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θ
r0≤ j≤ n,t
n)= q
n(
n _ na
j )
(10)
khe
nby(l)wehave
‖t
n‖ n ≤
1
‖w(t
‖ p,t
n)
n _ na
C1
(11)
ybvi
ous
l
y,t
hef
und
t
i
onal
z* st
nx
1
(2π)k
∫
t({)d{
kn
[- π,π]
(12)
*
*
*
*
s
at
i
s
f
i
e
sz _ n and‖ z ‖ n* W X YaFr
om |e
mma1we[now t
hatt
he
r
ee
}i
s
t
st =
(k,n)
qwj
r_p* s
at
i
e
s
yi
ne(7),and
‖t* ‖ p* ≤
1
‖z* ‖ n* WX Y,
C1
t
hati
s
1
(k,n) V
|wj
|
nk0Σ
≤j
≤n
(
1
V
) WX Ya
>0,t
Fori
nt
e
ee
rr
heee
ne
r
al
i
z
e
dJ
ad
[s
on[e
r
ne
li
sde
f
i
nde
dby
(1~)
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
356
n
r
t
s
i
n
2
=
s
i
n 1+
.20,No.3
Vol
t)2r
2
,
( [ ])
-1
kn,r(t
)= λ
n,r
=
)
whe
r
e[x]+1de
not
e
st
hes
mal
l
e
s
ti
nt
e
ge
rt
hate
xc
e
e
dsx.Le
t
1 π
kn,r(t
)dt
,
2π - π
∫
λ
n,r =
2r
-1
by[10]wehaveλ
n,r～n
and
π
∫tk (t)dt～ n ,1≤ k≤ 2r- 2.
k
-k
n,r
-π
I
nvi
e
w of[11],wege
t
t)2r
2
n
r
t
s
i
n
2
=
s
i
n 1+
( [ ])
=
(
=
Σ
n) i
kt
β(
|k|,re ,
n
k= - r
r
[]
)
n
[r])aregivenby
(n)
whe
r
eβk,r 0≤k≤r
n
[r]
r
r
-1
n)
β(
0,r =
j
=0
=
2r (r- j
) 1+
j
n
)
( [r])+ r- 1 ;
()
Σ (- 1)j
=
2r- 1
)
n
=
)
= r- 1
2r (r- j
) 1+
- k+ r- 1
j
r
(
1
)
,1≤ k≤
Σ
j
j
=0
=
2r- 1
)
()
r
-2
=
2r (r- j
) 1+
()
j
Σ (- 1) j
j
=0
( [ ])
n
n
[r]- (r- 1);
)
( [r])- k+ r- 1 ,
=
2r- 1
)
n
n
[r]- (r- 1)+ 1≤ k≤ 2[r]- (r- 1)+ 1;
(n)
...........................................
βk
,r = <.
=
r 1+
n
)
=
=
2r- 1
)
=
n
)
( [r])- k+ r- 1 - (21r) (r- 1)(1+ [r])- k+ r- 1 ,
n
2r- 1
)
n
[r]- 1≤ k≤ (r- 1)[r]- 1;
=
n
)
r 1+
- k+ r- 1
n
n
[
]
r
(
)
,(r- 1)
[r]≤ k≤ r[r].
(r- 2)
==
2r- 1
)
s
∈ Q ,wede
)= kn,r(t
...kn,r(t
Fort
f
i
nemul
t
i
ke
r
ne
lf
unc
t
i
onKn,r(t
kn,r(t
nd
1)
2)
s)a
ge
ne
r
at
e
dJ
ac
ks
onme
an
Jn(f,x): = Jn,r(f,x)=
r
1
(- 1)j f(x+ j
t
)Kn,r(t
)dt
.
s s
Σ
(2π) Q 1≤ j≤ r,j∈ Zs
j
∫
=(j
...,j
)asf
:
Al
s
owedi
f
i
nej
j
ol
l
ows
1,
2,
s
()
(14)
PERI
ODI
CNEURALNETWORK{
Zho
uGuanzhe
n
s
j
t=
r
i
=1
r
r
j
1
j
2
r
() ( )( ) ( )
Σ jt, j =
ii
357
...
j
s
,
(- 1)j = (- 1)j1(- 1)j2...(- 1)js.
,wec
Thus
ange
tt
hef
ol
l
owi
ngLe
mma.
[9]
p
≥1,f∈LQs,1≤p≤+∞,t
)>0s
he
r
ei
sac
ons
t
antC(r
at
i
s
f
yi
ng
Le
mma3. Forr
1
‖f- Jn,r(f)‖ *
(r
)wr f,
p,s≤ C
n
( )
s
,f∈ Lp
Q,
(15)
s
p,Q
whe
r
e
wr(f,t
)p,Q =
(∫
s
up
s
wr(f,t
)∞ ,Q =
Q
)
h
r
,xi
s
up ‖Δ (f,x)‖
s
1≤ i
≤ k,0< h≤ t
r
Δh
f,x)=
r
,xi(
s
1≤ i
≤ k,0< h≤ t
1
p
p
Δh
f,x) dx ,1≤ p<+ ∞,
r
,xi(
l
=0
r
,p=+ ∞,
()
r
-l
Σ (- 1)
*
∞ ,s
l
f(x1,x2,...,xi- 1,xi+ l
h,xi+ 1,...,xs).
Fr
om (14)wege
t
Jn,r(f,x)=
r
()
1
Σ (- 1) j (2π)∫K
j
1≤ j
≤r
r
()
s
1
Σ (- 1) j (2π)∫K
j
1≤ j
≤r
s
(t
)f(x+ j
t
)dt=
n,r
s
Q
x
(u- j)f(ju)du=
n,r
s
Q
r
()
j
1
Σ (- 1) j (2π) ×
1≤ j
≤r
∫[ [ ]Σ
x
s
Q
-r
s
n
n
≤ k≤ r
r
r
- s (n) i
k uλ
n,rβ
|k|,re( j)
]f(ju)du=
[]
r
() [ ]Σ [ ]
j
Σ (- 1) j
1≤ j
≤r
- s (n)
λ
n,rβ
|k|,r ×
n
n
-r
≤ k≤ r
r
r
kx
1
- i
i
ku
(j
u)e
du e j ,
s sf
Q
(2π)
( ∫
)
s
klxl
kx
=(j
...,j
),andde
.
f
ork=(k1,k2,...,ks),j
j
f
i
neβ =β β ...β , = Σ
1,
2,
s
l
=
1
j
j
l
2i
I
nvi
e
w oft
heLe
mma3.3n[10]ort
heLe
mma3.2i
n[12],wehavet
hati
ft
he
r
e
(n)
|k|
(n) (n)
|k1| |k2|
(n)
|ks|
1
s
i
ku
f(j
u)e
du= 0.The
doe
s
n'
te
xi
s
tα∈Z s
uc
ht
hatk=j
α,t
he
n
r
e
f
or
e,t
hi
sf
or
mul
a
(2π)s Qs
,he
i
sane
ve
nt
r
i
gonome
r
i
cpol
ynomi
nalofor
de
r≤nandhasse
l
e
me
nt
s
nc
e
y z
i
kx
Jn,r(f,x)= Σ Jn,r(f)(k)e .
∫
- n≤ k≤ n
{o
Jn,r(f,|lx- })=
Σ
- n≤ k≤ n
y z
i
k(|l
x- })
Jn,r(f)(k)e
=
Σ
y z
i
k|l
x- i
k}.
Jn,r(f)(k)e
- n≤ k≤ n
︵
[1]
*
p
*
p
*
)≠0,l
∈Zs.
Le
mma~. Le
tφ ∈LQs,t
he
n{panΔφ i
sde
ns
ei
nLQs i
fandonl
yi
fφ (l
︵
*
p
*
)≠ 0,l
∈ Zs andni
Le
mma5.Le
tφ ∈ LQs,1≤ p≤ + ∞.I
fφ (l
sl
ar
gee
nough,t
he
nwe
2<=v
2B
r--2
>2J2Q>?
n]
P
]@n?
92A]
@
Fc;
2Y/0DJ2F
CJL
have
* *
+
Jn(φ
,(-,. /0/1 -1 n2
* *
+
27
34
556
89
he
:
e;
<a-=h;
>
h<
a9
;
<
8
;
e
</1-1n<
?>
h9
ha9Jn0@(φ ,(-,A /2BCDe
EEaFaGH
K
9
heIJ
L
He
:;
Ge
M?aL
;
9
C0=ehave
S
*
*
*
NJn0@(φ
,O φ
N*
R@ φ
0
-0P1 Q
n
T
U
0
P
-0V
aGH
X
X
X
*
*
*
(-,WA Wφ
(-,O Jn0@ φ
(-,WA
Wφ
S
*
*
;
-\
Tφ (\,O Jn0@(φ 0\,U] H\ 1
(YZ,P VP
[
S
*
*
*
QNφ
O Jn0@(φ
,N *
R@ φ
0
-0P1 Q
n
T
U
P
^ /0(n^_ ‘,2
-0V
X
*
0φ
(-,A/0=h;
2
ah?<
>
h;
EbL
;
e
<9
ha9De
EEachJL
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iSj
o
o
de
ffgh2 De
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e
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9
;
JGJGV aGHa8
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9
;
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:
;
JH
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ve
:
Cva:
;
apL
e0rsJot P09
he
G=ehave
S
S
k(rv
,HvA
(Yu,P VP
(Yu,o
[
[k(v,Hv2
o
V
wx ygz
{4
e
|
}~
t
’
o
*
8<
baG)φ* 0J(l,;
<He
G<
e;
G*VP0pCiSj=ehave
W:
;
9
e#φ* A$%s& mφ (%,./(27
-
&P+$/(, $r %m%s
#
*
φ
0rs J(2
(S.,
P
2Whe
GoA PaGH#φ* / & +$/(0=eL
e
9J pea
ah;
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e
<
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EbJ:
9
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e
<
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9
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h;
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aL
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he9
:
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L
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;
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9
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9
:
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9
?:
e27
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he
:
-
P
*
s & +$/((2De
>
a<
e0=he
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9
eJA $2 m 2
9φ peaGe
0>
;
9
a9
;
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8
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9
;
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he
G3baG)φ* 0J(l,;
<aGe
?:
aLGe
9
=J:
1J?9
b?9J89
h:
e
eL
aCe
:
<=;
9
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G
2
L
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:
P
7
9;
<Jpv;
J?<9
ha99
he
:
e;
<JGe>
JJ:
H;
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eJ8ve
>
9
J:%s & =hJ<
eap<
JL
?9
evaL
?e;
<
P
s&+$/(0=e>
0aGH=e=:
p;
qqe
<
9
;
9
e;
9a<N%N24J:2
hJJ<
e%2s #φ* aGHr2sJ<
?>
h9
ha9
-
2A r2 %20
(S5,
=he
:
eN%2N ;
<9
heE;
G;
E?E aEJGqaL
L%2<
a9
;
<
8
C;
Gq(S5,24J:aGC;
G9
e
:
qe
:n6S0=e=:
;
9
e
7nm A Ea0$N%2Nm O n1 21 n02. /(2
-
P
s&+$/(0=eJp9
Ar2 %28
4J:a:
p;
9
:
a:
C2
a;
G2
:
JE (S5,24J:aGCbJ<
;
9
;
ve;
G9
e
qe
:
<86Y
aGH767n0=ehave
S
(YZ,o
[J
o
V
*
;
%2
9
(φ
0r2lO 9,]
H9A
708
S
(YZ,o
[:
o
X
*
;
%(r2
lO 9, ;
J708 (φ
,(%,]
]%29H9A
V O 71 %1 7
PERI
ODI
CuEURALuETWORKS
oyz
{|{}n~ye
n
Σ
︵
1
*
JN,α (φ
)(k)
(2π)d
∫e
- N≤ k≤ N
xw9
i
(kl- k)v
i
kAl
x
dve
=
d
Q
︵
︵
*
i
kl
Al
x
*
i
l
x
JN,α (φ
)(kl)e
= JN,α (φ
)(kl)e
.
Thus
i
l
x
=
e
1
︵
*
JN,α (φ )(kl)(2π)d
∫J
*
i
kl
v
(φ
,Alx- v)e
dv.
(18)
N,α
d
Q
*
i
k·
d
d
I
nvi
e
w ofJN,α(φ ,Alx-·)el ∈HN+ |kl|⊂HN+ Nn andLe
mma2,wege
t
i
l
x
e
=
1
︵
*
)(kl)(2π)d
JN,α (φ
∫J
d
Q
*
i
kl
v
(φ
,Alx- v)e
dv=
N,α
(d,N+ Nn)
wβ
1
Σ
︵
nd
*
JN,α (φ )(kl)0≤ β≤ N+ Nn
(d,N+ Nn)
*
(d,N+ Nn) i
JN,α(φ
,Alx- θ
)eklθβ
β
.
So
Jn,r(f,x)=
︵
Jn,r (f)(l
)
×
︵
*
- n≤ l
≤n
JN,α (φ )(kl)
Σ
d,N+ Nn)
(d,N+ Nn)
w(
β
*
(d,N+ Nn) i
JN,α(φ
,Alx- θ
)eklθβ
=
β
d
n
0≤ β≤ N+ Nn
Σ
︵
(d,N+ Nn)
wβ
Jn,r (f)(l
) iklθ(βd,N+ Nn)
e
×
d
Σ
Σ
︵
n
*
0≤ β≤ N+ Nn
- n≤ l
≤n
JN,α (φ
)(kl)
*
(d,N+ Nn)
JN,α(φ
,Alx- θ
).
β
(19)
:
Fr
om (19)wec
ande
f
i
nene
t
wor
kope
r
at
orasf
ol
l
ows
*
φ ,N
f,x)=
Jn
,r (
︵
(d,N+ Nn)
=
Jn,r (f)(l
) iklθ(βd,N+ Nn)
wβ
e
×
Σ
d
Σ
︵
*
n =- n≤ l≤ n JN,α (φ )(kl)
0≤ β≤ N+ Nn
*
(d,N+ Nn)
φ
(Alx- θ
)).
β
(20)
Thenumbe
roft
hene
ur
onsi
nvol
ve
di
nas
i
ngl
ehi
dde
nl
aye
ri
s
*
φ ,N
ne
ut
r
ons
(Jn
(N + Nn)dns).
,r )= O(
q
*
∈
I
nt
hene
ur
alne
t
wor
kc
as
e,d=1.Wehavemφ* =nk∈opφ (k)r0s,Nn=1,Al=l
s
1
:
o,kl=1,β∈o ,he
nc
ewege
tt
hene
ur
alne
t
wor
kope
r
at
or
sasf
ol
l
ows
*
φ ,N
Nn
f,x)=
,r (
(1,N+ 1)
wβ
1
Σ
︵
n
*
0≤ β≤ N+ 1
JN,α (φ )(1)
tΣ
︵
Jn,r (f)(l
)×
- n≤ l
≤n
(1,N+ 1)
*
(1,N+ 1)
i
θ
φ
(l
x- θ
β
)eβ
.
(21)
uow,t
henumbe
roft
hene
ur
onsi
nvol
ve
di
nas
i
ngl
ehi
dde
nl
aye
ri
s
*
φ
ne
ur
ons
(Nn
Nns).
,r)= O(
s
,Nn=n,Ali
∈o.Thus
,t
I
nt
het
r
ans
l
at
i
onne
t
wor
kc
as
e,d=s
sauni
tmat
r
i
v,kl=l
he
:
t
r
ans
l
at
i
onne
t
wor
kope
r
at
or
sar
easf
ol
l
ows
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
360
*
φ ,N
n,r
M
.20,No.3
Vol
︵
(s
,N+ n)
=
Jn,r (f)(l
) ilθ(βs,N+ n))
wβ
e
Σ
(f,x)= Σ
×
s
︵
*
n =- n≤ l≤ n JN,α (φ
0≤ β≤ N+ n
)(l
)
)
*
(s
,N+ n)
φ
(x- θ
).
β
(22)
Now,t
henumbe
roft
hene
ur
onsi
nvol
ve
di
nas
i
ngl
ehi
dde
nl
aye
ri
s
*
φ
t
r
ans
(Mn
(N + n)s).
,r)= O(
≥ d≥ 1,α≥ r≥ 2,n≥ 1,N> Nn,andal
.Le
The
or
e
m 1.Le
ts
loft
he
m ar
ei
nt
e
ge
r
s
tδ≤
^*
η
*
p
s
,1≤p≤+∞,f∈Lp
)≠0(l
∈Zd).I
andφ ∈LQd s
at
i
s
f
i
e
sφ (l
fN i
sbi
ge
nough,
Q,
N+Nn
t
he
nt
he
r
ee
xi
s
t
sac
ons
t
antC>0whi
c
hs
at
i
s
f
i
e
s
└
*
φ ,N
1
‖f(x)- Jn
f,x)‖ *
,r (
p,s≤ C
wr f,
n
L
1
*
nγwα φ
,
N
+
s
M*
p,Q
n
(
( )
)
┐
, (23)
‖f‖
┘
d
p,Q
*
p,s
^*
*
n
≤n},γ=s
/mi
whe
r
eM =mi
n{|φ (kl)|:-n≤l
n(p,2).
kc
as
e,whe
nN i
sbi
ge
nough,wehave
I
nt
hene
ur
alne
t
wor
*
1
φ ,N
,
‖f(x)- Nn
f,x)‖ *
,r (
p,s≤ C wr f
n
[( )
s
p,Q
1
*
+ nγwα φ
,
N
(
)
‖f‖ *
24)
p,s . (
]
1
p,Q
I
nt
het
r
ans
l
at
i
onne
t
wor
kc
as
e,whe
nN i
sbi
ge
nough,wehave
└
*
φ ,N
1
‖f(x)- Mn
f,x)‖ *
,r (
p,s≤ C
wr f,
n
L
1
┐
*
nγwα φ
,
N p,Qs
. (25)
*
+
‖f‖ p,s
^
s
p,Q
┘
mi
n φ* (l
)
( )
(
)
- n≤ l
≤n
.Fr
Pr
oof
om (19)wehave
*
N,α(φ )
Jn,r(f,x)= JJ
(f,x).
n,r
(26)
Soweus
et
heLe
mma3t
oobt
ai
n
*
φ ,N
‖Jn,r(f,x)- Jn
f,x)‖ *
,r (
p,s≤
︵
(d,N+ Nn) =
Jn,r (f)(l
) )
|wβ
|
×
d
Σ
Σ
︵
*
n
0≤ β≤ N+ Nn
=- n≤ l≤ n JN,α (φ )(kl) )
*
(d,N+ Nn)
*
(d,N+ Nn)
,Alx- θ
‖Jn,r(φ
‖*
β
p,s≤
)- φ (Alx- θβ
C(α)
Σ
0≤ β≤ N+ Nn
︵
d,N+ Nn) =
Jn,r (f)(l
) )
w(
1
β
*
wα φ
,
Σ
︵
*
N
nd
=- n≤ l≤ n JN,α (φ )(kl) )
(
1
*
C(α)wα φ
,
N
m*
n,N
(
*
n,N
whe
r
em
)
d
p,Q
Σ
)
d
≤
p,Q
︵
Jn,r (f)(l
),
- n≤ l
≤n
︵
*
=mi
≤n}.Us
n{|JN,α(φ )(kl)|:-n≤l
i
ngt
heCauc
hySc
hwar
t
zi
ne
qual
i
t
yand
t
hePar
s
e
vale
qual
i
t
y,wehave
Σ
- n≤ l
≤n
1
s
s
︵
︵
Jn,r (f)(l
) ≤ Cn2( Σ |Jn,r (f)(l
)|2)2 = Cn2‖Jn,r(f)‖ *
.
2,s
- n≤ l
≤n
Byt
heNi
kol
s
ki
i
t
ypei
ne
qual
i
t
ywede
duc
e
PERI
ODI
CNE^RA_NEvWORKS
Zho
uGumnzhe
n
1
1
1
361
1
‖Jn,r(f)‖ *
ns(p- 2)‖Jn,r(f)‖ *
ns(p- 2)‖f‖ *
.
2,s≤ C
p,s≤ C
p,s
/mi
Wr
i
t
i
ngγ=s
n(p,2),wehave
︵
Jn,r (f)(l
) ≤ Cnγ‖f‖ *
,
p,s
Σ
- n≤ l
≤n
t
he
n
1
*
,
C(α)nγwα φ
N
φ ,N
‖Jn,r(f,x)- Jn
f,x)‖ *
,r (
p,s≤
m*
n,N
(
*
)
d
p,Q
‖f‖ *
.
p,s
So
*
*
φ ,N
φ ,N
‖Jn
f,x)- f(x)‖ *
f,x)- Jn,r(f,x)‖ *
f,x)- f(x)‖ *
,r (
p,s≤ ‖J
n,r (
p,s+ ‖J
n,r(
p,s≤
1
┐
*
nγwα φ
,
N p,Qd
.
*
+
‖f‖ p,s
s
m*
┘
p,Q
n,N
\
]
*
*
i
ng_e
mma‘ana
I
nt
hene
ur
alc
as
e,kl=1.Wege
tmn,N=|JN,α(φ )(1)|.^s
\
]
d*
1 d*
*
*
l
i
m |JN,α(φ )(1)|=|φ (1)|,wehavemn,Ne |φ (1)|,whe
r
eN i
sfi
ge
nough.ge
nc
e,
Nb + c
2
(2h)i
.
sr
i
ght
\
]
*
.Wehavem*
)|.ir
I
nt
het
r
ans
l
at
i
onc
as
e,kl=l
n |JN,α(φ )(l
om _e
mma‘ana
n,N = mi
└
1
C
wr f,
n
L
(
( )
)
- n≤ l
≤n
︵
d
d*
1
*
*
)|=|φ* (l
)|,weal
)|,whe
s
oge
tmn,Ne
mi
n |φ (l
r
eN i
sfi
ge
nough.So
l
i
m |JN,αφ (l
Nb + c
2- n≤ l≤ n
(2‘)i
.
sr
i
ght
n,N
Wr
i
t
ejφ* ,J= q Σ
Σ
*
d
ml,lφ
(nlx+o
,o
nlpJrsq1r,ana
l)
lpQ ,
- n≤ l
≤ nk≤ l≤ N
*
φ ,s
tn
f)p,Qs =
,N (
*
φ ,s *
i
nu ‖f- o
,
n,N ‖ p,s
*
φ ,s
n,N
* ,J
o
n,N p jφ
t
he
nwege
tt
heu
ol
l
owi
ngvhe
or
e
m.
~ d~ 1,α~ r~ 2,n~ 1,Ne Nn,anaal
._e
wxy
z{
y
| }._e
ts
lout
he
m ar
ei
nt
e
ge
r
s
tδ≤
d*
η
*
p
s
,1≤p≤+c,fpLp
)≠k(l
pZd).I
anaφ pLQd s
at
i
s
u
i
e
sφ (l
uN i
sfi
ge
nough,
Q,
N+Nn
t
he
nt
he
r
ee
xi
s
t
sac
ons
t
antCekwhi
c
hs
at
i
s
u
i
e
s
*
1
φ ,s
tn
f)p,Qs ≤ Cwr f,
,N+ Nn(
n
( )
s
.
(27)
.
(28)
.
(29)
p,Q
I
nt
hene
ur
alc
as
e,
*
1
φ ,1
tn
f)p,Qs ≤ Cwr f,
,N+ 1(
n
p,Q
*
1
φ ,s
tn
f)p,Qs ≤ Cwr f,
,N+ n(
n
p,Q
( )
s
I
nt
het
r
ans
l
at
i
onc
as
e,
( )
s
* (α
)
p
.Whe
P{
zzf
nφ pLQd,weus
evhe
or
e
m 1t
ooft
ai
n
*
1
φ
tn
f)p,Qs ≤ C wr f,
,N+ Nn(
n
[( )
+
s
p,Q
nγ
* (α
)
d
‖φ
‖*
‖*
.
p,Q ‖f
p,s
Nα
(3k)
.Mat
.B
{ppl
h.J.Chi
ne
s
eUni
v.Se
r
362
*
.20,No.3
Vol
*
.I
I
f‖f‖ p,s=0,(27)i
sobvi
ous
l
yr
i
ght
f‖f‖ p,s>0,(27)i
sr
i
ghtagai
n.Be
c
aus
eN i
s
nr
1
* (α
)
*
*
i
r
r
e
l
e
vantt
onandN i
sbi
ge
nough,wehave α‖φ ‖ p,Qd‖f‖ p,s≤wr f,
n
N
( )
.
s
p,Q
Re
f
e
r
e
nc
e
s
1 Mha
s
ka
rH N,Mi
c
c
he
l
l
iCA.De
gr
e
eofa
ppr
oxi
ma
t
i
onbyne
ur
a
la
ndt
r
a
ns
l
a
t
i
onne
t
wor
kswi
t
hs
i
ngl
e
,Adva
183.
hi
dde
nl
a
ye
r
nc
e
di
nAppl
i
e
dMa
t
h,1995,16:1512 Di
,Ne
t
z
i
a
nZ,Tot
i
kV.Modul
iofSmoot
hne
s
s
w Yor
k:Spr
i
nge
r
Ve
r
l
a
g,1987.
3 Suz
,Ne
ukiShi
n.Cons
t
r
uc
t
i
onf
unc
t
i
ona
ppr
oxi
ma
t
i
onbyt
hr
e
el
a
ye
ra
r
t
i
f
i
c
i
a
lne
ur
a
lne
t
wor
ks
ur
a
l
,1998,11(5):10491058.
Ne
t
wor
ks
4 ChuiCK,LiXi
.I
,FIUt
,e
,Mul
n.Ne
ur
a
lne
t
wor
kswi
t
honehi
dde
nl
a
ye
r
n:K J
e
t
t
e
r
r
e
r
a
s
ds
t
i
va
r
i
a
t
e
,Si
,1993.
Appr
oxi
ma
t
i
on:Fr
om CAGD t
oWa
ve
l
e
t
s
nga
por
e:Wor
l
dSc
i
e
nt
i
f
i
cPr
e
s
s
5 XuZongbe
,Sc
n,Ca
oFe
i
l
ong.Thee
s
s
e
nt
i
a
lor
de
rofa
ppr
oxi
ma
t
i
onf
orne
ur
a
lne
t
wor
ks
i
e
nc
ei
nChi
na
112.
Se
rF,I
nf
or
ma
t
i
onSc
i
e
nc
e,2004,47(1):976 Mha
,Si
.Appr
,Pr
s
ka
rH N,Na
r
c
owi
c
hFJ
va
r
kuma
rN,e
ta
l
oxi
ma
t
i
onwi
t
hi
nt
e
r
pol
a
t
or
yc
ons
t
r
a
i
nt
s
oc
,2001,130(5):13551364.
Ame
rMa
t
hSoc
7 Er
,JAppr
72.
dy
l
yiT.Not
e
soni
ne
zua
l
i
t
i
e
swi
t
hdoubl
ngwe
i
ght
oxThe
or
y,1999,100(1):60-
8 Ma
,Cons
s
t
r
oi
a
nniG,Tot
i
k V.We
i
ght
e
dpol
ynomi
a
li
ne
zua
l
i
t
i
e
swi
t
h doubl
i
nga
nd{| we
i
ght
s
t
r
71.
Appr
ox,2000,16(1):379 Ti
ma
nA F.The
or
yofAppr
oxi
ma
t
i
onofFunc
t
i
onsofA }e
a
lVa
r
i
a
bl
e,Ne
w Yor
k:Ma
c
mi
l
l
a
nCo,
1963.(i
n}us
s
i
a
n).
10
Sun Yongs
he
ng.The
or
y ofAppr
oxi
ma
t
i
on ofFunc
t
i
ons(~ ),Be
i
j
i
ng:Be
i
j
i
ng Nor
ma
lUni
ve
r
s
i
t
y
,1988:133137(i
Pr
e
s
s
nChi
ne
s
e).
11 Wa
ngGua
nmi
n.Thea
s
ympt
ot
i
cr
e
pr
e
s
e
nt
a
t
i
on oft
hea
ppr
oxi
ma
t
i
on de
gr
e
ef
ort
heJ
a
c
ks
on t
ype
22(i
ope
r
a
t
orJn,p(f,x),JZha
ngz
houTe
a
c
he
r
sCol
l
e
ge,1993,7(2):15nChi
ne
s
e).
12
,Ha
Xi
eTi
ngf
a
n,Zhou Songpi
ng.The
or
y ofAppr
oxi
ma
t
i
on of}e
a
lFunc
t
i
ons
ngz
hou:Ha
ngz
hou
,1997:154157(i
Uni
ve
r
s
i
t
yPr
e
s
s
nChi
ne
s
e).
.ofMa
De
pt
t
h.,Ni
ngboUni
v.,Ni
ngbo315211,Chi
na.