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1,I,Daubechies,Ten lectures on wavelets,Siam,Philadelphia,PA 1992.
2,S,Mallat,A wavelet tour to signal processing,Academic Press,Boston,1998.
3,Y,Meyer,loD
0?B 
W?m
?
1992.
4.
C ?ú?los
W?m
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1995b
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1979.
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Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 3
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 ?fejg1j=11S??"5í k)
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Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 4
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7
Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 5
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kxk
Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 6
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 V, O
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 V?7 Of￥
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B
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 ??b
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 |jf(x)jT1fe?ix! f(x),! 2
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j
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Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 9
= 32?x21 +x22￠ = 32 hx;xi
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 è OB??
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j
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flfl
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j
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flfl
fl
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X
j
cjej
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flfl
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 O?1Bb
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Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 11
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j
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^>￥b
!fxng‰ Ran(S);xn ! x05ifyng
P¤Syn = xn:?
< xn?xm;yn?ym >? Akyn?ymk2
fkxnkgμ??fyng
^Cauchy
V7μKy0b?"
x0 = lim
n!1
xn = lim
n!1
Syn = Sy0;
[x0 2 Ran(S):
(iii) Ran(S)X?by?Ran(S),μkyk2? A?1 < Sy;y >= 0,
[S?1ib
(iv)??kS?1xkkxk?< S?1x;x >=< S?1x;SS?1x >? AkS?1xk2
[kS?1k? A?1b
(v)?(iv)μ< S?1xx >? A?1kxk27 OμkSS?1xk = kxk? kSkkS1xk#kS?1xk?
kSk?1kxkbV7< S?1xx >? AkS?1xk2? AkSk?2kxk2b
(vi) S1
a? O??< Sx;x >
^
L￥ ?x 2 Xb
!< Sx;x >
^
L￥5< Sx;x >= < Sx;x > =< x;Sx >=< S?x;x > ?x 2 X
[S1
abQ_2
V ?
￥￡
üb
(vii) ?μ? V
I
0S
@
(S?)?1 = (S?1)?x:
< (S?)?1x;y > < (S?)?1x;SS?1y >=< x;S?1y >=< (S?1)?x;y >,
Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 12
??? ? ? ?3:
!S1Hilbert bW?￥μ?L?
0< Sx;x >= 0 ?x 2 Xb5S = 0b
￡ ?x;y 2 Xμ
< S(x+y);x+y >= 0 )< Sx;y > + < Sy;x >= 0
< S(x+iy);S(x+iy) >= 0 )?i < Sx;y > +i < Sy;x >= 0:
[< Sx;y >= 0 |y = Sx¤Sx = 0b
??? ? ? ?6:
!fejg;f?ejg‰ X
@ O
?Hq
'Bessel?

X
j
j< f;e
j >j
2 +j< f;?e
j >j
2￠? Ckfk2 8f 2 X (10)
O
< f;g >=
X
j
< f;ej >< g;?ej > 8f;g 2 X0 (11)
?X0
^X￥

á0 bWb
?l
0
fTng
n?
Tn
o
Tnf =
X
jjj?n
< f;?ej > ej;?Tnf =
X
jjj?n
< f;ej >?ej:
5fTng;
n?
Tn
o
^Báμ?L?
0
 O
kfk2? C ￠min
X
j
j< f;ej >j2;
X
j
j< f;?ej >j2
!
(12)
f =
X
j
< f;?ej > ej =
X
j
< f;ej >?ej 8f 2 X0 (13)
￡?? ?1
10w
X
j
cjej
+
X
j
cj?ej
Ckfcjgk2 8fcjg2 l
2
|cj =< f;?ej >i ?¨
10¤
kTnfk =
X
jjj?n
< f;?ej > ej
C
0
@X
jjj?n
j< f;?ej >j2
1
A
1
2
Ckfk
[fTng ?
1
n?
Tn
o
^Báμ?￥b
Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 13
?
10
118f 2 X0μ
f?
P
jjj?n
< f;?ej > ej
= sup
g:kgk?1
flfl
flfl
fl< f?
P
jjj?n
< f;?ej > ej;g >
flfl
flfl
fl
= sup
g
flfl
flfl
fl
P
jjj>n
< f;?ej > < g;ej >
flfl
flfl
fl
sup
g
P
jjj>n
j< g;ej >j2
!1
2
P
jjj?n
j< f;?ej >j2
!1
2
C ￠ sup
g:kgk?1
kgk￠
P
jjj>n
j< f;?ej >j2
!1
2
= C
P
j>n
j< f;?ej >j2
!1
2
! 0
y1)
P
j
j< f;?ej >j2
l ?

T
l ??0b
[
13f 2 X0? ?b
??fTng
n?
Tn
o
^Báμ??
X0

ìsYμK
Tf =
X
j
< f;?ej > ej?Tf =
X
j
< f;?ej >?ej
?? ?2 V￡

T ?if 2 Xg? ?b?ü￡
ü

13b
 ?if;g 2 X?
10μ
kfk = sup
g2X;kgk?1
j< f;g >j = sup
g
flfl
flfl
fl
X
j
< f;ej > < g;?ej >
flfl
flfl
fl
sup
g
Ckgk￠(
X
j
j< f;ej >j2)1=2? C(
X
j
j< f;ej >j2)1=2
?N￡¤
12b
?
1(11)è?1(13)￥ D?
T O(11)o30"
? ?b
2(10)
^ O
?(12)
^ O/?'? ?￡
ü O/? V?
?D D?
T￥?
Tw
b9' ?Bessel?

@ D?
T(13)5
1 Ob
/
a
ü?fejgó?
H8"/f?ejgb
??? ? ? ?7:
!fejg
^X?￥ O 
a/?sY1A;Bb
7
Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 14
e
j = (T?T)?1ej (14)
T,X ! l2
f !f< f;?ej >gj (15)
?TDT??(7)(8)
?lb5
(i) f?ejg
^X￥ O 
a/?sY1B?1A?1b
(ii)
T = T(T?T)?1;?TT = (T?T)?1 (16)
(iii)?T T? = T?T?
^l2?Ran(T) = Ran(?T)
￥??g?
0b
(iv)?T?T = I = TT'
f =
X
j
< f;ej >?ej =
X
j
< f;?ej > ejX?b(17)
￡,(i)(ii)?
9? ?2(T?T)?1
^1
a
0bV(?Tf)j =< f;?ej >=< f;(T?T)?1ej >=<
(T?T)?1f;ej >(T(T?T)?1f)j =< (T?T)?1f;ej >¤?T = T(T?T)?1b?Pj j < f;?ej >
j2 = Pj j < (T?T)?1f;ej > j2 = kT(T?T)?1fk2 =< T(T?T)?1f;T(T?T)?1f >=< (T?T)?1f;f >b
???
T(9)? ?2w¤f?ejg
^ Ob 
?1A?1
/?1AkT?Tk?2b
(iii)T￥?lw¤?T T? = T?T?'?T T?
^1
a￥bT = T(T?T)?1¤Ran(?T) ‰
Ran(T)??T =?T(T?T)¤Ran(T) = Ran(?T)
[Ran(T) = Ran(?T)b
:P1l2?Ran(T)
￥??g?b1￡?TT?"T?T?=P,á
ì￡Ran(T)
^P￥???
Ran(T)?
^
￥
,?"'
TT?Tf = Tf;8f 2 X;?TT?(fcjg) = 0;8fcjg?Ran(T)
?B
TTT?Tf = T(T?T)?1T?Tf = Tf°¤wb?=
T?
fcjg?Ran(T)w¤hc;Tfi = 0;8f 2 X;9'T?(fcjg) = 0:
[?TT?(fcjg) = 0.
(iv)??lμ
T?T = (T(T?T)?1)?T = (T?T)?1T?T = I
TT = T?T(T?T)?1 = I
?N'¤(17)b
1?(17)?)
￥
l ??￡
ü V? I? ?6b
:/á
ìw? Of?ejg￥/?9bá
ìμ
Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 15
kfk2 =
X
j
hf;ejihf;?eji? (
X
j
jhf;ejij212
X
j
jhf;?ejij2)
1
2
Bkfk
X
j
jhf;?ejij2)
1
2
#f?ejg￥ O/?1B?1b
?
1?T;T?￥?lμ
TT?f =
X
j
hf;ejiej
y1fejg
^ O?
9
TTT? V
I# V
f =
X
j
hf;eji(T?T)?1ej
'á
ìê4?ej = (T?T)?1ej
2) f?ejg￥ê4B?
^?·B￥b
è
!"1 =
1
0
!;"2 =
0
1
!
1R2?￥S?? I
n ?
A =
1 0
p2
2
p2
2
0 1
p2
2?
p2
2
!
7(e1;e2;e3;e4) = ("1;"2)A5fejg1? Ob
!(?e1;?e2;?e3;?e4) = ("1;"2)BB12￡4 ? OA￠BT = I2￡2.5 ?x 2 R2μ
x =
X
j
hx;?ejiej = ABTx =
4X
j=1
fijej:
A ?B?
^·B￥'f?ejg?
^·B￥b
:?? ?7￥ZEsf?ejg
!x =
1
0
!
ù pf?ejgsY
Pa) kfikKl b) fiK

'μK￥d
,í.
1 Y Oá
ì￡
ü
?￡ù2T
'
&9μ??￥ilb
??? ? ? ?4
!S,X ! Y
^L?μ?
5iμ?
0Sr,Y ! X
P¤
SSrf = f;8f 2 X
￡¨N:S￥
, bW'N = fx,Sx = 0gbN?1NX?￥???|SK?N?

¤SN?,N? ! Y5
Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 16
i) SN?
^L?μ?￥
ii) SN?
^?
 ?SN?x = 05x 2 N \N? = 0
iii) SN?
^
 ?óy 2 Yix 2 X
P¤Sx = y:x = x1 + x2; x1 2 N?;x2 2
Ná
ìμ
SN?x1 = Sx1 = S(x1 +x2) = Sx = y0
[SN?μμ?
I?

Sr = S?1N?,Y ! N?
V7SSrf = f;8f 2 X.
??? ? ? ?5:
!S,X ! Y1L?μ?
0bó?y 2 Y I
n
0Z?Sx = y0
μ·B￥
l
3x = Sry.
￡:?? ?4Sry
^3b
[ 
e3 V1
x = Sry +z;
?z 2 Nb??Sry 2 N?#B?3x￥

@
jjxjj2 = jjSry +zjj2 = jjSryjj2 +jjzjj2
?z = 0
H |lb
?Srè?1S￥
E
I
psedo-inverseb
??? ? ? ?8:?
fejg1X￥ O? O???

R,fcjg!
X
j
cjej
^l2?X￥
?
b
?iR'?
8?l￥T?
￡
!fejg1 O?1?Bessel?
￥2TR
^l2?X￥μ?L?
0b?
9RT1
?

[R1
?
bQ-
!R
^
?
??jjRjjμ? 
a
0R?9μ?,i
OjjRjj = jjR?jj OR? V1
R?f = f< f;ej >g
?
H
jjR?fjj2 =
X
j
j < f;ej > j2? (jjR?jj￠jjfjj)2 = jjR?jj2 ￠jjfjj2 = jjRjj2 ￠jjfjj2
[fejg
^Bessel?
b
/
? O/?.
Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 17
!Rr,X ! l2
^R￥
E
If 2 X,μ
f = RRrf =
X
j
(Rrf)jej;
?(Rrf)j
^Rrf￥?j?s

[
jjfjj4 =< f;f >2=< P
j
(Rrf)jej;f >2= (P
j
(Rrf)j < ej;f >)2
P
j
j(Rrf)jj2P
j
j < f;ej > j2?jjRrjj2jjfjj2P
j
j < f;ej > j2
#P
j
j < f;ej > j2? 1jjRrjj2jjfjj2
Cá
ì)

} Ob
???lll:
!fejg;f?ejg
^X? ?i
??
 ?(17)X?
l ?￥il/? ?5?fej;?ejgo1

}b ?Tfejg;f?ejg?
^ O OoM
}5?fej;?ejg
^
} Ob
?μ
hej;?eki = –j;k;
5?fej;?ejg
^B
?? Ob
???lllX?￥?
fejgX?L?? ? ?Toμ
,?
fcjg?
PP
j
cjejX?
l ??
,b
??? ? ? ?9:
!fej;?ejg
^
???
5fejgDf?ejg?
^? ?￥.
￡
!fcjg
PPcjejX?
l ??
,5
0 =<
X
k
ckek;?ej >=
X
k
ck < ek;?ej > = cj;8j
#fejg? ?b ?
1￡
üf?ejg9? ?.
??? ? ? ?10
!fejg
^X?￥ O OL?? ?5
￥
}f?ejg
^·B￥ Ofej;?ejg
^
??
￥b
￡
!fejg
^B? O O? ?b
!f?e1jg;f?e2jg
^fejg￥
?
}b5X?
l ?il/μ
f =
X
j
< f;
e1j > ej =
X
j
< f;
e2j > ej
#X
j
(< f;
e1j >? < f;
e2j >)ej = 0 (18)
?
f< f;e1j? >? < f;e2j? >g
P(18)?)
X?
l ??0?fejg-? ??μ
< f;
e2j?
e1j >= 0;8f 2 X
?w
e2j =
e1j 8j·B?¤￡.
Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 18
!f?ejg
^fejg￥·B
}
?? ?7
ó.5X?
l ?il/μ
ej =
X
k
< ej;?ek > ek
?fejg￥? ??μ< ej;?ej >= –j;k?? ?7,f?ejg9
^ O#fej;?ejg
^
?? O.
??? ? ? ?11:
!fejg
^ O,? ?,f?ejg
^ ·B
}.5?
f ! f< f;ej >gDf ! '< f;e?j >“
?
^X?l2
￥].
￡:??fejg
^ O O? ?,f?ejg9
^ O.#

=?
?
^μ?l￥.C￡
ì?
^
.
!fcjg2 l2?l
f =
X
j
cjej;g =
X
j
cj?ej
?? ?1,f;g 2 X Okfk;kgk? ckfcjgk2,?fej,?ejg￥
???,?
cj =< f;?ej >=< g;ej > ; 8j
?￡
ü
?
￥′×
^
￥b
? O?é,μ
A(kfk2 ;kgk2)?
X
j
jcjj2? B(kfk2 ;kgk2)
#N?
1X?l2
￥?
ê].
á
ìC)
? ?￥ Ob
???lll::span(fejg) = ff,f
^fejg￥μKL?Fgb
RieszHilbert bWX?￥?
fejg1Riesz ?span(fejg) = X O
A(
X
j
jcjj2)1=2?
X
j
cjej
B(
X
j
jcjj2)1=2;8fcjg2 l2 (19)
?
1(19)i?"?
fcjg ! P
j
cjej
^l2?span(fejg)
￥]b
V1? ?8
* úX￡
ü
1
?
b
2? ?11￡
ü? ?￥ O?
f ! f< f;ej >g
^X?l2
￥]b????
0
D
0fcjg ! P
j
cjejo1
ab#f ! f< f;ej >g
^X?l2
￥]?N?fcjg !
P
j
cjej
^l2?X
￥]b?￡
ü? ?￥ ODRiesz
^]Bà
Qb
Lecture Notes on Wavelets,Chapter 1,by D.Q,Dai,2003 19
1?
üM.
!g 2 L2(R)bá
ì I
n
üM
?￥"fg(t?j)gj2Zb5μ
??? ? ? ?12:
a ?gfejg(ej(t) = g(t?j))? V
^L2(R)￥ O
nChristensen O.,etc.ACHA
bfejg
^span(fejg)￥Riesz? O??9A;B > 0;
P¤
A?
X
k
j?g(w +2…k)j2? B;a:e:
g1g￥FourierMDb