Lecture Notes on Wavelets,Chapter 6,by D.Q,Dai,2003 1
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f(t) = f(c?t):
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m0(!) = 12
X
k
hke?ik!
O1?c2|?5μ
b’(!) = eic!b’(!);
hk = hc?k; 8k;?c1?
H
m0(!) = eic!m0(!);?c1?
H
O ??-B?w’(t)1?c2?b
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C ?pp,94-95b
á
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L
?
 Oc = 0c = 1￥ f?b
c = 0
Hμ
m0(?!) = m0(!) (1)
c = 1
Hμ
m0(?!) = ei!m0(!) (2)
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6?B?f
e’
@
e’(t) =
X
k
ehke’(2t?k) (3)
Lecture Notes on Wavelets,Chapter 6,by D.Q,Dai,2003 2
i:
fm0(!) = 12
X
k
ehkeik!:
á
ìL?’De’ ?/il/
^
??￥'
h’0;k; e’0;li = –k;l:
51?
fhkgDfehkgμ
? ?X
k
hkehk?2l = 2–l;0 (4)
￡
–l; 0 =
Z
’(t)e’(t?l)dt
=
X
k; n
hkehn
Z
’(2t?k)e’(2t?2l?n)dt
= 12
X
k; n
hkehn
Z
’(t?k)e’(t?2l?n)dt
= 12
X
k; n
hkehn–k; 2l+n
= 12
X
k
hkehk?2l:
? ?á
ìμ
m0(!)fm0(!)+m0(! +…)fm0(! +…) = 1 (5)
? ? ?m0(!)
@Hq(1)(2)5Z?(5)i
@Hq(1)(2)￥3fm0(!)b
￡n
C ?p.145b
1 ￡’De’μB?￥?5?á
ì1 p
m0(!) =
1+e?i!
2
N
M0(!)

fm0(!) =
1+e?i!
2
eN
fM0(!):
Hq(1)? ?
Hm0(!)Dfm0(!)9cμ(1 + ei!)ND(1 + ei!)eN￥y0#m0(!)?
cμ
(1+e?i!)N(1+ei!)N = c
cos2 !2
·N
Lecture Notes on Wavelets,Chapter 6,by D.Q,Dai,2003 3
￥y0 ?
1fm0(!)?cμ(cos2 !2)eN￥y0b
[μ
? ?
!m0(!)
^
L"
??[
T
@Hq(1)5i
L"
[
TP(t)
P
¤P(?1) 6= 0
m0(!) = (cos2 !2)NP0(cos!):
7(2)? ?
Hμ
m0(!) = e?i!2 (cos !2)2N+1P0(cos!):
1?fm0(!)gμ ?
￥2Tb
/
o)
Hq(1)? ?
H￥ f??
H
m0(!) = (cos2 !2)NP0(cos!); fm0(!) = (cos2 !2)eNfP0(cos!)
??
T¤
(cos2 !2)N+eNP0(cos!)fP0(cos!)+(sin2 !2)N+eNP0(?cos!)fP0(?cos!) = 1:
:y = sin2 !2
P0(cos!)fP0(cos!) = P0(1?2sin2 !2)fP0(1?2sin2 !2) = P(sin2 !2)
5μ
(1?y)N+eNP(y)+yN+eNP(1?y) = 1:
3NZ?¤
P0(cos!)fP0(cos!) =
N+eN?1X
j=1
Cjn+j?1(sin2 !2)j +(sin2 !2)kr(sin2 !2):
?r(u) = R(12?u)R(u)
^ Q[
Tb
B?+ è
^
P0(u) · 1; |r(u) · 0;
'm0(!) = (cos2 !2)N
"Hf

?
H
fm0(!) = (cos2 !2)eN
N+eN?1X
j=1
Cjn+j?1(sin2 !2)j:
Lecture Notes on Wavelets,Chapter 6,by D.Q,Dai,2003 4
?f
’De’M?￥lof
De? V |1
b?(!) = e?i!2 fm0(!
2 +…)b’(
!
2)

be?(!) = e?i!2 m
0(
!
2 +…)b’(
!
2)
1?m0Dfm0￥BtHq/
?
?
P’0; kS??μ
f =
X
j; k
hf; e?j; ki?j; k =
X
j; k
hf;?j; kie?j; k:


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f
ZE÷B?
￥ f ? V? I
C ?￥÷Tb
!μL2(R)￥MRA fVjgj2Z?l
V0 = V0?V0 = spanfF(x;y) = f(x)g(y);f;g 2 V0g:
Vj
F(x;y) 2Vj () F(2jx;2jy) 2V0
5fVjg
@
￠￠￠‰V?2 ‰V?1 ‰V0 ‰V1 ‰V2 ‰￠￠￠
\
j2Z
Vj = f0g;
[
j2Z
Vj = L2(R2):
??f’(x?n)gn2Z
^V0￥S??e
Φ0;n1;n2(x;y) = ’(x?n1)’(y?n2); n1;n2 2Z
?V0￥S??b
Φj;n1;n2(x;y) = ’j;n1(x)’j;n2(y)
1Vj￥S??b
Lecture Notes on Wavelets,Chapter 6,by D.Q,Dai,2003 5
?lWj1VjVj+1?￥???'
Vj+1 =Vj 'Wj:
??
Vj+1 = Vj+1?Vj+1
= (Vj 'Wj)?(Vj 'Wj)
= (Vj 'Vj)'[(Wj?Vj)'(Vj?Wj)'(Wj?Wj)]
= Vj 'Wj
Wj? ??sF?
j;n1(x)’j;n1(y); Wj?Vj;
’j;n1(x)?j;n2(y); Vj?Wj;
j;n1(x)?j;n2(y); Wj?Wj:
# V?l ??lo
h(x;y) = ’(x)?(y);
￡
ü
v(x;y) =?(x)’(y);<°
d(x;y) =?(x)?(y):?
5fj;n1;n2(x;y),n1;n2 2Z;? = h;v or dg?Wj￥S?? O
fj; n(x; y),n 2Z2;? = h;v or dg
^
'j2ZWj = L2(R2)
￥S??b
!’(t)
@
Z?
’(t) =
X
k
hk’(2t?k):
5=?f
Φ(x; y)μ
Φ(x; y) = ’(x)’(y)
=
X
n1
hn1’(2x?n1)
!
￠
X
n2
hn2’(2y?n2)
!
=
X
n1;n2
hn1hn2Φ(2x?n1;2y?n2)
Lecture Notes on Wavelets,Chapter 6,by D.Q,Dai,2003 6
V7Φj;k1;k2(x;y)μ
Φj;k1;k2(x;y) = 2j’(2jx?k1)’(2jy?k2)
= 2jΦ(2jx?k1;2jy?k2)
= 2j
X
n1;n2
hn1hn2Φ(2j+1x?2k1?n1;2j+1y?2k2?n2)
= 12
X
n1;n2
hn1hn2Φj+1;n1+2k1;n2+2k2(x; y)
= 12
X
n1;n2
hn1hn2Φj+1;n1;n2(x;y):
f 2 L2(R2)?l
cj;k1;k2 = hf; Φj;k1;k2i;

d?j;k1;k2 = hf;j;k1;k2i
5á
ìμs3
E:
cj;k1;k2 = 12
X
n1;n2
hn1?2k1hn2?2k2cj+1;n1;n2
= 12
X
n1
hn1?2k1
X
n2
hn2?2k2cj+1;n1;n2
!
dhj;k1;k2 = 12
X
n1;n2
hn1?2k1gn2?2k2cj+1;n1;n2
dvj;k1;k2 = 12
X
n1;n2
gn1?2k1hn2?2k2cj+1;n1;n2
ddj;k1;k2 = 12
X
n1;n2
gn1?2k1gn2?2k2cj+1;n1;n2
?gk = (?1)kh1?k.

TV
ü=?f
los3V? V[5o?pTMD ?ao
pT
MDb?
^
P¨f
lo￥a?b
á
ì9 V[YVfcj;k1;k2; d?j;k1;k2;k1;k2 2 Z;? = h; v; dg×cj+1; k1; k2b h1-w
?b


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