2">m
j


1,12 m
p
hn
From (11), (12) and (3), we have
2.
m
p
h
Copyright © 2012 SciRes. APM
R. S. LI, Y. P. LIU
188
So


 
221
:,
sup
q
p
fU
B L
SSfh
12.
m
p
nh


 (13)
For sufficiently large m, by (12), (13) and the
mcity of ,
m
onotoni
we have



.
d
UB Lm


(14)
mqp
r cases can be ob-
tai In detail, we may
show them in the following.
,en
The upper estimations for the othe
ned by the embedding Theore.m
If 2qp th.
p
 So for any
B


q
L, by fU
0
21,
j
jqjl
f
we have
s
21,
j
jq
f
for all .j Thus,
2
221.
jj
jj
q
ff

 Hence we have

20
21,
j
jjl
f
i.e.,


2.
f
UB L
So, we have
following embedding relation

2.
q
L L

ave


UB
UB
By (10), we h



2
p
LUBL

then for any j and
(3), we hav
.
d
m
02 ,qp


,
q
L by
mq
m
UB



If
fUB
e21
j
jq
f
different valu, repl
(if q takes
es acing
j
f
by ,
j
T does not influ-
ii inequality (see [1], p.
102) for the inequality), we have
ence the proof). So by Nikol’sk

112
2
22
jd q 2 1.
jj
jj
q
ff

Hence



112dq
sq
L
 and we have folw-
ing embedding formula
fUB
lo





112dq
q sL

2.
s
UB LUB
By (10) we can get







11
2
.
dq
m
112
2
dq
msq ms
p
UB LUBL




0< 2,qp then for any j and

L
If
sq
UB
, we have f
=0
21
s
jqjl
.
jf
By
Nikol’skii inequality we have

11
00
222 .
s
s
jd qp
jj
jj
pq
nj
ll
ff




Thus we have following embedding formula


11 .
dq p
sq sp
UB LUBL

By (14), we have





11
/11 .
dqp
ms pmsq
p
dqp
UB L
m


If 1,pq
UBL

,
sq
fUBL
by

then, for any
=0 s
jqjl
f
21
j
, we have 2
j
n
f
1,
q for any
.jN 221
j
jj
pq
ff and
j

So there hold
0
21.
j
jpjl
f
Therefore, we have


.
sqsp
UB LUB L

By (14) we have



.
d
mspmsq
pp
UB LUB Lm


 

The upper estimation is finished.
By the definition of m
and m
, the lower estima-
tion can be gotten from Theorem 2.4, and the following
relation


2.
msqmsq
p
p
UBB L


Proof of Theorem.
Proof of Theorem 2.6. First, we consider the case
1p
L U
2.5 is finished
2 .qition 2.2 and 2.3, we ha By Definve


q msq






2
2
.
ms
p
p
ms
msq
UBL
UBL
UBL



(15)
By Theorem 2.5, we have
q
UB L

11 .
dqp
UB Lm

ms
q
p
When 12,p
for 2q1 ,
by Theorem 2.5, the
upper estimation is

112
2
.
dq
msq
UB Lm

From (15) we can get
Copyright © 2012 SciRes. APM
R. S. LI, Y. P. LIU
Copyright © 2012 SciRes. APM
189





112dq
2
.
ms
q msq
p
mUB LUBL



)
, 12q, by the relation between
-tximation and Greedy algorithm [7], we
have

(16
Whe
best m
n 2p
erm appro
 
1.
p
mm
12
p
p
f
gf mf
(17)
For
,
sq
fUBL
and any l, by (16), (17) and
Theorem 2.4, we have
 
|2π|π
121 12
sup
lm
ylnn
dq
gfy


(18)
:,
.
m
afy
fgmmm

From Lemma 3.2 and relation (17), (18), w


e have

1
2πsup
n
n
|2π|π
|| 0
1 112/1 121
sup
.
sq
ms
q mm
p
pylnn
l
fUB L
p
dqp dq dq
f gf g
mm mmmm


  






r the case 2.q
xy ln

UBL
When 2,p we conside

By the 2,
jj
q
ff we have



2.
ssq
UB LUB L

(20)
By (19) and (20), we can get





12
2.
d
msqms p
p
UB LUB Lm


 
(21)
In the following we will give the lower estimation. By
on 2.3, we have
.
Definiti



2msqmsq
p
p
UBL
UBL


And by Theorem 2.4, we have




11dqp

when 12,p and



ms
q
p
UB Lm

112dq
msq
p
UB Lm

w
2.6.
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