ?5μ
Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 9
jf(t)?f(s)j? C
X
j;k
2?(12+fi)j j?j;k(t)j;k(s)j (12)
|j0
P¤2?j0?jt?sj? 2?j0+15?±s?′? ?μ
P
j?j0
P
k
2?(12+fi)j j?j;k(t)j;k(s)j
P
j?j0
2?fij P
k
j?(2jt?k)(2js?k)j
P
j?j0
2(1?fi)j jt?sjP
k
j?0(2j?k)j( ê?tDs-W)
C P
j?j0
2(1?fi)j jt?sj
= C2(1?fi)j0 jt?sj
= Cjt?sjfi
7
P
j>j0
P
k
2?(12+fi)j j?j;k(t)j;k(s)j
P
j>j0
2?fij P
k
(j?(2jt?k)j+j?(2js?k)j)
C P
j>j0
2?fij
= C2?fij0
= Cjt?sjfi
[
9? ? b
1? ??5?'
?t0?íμ
jf(t)?f(t0)j? cjt?t0jfi (13)
á
ìμ
??? ? ? ?6
!?μ?|" O
13? ?5μ
max
k;t02supp(?j;k)
jdj;kj? C2?(1=2+fi)j (14)
Q- ?
14? ? O
? > 0f
^?-H¨older ??￥5
Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 10
jf(t)?f(t0)j? cjt?t0jfi ln2=jt?t0j
D ??loMD
H￥ f?B"? ?6￥?Q
? f ?
^??￥ b7 O?
^?
?é￥ b? ?|Hq
14
T??5μ ?/￥2T b
???lll? > 0?l"
s(t0;j;?) = fk,k 2Z;supp?j;k \(t0;t0 +?) 6= ;g:
??? ? ? ?7:
!?μ?|" b ?
? > 00 < fi < 1;μ
max
k2s(t0;j;?)
jdj;kj? c2?(1=2+fi)j
5
jf(t)?f(t0)j?jt?t0jfi,
￡￡￡ ?t 2 (t0;t0 +?);yj;k(t) 6= 0j;k(t0) 6= 0 Vw¤k 2 s(t0;j;?);#μ
jf(t)?f(t0)j =
flfl
flfl
fl
P
j;k
dj;k(?j;k(t)j;k(t0))
flfl
flfl
fl
=
flfl
flfl
fl
P
j
P
k2S(t0;j;")
dj;k(?j;k(t)j;k(t0))
flfl
flfl
fl
CP
j
2?(12+fi)j P
k2S(t0;j;")
j?j;k(t)j;k(t0)j
CP
j;k
2?(12+fi)j j?j;k(k)j;k(k0)j
?
12
T# a
M]￥w ?' V￡
üf t0) ?fi-H¨older ?? b
á
ìC )
d
QH¨olderf
?Cfi(0 < fi < 1)'
Cfi(R) = ff,f 2 L1(R);jf(t)?f(s)j? Cjt?sjfi ;8t;s 2Rg
f
?Cfi(R)¨lo"f?j;kg
^?
a￥y1
￥L?FD [
T??N
[
T￥Q
??lo￥h
> ￥?
bá
ì|
P¨V
U
T
2'¨S??
Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 11
f’0;k(t);?j;k(t)gk2Z;j?0 (15)
á
ìμ ?/￥ Y b
??? ? ? ?8
!
15
^L2(R)￥r-?5S?? b0 < fi < r?2.5?f 2 Cfi(R)
Hμ
C0;k =
+1Z
1
f(t)’0;k(t)dt = O (16)
dj;k =< f;?j;k >= O(2?(fi+1=2)j); j? 0 (17)
Q- ?"
fc0;kg;fdj;kg
@Hq
16
175f
f(t) =
X
k
c0;k’0;k (t)+
X
j?0
X
k
dj;k?j;k(t) (18)
?Cfi(R).
￡￡￡
!f 2 Cfi(R) b5?f 2 L1(R)w¤c0;k = O(1)'
16? ? b?? ?5w¤
17 b
Q-
!
16
17? ?5
17?￥?B)
??c0;kμ? O’
^r-
?5￥?P
k
c0;k’0;k (t)
^rQ ?? V±￥#
?Cfi(R) b7
18?￥?=?)
?? ?5
^fi- H¨older ??￥ b
:/￡
ü
?L1(R) bá
ìμ
flfl
flfl
fl
P
j?0
P
k
dj;k?j;k (t)
flfl
flfl
fl
C P
j?0
2?(fi+1=2)j P
k
2j=2j?(2jt?k)j
C P
j?0
2?fij P
k
(1+j2jt?kj)?r
C P
j?0
2?fij P
k
(1+j2jt?kj)?2
C P
j?0
2?fij < C0
á
ì??)
0 < fi < 1￥ f? b?fi? 1
H ˉμ ?
￥ Y b V? IMeyer￥÷
T b
Lecture Notes on Wavelets,Chapter 5,by D.Q,Dai,2003 12
lo V[T1
f
bW￥iYVZ 7"
é? Y b è ?
s¨Sobolev bW
Ws(R) =
8
<
:f,
+1Z
1
(1+j!j2)s
flfl
fl?f(!)
flfl
fl
2 d! < +1
9
=;
á
ìμ ?/￥2T
f 2 Ws(R),
X
j;k
jhf;?j;kij2 (1+2?js) < +1:
