TELKOM
NIKA
, Vol.11, No
.1, Janua
ry 2013, pp. 213
~22
6
ISSN: 2302-4
046
213
Re
cei
v
ed O
c
t
ober 1, 20
12;
Revi
se
d No
vem
ber 29, 20
12; Accepted
De
cem
ber 5,
2012
Lebesgue-type Inequality for Orthogonal Matching
Pursuit for
μ
-coherent
D
ictionaries
Ye Peixin*,
Wei Xiujie
Schoo
l of Math
ematics an
d L
P
MC, Nanka
i
Univers
i
t
y
, T
i
anjin
300
07
1, Chin
a
*corres
pon
di
ng
author, e-mai
l
:
y
e
p
x
@n
anka
i
.edu.cn
A
b
st
r
a
ct
In this paper,
w
e
investigate
the efficiency
of so
me ki
nd
of Greedy Algorith
m
s w
i
th respect t
o
dictio
nari
e
s fro
m
H
ilb
ert spac
es. W
e
establ
i
s
h id
eal
Le
be
sgue-typ
e
in
eq
uality for Orth
ogo
nal M
a
tchi
n
g
Pursuit a
l
so kn
ow
n in l
i
teratur
e
as the Orth
o
gon
al Gree
dy
Algorit
h
m
for
-coher
ent dicti
o
nari
e
s. W
e
sho
w
that the Orthog
ona
l Matchin
g
Pursuit prov
ide
s
an al
most opt
imal a
pprox
i
m
a
t
ion on th
e first
[1
/
2
0
]
.
Ke
y
w
ords
:
ort
hog
on
al match
i
ng pursu
it
(OMP),
orthogo
n
a
l gr
eedy
al
go
rithms, b
e
st
m
-t
e
r
m
,
Le
be
sgu
e
-
type ine
q
u
a
liti
e
s, dictionar
ies
Copyrig
h
t
©
2013
Univer
sitas Ahmad
Dahlan. All rights res
e
rv
ed.
1. Introduc
tion
In this paper
we co
ntinue t
o
study the conver
g
e
n
c
e o
f
greedy algo
rithms with re
gard
s
to
dict
ion
a
rie
s
w
i
t
h
small coh
e
r
en
ce (
s
ee [
1
-9]
)
.
T
he pu
rp
ose of the re
search of app
roximation with
rega
rd
to in
coherent
dictio
narie
s wa
s to
apply
in
co
mpre
ssed
se
nsin
g. In [1,
8, 10,
11], it
wa
s
sho
w
n th
at the Orth
ogo
nal
Matchi
ng Pu
rsuit i
s
effe
ctive for si
gnal
recoveri
ng. In
this pa
pe
r, we
will discu
ss t
h
is problem
from the poi
nt of vi
ew of Approximati
on Theo
ry. We obtain u
pper
estimate for
the error of approximatio
n by
OMP in terms of the error of the be
st
m
-ter
m
approximatio
n. This arti
cle
is a develop
ment of re
cen
t
result
s obtai
ns by Eugen
e
Livshitz in [1
2].
Let us recall t
he stan
dard d
e
finitions of G
r
eedy Algo
rithms. We say
that a set
D
from a
Hilbert Space
H
is a dictio
nary
if
||
|
|
1
,
.
and
span
H
DD
The co
he
ren
c
e of a diction
a
ry is define
d
as
,,
:s
u
p
|
,
|
D
(1)
Dictio
nari
e
s with
co
heren
ce
are call
ed
-
c
oh
er
e
n
t
.
ORT
H
O
G
O
N
AL MATC
HIN
G
PU
RSUIT
(
O
MP) Set
0
:,
f
fH
an
d
0
(,
)
:
0
OMP
f
D
.
For ea
ch
0
m
, we can find a
1
m
g
D
su
ch t
hat
1
|,
|
s
u
p
|,
|
mm
m
g
f
gf
g
D
,
and defin
e
11
1(
,
,
)
(,
)
:
(
)
,
m
ms
p
a
n
g
g
OMP
f
Proj
f
D
11
:(
,
)
.
mm
ff
O
M
P
f
D
The be
st
m
-term approximation for a funct
i
on
,
f
H
is define
d
as
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 1, Janua
ry 2013 : 213 – 2
2
6
214
,,
1
1
()
:
(
,
)
:
i
n
f
.
ii
m
mm
i
i
ci
m
i
ff
f
c
D
D
Suppo
se that
dictiona
ry
D
is
-co
h
e
r
ent an
d
1
1
(1
)
.
2
m
It is
well k
n
own (s
ee [3,4] ) that
if
f
H
is
m
-sparse,
that is
(,
)
0
m
f
D
, then
(,
)
.
m
fO
M
P
f
D
(2)
More
over, Te
mlyakov an
d
Zheltov sho
w
ed that if
1
1
(1
)
2
m
, then eq
uality (2) d
o
e
s
not
hold for all
-coherent dictio
narie
s
D
and all
m
-spa
r
s
e
f
:
1
1
,(
(
)
1
)
,
:
(
,
)
0
,
2
m
mf
H
f
O
M
P
f
DD
D
(3)
(,
)
0
.
m
f
D
Followi
ng T
e
mlyakov, we recall result
s of
L
e
b
e
s
gue
ty
pe
inequalitie
s
whic
h
con
n
e
c
ting the error of
Gree
dy app
roximation an
d the best
m
-term ap
proxi
m
ation. The
s
e
inequ
alities d
o
hold not for all
m
, but only
for
()
mC
; they provide an estim
a
te for the
quality of app
roximation of
A(m) iteration
of OMP by th
e best
m
-term approximatio
n:
()
(,
)
(
)
(
,
)
,
(
)
Am
m
fO
M
P
f
B
m
f
m
C
DD
(4)
with s
o
me
()
,
(
)
,
(
)
Am
B
m
C
.
The first Leb
esg
ue type inequality for Greedy Algo
rithms was obt
ained by Gilb
ert et al.
in [3] They establish
ed (4
) for an optim
al
()
:
,
A
mm
an orde
r-opti
m
al
1
()
1
,
82
C
and fast g
r
owing
1/
2
()
:
8
.
B
mm
Don
oho et al.
[2] obtained
inequ
ality (4)
with opt
imal
(up to a con
s
t
ant factor) B(m)=24,
but not optimal A(m):=
lo
g
mm
and
2/
3
1
()
20
C
.
Temlyakov and Zheltov [1
0] proved in
e
quality (4)
wi
th
lo
g
()
:
2
,
m
Am
m
B(m):=
3
and
()
,
C
which g
uara
n
tee
s
ine
quality
2l
o
g
1
2
26
m
m
In other wo
rd
s, the
y
proved
.
T
h
eorem
1
,
For
e
ve
ry
cohe
rent
dictionary
and
any
function
f
H
D
lo
g
2l
o
g
2
1
(,
)
3
(,
)
,
2
.
26
m
m
m
m
fO
M
P
f
f
i
f
m
DD
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Lebe
sg
ue-t
y
p
e
Inequalit
y for Orth
ogo
nal
Matching Pu
rsuit for
µ
-
coherent … (Y
e Peixin
)
215
Re
cently Eug
ene Livshitz [
6
] prove (4)
with
1
(
)
:
2
,
(
)
:
2.7,
(
)
20
Am
m
B
m
C
. I
n
other word
s,
he provedh
ich gua
ra
nte
e
s ineq
uality
2l
o
g
1
2
26
m
m
.
In other words, they
proved
2.
T
h
eorem
,
For
e
ve
ry
cohe
rent
dictionary
and
any
function
f
H
D
22
(,
)
2
.
7
(,
)
mm
m
fO
M
P
f
f
f
DD
f
or
all
1
1.
20
m
The aim of this pap
er is to
prove (4) with
1
(
)
:
2
,
(
)
:
2.47
(
1
)
,
0
,
(
)
,
20
Am
m
B
m
C
(5)
and thereby improve
s
ab
o
v
e result.
.
T
h
eorem
3
,
For
e
ve
ry
cohe
rent
dictionary
and
any
function
f
H
D
0
, we
have
22
(
,
)
2.47
(
1
)
(
,
)
mm
m
ff
O
M
P
f
f
DD
f
or
all
1
1.
20
m
2. Preliminary
lemmas
By condition
s of Theore
m
3 , we have
1
/
20.
m
(6)
We u
s
e several stand
ard le
mmas to p
r
ov
e Theo
rem 3.
Lemma 1.
,1
2
,
For
any
n
n
m
and
1
,,
,
n
ii
i
i
i
hc
c
D
we
h
a
v
e
11
max
|
,
|
max
|
|
(
1
2
)
ii
in
in
hc
m
, (7)
11
ma
x
|
,
|
ma
x
|
|
[
1
(
2
1
)
]
,
ii
in
in
hc
m
(8)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 1, Janua
ry 2013 : 213 – 2
2
6
216
1
11
ma
x
|
|
m
a
x
|
,
|
,
ii
in
in
cK
h
(9)
whe
r
e
1
11
0
1
0
:
1
(
2
1
)
9
10
9
K
m
(10
)
pro
o
f
. For any
1
in
, using (1
), we ha
ve
1,
,,
,
ii
i
i
j
j
i
jn
i
j
hc
c
1
(1
)
(
m
a
x
|
|
)
ii
in
cn
c
1
(m
a
x
|
|
)2
ii
in
cc
m
1
(m
a
x
|
|
)
(
1
2
)
.
i
in
cm
Then we get inequ
ality (7).
Similarly
1
,(
1
)
(
m
a
x
|
|
)
ii
i
in
hc
n
c
This imply inequality (8).
Inequality (9
) follows from (8).
As a co
nse
q
u
ence of Lem
ma 1, we de
ri
ve the followi
ng lemma.
Lemma 2.
2,
,
,
1
.
i
Le
t
n
m
h
H
i
n
D
Suppo
se that
1
(,
,
)
1
()
.
n
n
s
pa
n
i
i
i
Proj
h
c
Then
1
11
max
|
|
m
ax
|
,
|
.
ii
in
i
n
cK
h
pro
o
f
.Set
1
(,
,
)
1
()
.
n
n
s
pa
n
i
i
i
hP
r
o
j
h
c
It is easy to see that
,,
,
1
.
ii
hh
i
n
Thus th
e lem
m
a follows from inequ
ality (9) for
h
.
For
1
n
we defin
e
1
:,
.
nn
n
df
g
(11
)
Let numbe
r
,
,1
,
in
xn
1
in
satisfy the equ
ality
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Lebe
sg
ue-t
y
p
e
Inequalit
y for Orth
ogo
nal
Matching Pu
rsuit for
µ
-
coherent … (Y
e Peixin
)
217
1,
1
.
n
nn
i
n
i
i
f
fx
g
(12
)
The followi
ng
lemma tell us ho
w the value of
,
in
x
depen
d
s
on the cohe
ren
c
e of the d
i
ctiona
ry.
Lemma 3.
2,
:
For
any
n
m
w
e
c
a
n
g
e
t
the
f
ollow
i
ng
e
s
timates
,1
||
|
|
,
1
1
,
in
n
xK
d
i
n
(13
)
,1
||
|
|
.
nn
n
n
x
dK
d
(14
)
pro
o
f
.By the definition of OMP
1
(,
,
)
:(
,
)
(
)
.
l
l
l
span
g
g
f
fO
M
P
f
f
P
r
o
j
f
D
Then we ca
n see
,0
,
1
.
li
f
gi
l
(15
)
H
e
nc
e
1
1(
,
,
)
1
()
n
nn
s
p
a
n
g
g
n
f
fP
r
o
j
f
1
1(
,
,
)
1
()
.
n
nn
n
s
p
a
n
g
g
n
n
n
f
dg
P
r
o
j
f
d
g
(16
)
Usi
ng (1
) an
d
(15)
we have
for
1
:,
nn
n
hf
d
g
1
|,
|
|
,
|
|
,
|
|
|
,
1
1
,
in
i
n
i
n
n
hg
f
g
d
g
g
d
i
n
(1
1
)
1
|,
|
|
,
|
|
,
|
|
|
|
|
0
.
nn
n
n
n
n
n
n
hg
f
g
d
g
g
d
d
Suppo
se that
,
,1
,
in
x
in
sat
i
sf
y
11
(,
,
)
(
,
,
)
1
,
1
()
(
)
.
nn
n
s
p
a
n
gg
s
p
a
n
gg
n
n
n
i
n
i
i
Proj
h
P
roj
f
d
g
x
g
By Lemma 2
,1
1
11
ma
x
|
|
m
a
x
|
,
|
|
|
.
in
i
n
in
i
n
x
Kh
g
K
d
(17
)
It follows fro
m
(16)
and (1
2) that
(1
6
)
(
1
2
)
1,
1
,
11
.
nn
nn
n
i
n
i
n
n
i
n
i
ii
f
dg
x
g
f
f
x
g
Then we get the relatio
n
be
tween
,,
,1
,
in
in
x
and
x
i
n
,,
,
,
,1
1
,
.
in
i
n
n
n
n
n
n
x
xi
n
x
d
x
This an
d (1
7) complete the
proof.
The followi
ng
lemma provi
des a
n
estim
a
te of
{|
|
}
n
d
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 1, Janua
ry 2013 : 213 – 2
2
6
218
Lemma 4.
12
1
For
a
n
y
l
n
m
,
we have
2
||
|
|
,
nl
dK
d
whe
r
e
2
10
:
e
xp(
2
)
exp
(
1
/
9).
9
Km
(18
)
pro
o
f
. Using Lem
ma 3, for
12
ln
m
, we have
11
1
,
1
1
||
,
|
|
,
|
n
nn
n
n
i
n
i
n
i
df
g
f
x
g
g
11
,
1
1
|,
|
|
,
|
n
nn
i
n
i
n
i
fg
x
g
g
1
(1
)
,,
1
||
(
|
|
|
|
)
n
nn
n
i
n
i
dx
x
1
|
|
[1
(1
2
)
]
n
dm
K
(6)
,
(
1
0
)
21
0
|
|
[1
(1
)
]
20
9
n
d
10
||
(
1
)
.
9
n
d
H
e
nc
e
fo
r
any
,1
2
1
nl
n
m
, we can get
2
2
10
2
10
10
9
|
|
||
(
1
)
|
|
(
1
)
||
(
2
)
|
|
.
92
9
nl
m
nl
l
l
l
m
dd
d
d
e
x
p
m
K
d
m
3. Nota
tions
By the definition of the best
m
-term ap
prox
imation there exist
,0
,
j
a
,
j
D
1,
jm
and
0
H
su
ch t
h
at
0,
0
0
0
1
,,
0
,
1
,
m
jj
j
j
f
fa
j
m
0
(1
)
(
,
)
(1
)
(
)
,
0
.
mm
ff
‖‖
D
(19
)
Set
1
:
(
,
,
),
(.)
:
(.),
(.)
:
(.),
mL
L
L
L
L
s
pan
P
P
roj
P
Proj
:(
)
,
0
2
.
nL
n
Pf
n
m
Let the numb
e
r
,
,0
,
1
,
jn
an
j
m
satisfy equalities
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Lebe
sg
ue-t
y
p
e
Inequalit
y for Orth
ogo
nal
Matching Pu
rsuit for
µ
-
coherent … (Y
e Peixin
)
219
,
1
:(
)
(
)
.
m
nL
n
L
n
j
n
j
n
j
fP
f
P
f
a
(20
)
Define
11
2
1
:
{
{
1
,,
2
}
:
{
}
}
,
:
{
1
,,
2
}
\
.
m
ij
j
Ti
m
g
T
m
T
Then, for
1
n
, we let
2
2
22
:{
1
,
,
}
,
:
.
n
n
nT
TT
n
D
d
(21
)
4. Main lemmas
Lemma 5.
2
12
,
,
.
Le
t
i
n
m
i
n
T
T
hen we have
19
|(
)
,
|
.
18
Ln
i
Pg
g
pro
o
f
.Let
1
()
.
m
Ln
j
j
j
Pg
c
Since
2
nT
and
,|
,
|
,
1
,
nj
n
j
gg
j
m
it follows
from Lemma 2 that
1
1
ma
x
|
|
.
j
jm
cK
Therefore, we have
|
(
)
,
|
|
(
)
,|
|
,
|
|
(
)
,|
L
ni
n
L
ni
n
i
L
n
i
Pg
g
g
P
g
g
g
g
P
g
g
11
1
|,
|
(
m
a
x
|
|
)
(
m
a
x
|
,
|
)
n
jj
i
j
j
i
jm
jm
i
cg
c
g
1
19
()
.
18
mK
□
Lemma 6.
1
;
Le
t
n
T
then we h
a
ve
22
1
0.20
.
nn
D
‖‖
‖
‖
pro
o
f
. Let
2
:.
n
n
tT
#
(22
)
If
2
n
T
, then
10
nn
and no prove i
s
nee
ded, so
we ca
n assume that
1
n
t
. B
y
Lemma 4
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 1, Janua
ry 2013 : 213 – 2
2
6
220
2
2
||
m
i
n
|
|
.
n
ni
iT
dK
d
(23
)
On the othe
r hand, by defi
n
ition (21
)
an
d (22
)
, we ha
ve
2
2
2
22
2
(m
i
n
|
|
)
,
n
n
in
i
i
iT
iT
iT
dt
d
d
D
so we get
2
1/
2
mi
n
|
|
(
)
.
n
i
iT
n
D
d
t
Combi
n
ing thi
s
with (23), we obtain
1/
2
2
||
(
)
,
n
n
D
dK
t
22
2
.
nn
dt
K
D
(24
)
Define
1
1
22
,,
:.
nn
nT
in
i
i
n
i
iT
i
T
hx
g
x
g
(25
)
Acco
rdi
ng to the definition
of
n
, we have
1,
1
()
(
)
n
nL
n
L
n
i
n
i
i
Pf
P
f
x
g
2
1,
1
()
(
)
.
n
Ln
i
n
i
n
L
iT
Pf
x
g
P
h
So we get
22
2
2
11
1
||
||
||
(
)
||
||
||
2
|
,
(
)
|
||
(
)
||
nn
L
n
n
L
L
Ph
Ph
Ph
22
11
2|
,
|
|
|
|
|
.
nn
hh
(26
)
Thus to p
r
ov
e the lem
m
a,
we m
u
st e
s
t
i
mate
1
|,
|
n
h
and
2
||
||
h
. Usi
ng
(15
)
a
nd (25),
we
obtain
(1
5
)
1
1
,1
,1
11
|,
|
|
,
|
|
,
|
mm
nn
j
n
j
j
n
j
jj
h
f
ah
ah
1
2
(2
5
)
,1
,
1
|,
|
n
m
jn
j
i
n
i
j
iT
ax
g
1
2
,1
,
1
||
|
,
|
.
n
m
jn
i
n
j
i
j
iT
ax
g
(27
)
For any
1
lm
, we have
(2
0
)
,1
,
1
1
1
11
|,
|
|
,
|
|
,
|
|
|
.
mm
jn
j
l
j
n
j
n
l
n
l
n
jj
aa
f
d
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Lebe
sg
ue-t
y
p
e
Inequalit
y for Orth
ogo
nal
Matching Pu
rsuit for
µ
-
coherent … (Y
e Peixin
)
221
By Lemma 1, we get
,1
1
,
1
1
11
1
max
|
|
(
max
|
,
|
|
|
.
m
jn
jn
j
l
n
im
l
m
j
aK
a
K
d
(28
)
Then we obta
i
n the estimat
e
,1
1
1
||
|
|
.
m
jn
n
j
am
K
d
It follows
from (1) and Lemma 3 that for
1
jm
,
1
1
2
2
(1
3
)
(
2
2
)
12
,2
,
2
1
1
|,
|
#
m
a
x
|
|
#
|
|
|
|
.
n
n
nn
in
j
i
in
n
n
n
iT
iT
x
gT
x
T
K
d
K
d
t
Thus,
we can
continu
e
(27
)
as
1
2
22
2
1,
1
,
1
1
|,
|
|
|
|
,
|
n
m
nj
n
i
n
j
i
n
n
j
iT
ha
x
g
m
K
d
t
22
2
2
2
2
12
12
2
5
81
m
KKD
m
K
KD
KD
By condition (6), we
can
su
ppo
se that
01
/
4
0
, s
o
1
10
1
1
.
9
91
0
3
7
1
10
K
(29
)
Acco
rdi
ng to Lemma 3, we
can write
1
1
2
2
22
2
1
1
2
,,
2
2
||
|
|
#
#
(m
a
x
)
[
(
)
]
n
n
nn
in
i
i
n
iT
iT
hx
g
x
T
T
‖‖
22
2
1
1
2
12
2
#
[)
]
#
(
nn
n
Kd
T
T
22
2
2
22
2
11
2
()
(
1
2
)
nn
n
K
dt
t
K
K
D
m
22
2
11
2
2
2
10
1
1
1
()
(
1
2
)
1
.
1
.
9
3
7
333
KK
K
m
D
K
D
K
D
(30
)
No
w usi
ng th
e estimate
s for
1
|,
|
n
h
and
2
||
||
h
, we can co
ntinue i
nequ
ality (26):
22
2
11
||
||
||
||
2
|
,
|
|
|
||
nn
n
hh
22
2
12
2
51
1
||
|
|
2
81
333
n
K
DK
D
2
1
||
||
0.2
0
.
n
D
This
es
timate c
o
mpletes
the proof of the lemma.
No
w we p
r
o
c
eed to the est
i
mate of
||
||
n
for
2
.
nT
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 1, Janua
ry 2013 : 213 – 2
2
6
222
Lemma 7
.
2
;
Le
t
n
T
then we h
a
ve
22
2
1
0.76
.
nn
n
d
‖‖
‖
‖
pro
o
f
. Just as in th
e proof of Le
mma 6, we u
s
e the elem
e
n
t
1
2
,
:.
n
in
i
iT
hx
g
Set
1,
:(
)
.
nL
n
n
n
n
Pf
x
g
Then we ca
n write
1,
1
()
(
)
n
nL
n
L
n
i
n
i
i
Pf
P
f
x
g
1
1,
,
1
()
(
)
n
Ln
n
n
n
L
i
n
i
i
Pf
x
g
P
x
g
1
2
,
()
(
)
,
n
nL
i
n
i
n
L
iT
Px
g
P
h
22
2
2
2
||
||
||
||
2
,
(
)
||
(
)
|
|
||
||
2
|
,
|
||
|
|
.
nn
n
L
L
n
n
Ph
Ph
h
h
(31
)
Therefore, to
prove th
e le
mma it suffices to
obtain
uppe
r b
oun
d
s
for
22
|
|
|
|
,|
,
|
,|
|
|
|
nn
hh
. To
es
timate
2
||
||
,
h
we can u
s
e in
eq
uality (31) fro
m
Lemma 6
22
2
2
1
1
2
12
2
||
||
[
(
#)
]
#
nn
n
hK
d
T
T
22
2
2
2
11
1
[2
(2
)
]
(
)
2
(
1
2
)
nn
K
dm
m
K
K
m
m
d
22
2
10
1
1
11
1
.
1
0
.0
04
.
9
1
0
3
7
3330
nn
n
dd
d
(32
)
Then we proceed to the est
i
mate of
2
||
||
n
.
Usi
ng (2
0), (2
8) and the in
clusio
n
2
nT
, we ca
n write
11
,
1
1
|,
|
|
,
|
m
nn
n
n
j
n
j
n
n
j
gd
f
a
gd
1,
1
,
1
11
|,
,
|
|
,
|
mm
nn
j
n
j
n
n
j
n
j
n
jj
f
ga
g
d
a
g
,1
1
11
(m
a
x
|
|
)
m
a
x
|
,
|
|
|
.
jn
j
n
n
jm
j
m
am
g
K
d
m
(33
)
Then u
s
ing L
e
mma 3 an
d (29
)
,(6
)
, we o
b
tain the esti
mate
,1
,
1
2,
2
[
(
)
]
[
(
,
)
]
nn
n
n
n
n
n
n
n
n
n
n
x
gd
x
d
d
g
d
11
2
(
||
||
)
(
||
||
)
nn
n
n
dK
d
d
K
d
m
Evaluation Warning : The document was created with Spire.PDF for Python.