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Периодическая оптимальная кубатурная формула на пространстве Wm-wm-p.p

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p
ÐÑ¿ÒÓԗÕÓÖØ×-Ù%Ú-ÓÖÛÜÕÝ-Þ%ßÌàáÞÛ W
f2µ âã­äã Ú-ÒۉÚ
ßÌàæå%ßÌçۉÚ
ß â Þ%ßÌàéèf× â ÖrßêÕߍëÕìÓÞ%Ú zã ízâ Ó â ×-îÛ
ÞÓ%ì
ì ë2×-îÂô!ß Ý
Ù× Q = [0, 1)n ï
ð ë Ôzݍ٠×-Ô­Ú Û
ÞÔ­Ú-î-× W
f µ ×-Ù ë Õì Ú-Ô¶ìñåÌÛ
åcòêÛ
ÖnÑ¿åÌÛ
ÞÓ å#×-Þ ÒÞÑó #
ã
â
â
2
â
ã
ã
ã
ã
ã
â
â ã
P
îÞ× Ö
f (x) = fk e2πikx
â
ã
k

f2µ k = 
kf |W
Z
Q
dx|
X
k
1/2
fk µ(k)e2πikx |2 
,
ë Ù׉Õ2Û Û Öáå#×-ÖnÙÌÕ å#ÔzÞ×
ò‚ÞÛêÒÞ×-õXÞ ×-ç ۉöñÛ
à_ö õÔ¶ìwî¬Þ%ßêÕݬèÂßÌÞå÷Ó õgø2ÓÙ%ß-Ô­Ú-Ý µ(0) = 1.
íÐÕã ×êµù ÞÓ í ã µ
×-ã ç ÔzÙ ÒÓîÛ Ú-Ô¶ìXã Ú ç-â ×-îÛ
ÞÓ Ö ã P |1/µ(k)|2 < ∞. ô!ã ßÌÞå÷Ó×-ÞÛÜÕ~Ù×
f
W
⊂⊂
C
ã
ã
ízâã­äcú
Þ×-Ô­Ú-Óã û­Ú-×-ã õdå%ßÌ2 çۉÚ
ß Þ×-õèfã × ã ÖrßêÕÑ ã Ô­Ú-ÝXâü 1/h
ý ÷ ã ÕÑ ã ÒÓԗÕ2Ûþ
â
â
ã
lhQ,opt (x) = χQ (x) − hn
X
copt
k (h)δ(x − hk) =
hk∈Q
= χQ (x) − hn c0 (h)
X
δ(x − hk).
hk∈Q
ÿÕ2Û ×zë+Û X
ì Ô­Ú × ×-õØî-Ñ¿Ù%ßÌåÌÕ×-Ô­Ú-Ó ë2ÓÞÓÒÞ× × Û Û î fµ)∗ = W
ÖrßêÕ2Û
1/µ ×-Ù%Ú-ÓÖÛÜÕÝ-ÞۉìXèf×
ë2ÓÞí Ô­Ú-î â ÞÞ Û ï â fÛ
í å‡Ù Ó×zë2ÓÒ ÔzåÌۉìwã ×-ç-×-ç
ö Þí Þۉì‡ä èÂâ ßÌÞå÷(Ó%W
ì2×-ÞÛ Û‰fò­Õ22 Û Û Ú-Ô¶ìwî ì#ë ô!ß Ý â
ã
ã
ãzâ
ã
ã
â
í ã
â
â ã
lhQ,opt (x) =
X
lt = (lhQ,opt (x), e−2πitx ).
lt e2πitx ,
t
∗ !#"%$&(')**+-,/.01'2&!34*57698:;+#$&)
&<=3&&<*+
>@?ABB!#(CD)=,FEG
HIE.HIEE#JK&LMBN
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\<]
^(_`baTcdaeZ_#f
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`hrTf&m&{|_ fm
Wp
~Â×
ÖÛñèÂßÌÞå÷Ó×-ÞÛÜÕ2ÛÙ×
â
ízâã­ä
Þ×-Ô­Ú-Ówî
f21/µ
W
f 1/µ k =
klhopt |W
2
ÐÑ¿ÒÓԗÕÓÖáå#×-ûzèfèfÓ÷Ó %Þ Ú-Ñ@ô!ß Ý
ã
â ã
ã
"
\i}
Ô­Ú-Ý
X
|lt |2 /|µ(t)|2
t
#1/2
.
{lt } 
l0 = (lhQ,opt (x), 1) = 1 − c0 (h),
Ù Ó
â
t 6= 0(mod 1/h)
lt = (lhQ,opt , e2πitx ) = −c0 (h)hn
X
e2πithk =
hk∈Q
= −c0 (h)hn
ÛñÙ Ó
â
t=
rø
ø
r ∈ Rn \ 0
h
n
Y
1 − e2πihkj tj
= 0,
1 − e2πihkj
j=1
lr/h = (lhQ,opt (x), e2πirx/h ) = −c0 (h)hn
€Ú‰Û
ågø
X
e2πirk = −c0 (h).
hk∈Q
f 1/µ k2 = |1 − c0 (h)|2 + |c0 (h)|2
klhopt |W
2
P
(1/|µ(2πri/h)|2 ).
Ò î-Ó#ë2Þ×ørÖnÓÞÓÖrßÌÖ|û­Ú-× ×Øî-Ñ Û‰ù ÞÓ%ì r6=ë20×-Ô­Ú-Ó Û Ú-Ô¶ì Ù Óyî ö ­Ô Ú-î ÞÞ×-Ö
ډÛ
å#×-Ö¿øã Ò%Ú-׬٠×-Ó%ò‚î-×zë2ÞۉìdÙ× cí (h) ×-â ç ۉã öñÛ Ú-Ô¶ìîÞ%ßêí ÕãÝ ïT‚ Ú-×ñâ +ë Û Ú ã ã ã
0
â
â ã
ã

c0 (h) =
Ð×
ò‚Ý-Ö Ö
ã
lh1 (x)
î¬î-Ó#ë
1+
P
1
.
(1/|µ(2πri/h)|2 )
ã
= χQ (x) − h
n
X
δ(x − hk) =
hk∈Q
ë
û—Õ Ö Þ%Ú‰Û ÞÑ¿õèÂßÌÞå÷Ó×-ÞÛÜÕdÙ× ì#ë2åÌÛ
íÒÓã ԗÕλ×-MÖ ý ã ã þ â
â
M ≥m 
λM (x) = χQ (x) −
ãzã
Ó ÓÖ Þ Þ×
ã
r6=0
lh1 (x)
€ÂÖ
c0 (h)
X
X
hk∈Q
M
λM
x − hk
h
üÍò—ë zÔ ÝñÓ¬ë+ÛÜÕÝ
ã
äã
as δ(x − s),
M
,
ç
߂ë ñ
Ú ÷ ÕÑ¿Ö™Ò Ú-ÞÑ¿Ö
ã ã
ã
(λM (x), xα ) = 0, |α| ≤ M.
s∈Z n ,|s|≤S
Ö
f k = k[1 − cƒB„ … (h)]hn
klh1 − lhQ,opt |W
2
1/µ
=
X
X
hk∈Q
1/µ
f k=
δ(x − hk)|W
2
f 1/µ k),
(1/|µ(2πri/h)|2)(1 + o(1)) = o(klhQ,ƒB„ … |W
2
Ú ï ï {l1 } ìî‰Õì Ú-Ô¶ì~Û
ÔzÓÖnÙ%Ú-×
Ú-ÓÒ ÔzåÓq×-Ù%Ú-ÓÖÛÜÕÝ-Þ×-õ Ù×-Ô—Õ ë2×-îÛ‰Ú ÕÝ-Þ×-Ô­Ú-Ý-àNèÂßÌÞå÷Ó×-ÞÛÜÕ×-î‡Ù×
ã Þ h×-Ô­Ú õ ï‚ ã Ú-׬î Þ׬Ӈî¬×-ç
ö ã Ö Ô—ÕßÌÒÛ ï
ã
ã
ú
ízâã­ä
ã
ãzâ
ã
ã
r6=0
‡‰ˆ‹ŠˆsŒ hnoh!&hrTc{
\7†
#sŽ#Â$@S‘Z’“y”5•–!—–4˜s™š!™x›!œT™’“VœZ–!—s’“š!™
‘7Ÿ™š#§T“š!™ œi¨s“©‘Z’§T™šž|¨|ª
e ⊂⊂ C
B
e ›sBœsžs™4Ÿž i#’¡7ž|˜£¢¤‘7—¡7¥ž¦ f (x) = P fk e2πikx
B
k
∀a ∀f
e = kf (x)|Bk,
e
kf (x + a)|Bk
e ≤ kf |Bk.
e
kf − f0 |Bk
«™¬
Ÿ–1›s™’§TŸ™š–!“§s”7—s™’“y”­¢¤‘7—¡7¥žs™4—–&§T™š5›s™¬®œT¯y—s™’“¦
lh1 (x) = χQ (x) − hn
X
{lh1 (x)}h→0 °
δ(x − kh),
kh∈Q
–<’ž|ª±›“™4“yž i#’¡7ž²™4›“yž|ª±–&§s”7—–5—–!Ÿy›!œT™’“VœZ–!—s’“š!™
ª e
B °@³
š´¤›s™X§s—|¨s“’X¨£•™X§T#wµ¶#’“y¡Z™&·
“™
#’§sž£š&ªw#’“™x›s™’§TŸ—s#¬™y‘Z’§T™šž|¨
kf kBe = max{|f0 |, kf − f0 kBe },
’“y–!—s™šž“’X¨£™4›“yž|ª±–&§s”7—´9ª
¸ ,g$i¹#$2-'zbº ×-î ÓÖyÔzÞÛêÒ° ÛÜÕ2Û%øÒ%Ú-×
lh1 (x)
Ú × ÖnÑ —Ô Õ ëß ÚÜø+Ò%Ú-׬î‰Õ×êâ ù
ã â»rã Û
å‡åÌÛ
å ã 1ã
Û
î-Þ×-Ö
{lh (x)} â
Ù Ó
€òÂÙ î-× ×ñß-ԗÕ×-î-Ó%ì
0 â h → 0.
ãzÞâ Ó Ôz×-Ù ì%ù ÞÞÑódÙklh1×-|BÔ­eÚ ∗kÛ
→
ãzâ í ∗ þ ï
ÞÔ­Ú-î å#-× ÖnÙÛ
å%Ú-Þ×Xü Ce∗ ⊂⊂
e
B
ã Þ׬ã × Û
â ÞÓÒ ã Þ׬î ∗â â
e ,
C
ãzâ
ízâ
ã
e ∗ k ≤ kχQ |C
e ∗ k + hn
klh1 |C
X
kh∈Q
e ∗ k = 2,
kδ(x − kh)|C
Ú-×¬Ô Ö õÔ­Ú-î-× {l1 (x)} å#×-ÖnÙÛ
å%Ú-Þ׬î Be∗ .
º ã ã ë2Ù׉Õ×êù ÓhÖ¿øÌÒ%Ú-h∈H
×Þ× ÖÛ kl1 |Be ∗k Þ Ô­Ú ÖnÓ%Ú-Ô¶ì‡å Þ%ßêÕà ï Ð û­Ú-×-ÖyԗÕßÌÒÛ Ô­ß#ö Ô­Ú-î
ß Ú Ù×zë
â
ã
Ù×-Ô—Õ ë2×-îÛ‰Ú ÕÝ-Þ×-Ô­Ú-Ý h →â 0 øÌëÕì‡h å#×
Ú-× ×-ã õ l1âã (x) → l(x) Ó kl|Be∗ k = a > 0 ï ã»× ë+Ûñã Ô­ß#ö ã Ô­Ú-î
ß Ú ú
Ó èÂßÌã Þå÷Ó%ì ã ϕ ∈ Be øÞ j òêÛ
î-ÓÔ¶ì%öñۉì ×
Ú hâ ø2ëÕìh å#×
Ú-× ×-õ hl, ϕi ≥ a/2 ïð ÞÛêÒÓ%ÚÜøÙ í ÓXë2×-ԭډã ۉÚ-×-ÒÞã ×
ã ï ~Â×
ç-׉ÕÝ ÓÌó j hl1 , ϕi ≥ a/4
Ódî-Ô—Õ ë2Ô­Ú-â î-ÓÓwÞ Ù Ñ¿î-Þ×-Ô­Ú-ÓwèÂßÌÞå÷â ÓÓ ϕ
e
ϕ
∈
B
⊂
C
ä
ã
ã âãzâ
h
j
j
hlh1 j , ϕi
=
Z
Q
º
ϕ(x)dx − hnj
×
Ú-Óî-× ÒÓ ë2×-åÌۉò‚Ñ¿îÛ ÚÜøÒ%Ú-×
⠀ò‚î Ô­âÚ-ã Þ×øã Ò%Ú-׬×-Ù%Ú-ÓÖÛÜã ÕÝ-ÞÑ¿õklèÂh1ßÌ|BÞeå∗÷kÓ→×-Þ0ÛÜÕ ï
ã
X
ϕ(khj ) → 0.
khj ∈Q
ÓÖ Ú î-Ó#ë
lhQ,opt (x) ≡ lh0 (x)
ãzã
X
lh0 (x) = χQ (x) − c0 (h)hn
δ(x − hk).
hk∈Q
» Ù Ý Ù×-åÌۉù Ö¿ønÒ%Ú-×
Ù Ó
Õì™û­Ú-× ×~×-Ù%Ú-ÓÖÛÜÕÝ-ÞÑ¿õáèÂßÌÞå÷Ó×-ÞÛÜÕ
1 â h → 0. ¼
òêÛ
ã ÙÓ ãzâ Öáîî-Ó#ë ã l0 (x) =c0(1(h)− →
Â
€
Ö Ö í
c0 )χQ (x) + c0 lh1 (x).
äã
ã h
ãzã
e ∗ k = sup |h(1 − c0 )χQ + c0 l1 , f i/kf |B
e ∗k| ≥
klh0 |B
h
e
f ∈B
≥
sup
e 0 =0
f (x)∈B,f
= |c0 |
sup
e 0 =0
f (x)∈B,f
e ∗ k| =
|h(1 − c0 )χQ + c0 lh1 , f i/kf |B
e ∗ k] = |c0 | sup |hlh1 , f − f0 i|/kf − f0 |B
e∗k =
[|hlh1 , f i|/kf |B
e
f (x)∈B
lh0 (x)
^(_`baTcdaeZ_#f
gihjkclbm&a+nohpq<rshj;gZtTu!hZmt7`brshjkvwc`bn9t!pThxrshyl7`zc<f&m
`hrTf&m&{|_ fm
Wp
\<½
üÍò—ë zÔ Ý Ù Ó Ö Þ%ì ™
Ö Ú Ú-Ý ß-ԗÕ×-î-Ó Ú × ÖnÑ_þ
ã â ã ã â ã ã
ã ã âã
e∗k
kf |B
|hlh1 , f i|
·
≥
= |c0 | sup
e ∗ k kf − f0 |B
e∗k
e kf |B
f (x)∈B
e ∗ k.
≥ |c0 | · klh1 (x)|B
ð ÞÛêÒÓ%ÚÜø |c0| ≤ 1 ï ¼ ÛÜÕ ãzã ø |1 − c0| · kχQ|Be ∗k = klh0,Q − c0lh1 |Be∗ k ≤ 2klh1 |Be∗ k → 0 Ù â Ó h → 0 ï ~Â×
e ∗ k = const 6= 0 ï ºÂ×
Ú-×-Örß |1 − c0 | ≤ C · kl1 |B
e∗k → 0 Ù â Ó h → 0 ï
kχQ |B
h
»rÛ
åÓ֙×-ç ۉò‚×-Ö¿ø
â
e ∗ k ≥ |c0 | · klh1 |B
e ∗ k = (1 + o(1))klh1 |B
e∗k
klh0 |B
Ó
e ∗ k/kl1 |B
e ∗ k ≥ 1 + o(1) Ù â Ó h → 0.
1 ≥ klh0 |B
h
‚ Ú-×_ÓÂ×
ò‚ÞÛêÒÛ Ú„Û
ÔzÓÖnÙ%Ú-×
Ú-ÓÒ Ôzå%ßÌàá×-Ù%Ú-ÓÖÛÜÕÝ-Þ×-Ô­Ú-Ý_Ù×-Ô—Õ ë2×-îÛ‰Ú ÕÝ-Þ×-Ô­Ú-Ó!èÂßÌÞå÷Ó×-ÞÛÜÕ×-î {l1 }h→0.
ã
ã
h
ºß-Ô­Ú-Ý¬Ú Ù ã Ý
»× ë+Û ã
kf
k
=
max{|f
|,
kf
−
f
k
}.
e
B
0
0 B
ã ãzâ
í
|h(1 − c0 )χQ + c0 lh1 , f i|
=
e
max{|f0 |, kf − f0 |Bk}
e ∗ k = sup
klh0 |B
f
|(1 − c0 )f0 + c0 hlh1 ,
= sup
f − f0
e
ikf − f0 |Bk|
e
kf − f0 |Bk
e
max{|f0 |, kf − f0 |Bk}
f
= sup[|1 − c0 | + |c0 | · |hlh1 ,
f
=
f − f0
e∗k ≥
i|] = |1 − c0 | + |c0 | · klh1 |B
e
kf − f0 |Bk
e ∗ k.
≥ |1 − c0 | + |c0 | · klh0 |B
-Ú Ôzà.ë+ÛdÔ—Õ ëß Ú (1 − |c |)kl1 |Be ∗k ≥ |1 − c | ≥ 1 − |c |, Ò%Ú-×dî-×
ò‚Ön×êù Þ× Ú-׉ÕÝ-å#×dÙ Ó |c | = 1.
â 0
ð ÞÛêÒÓ%ÚÜø klh0ã |Be∗ã k = klh1 |Be0∗k, Ò%hÚ-×¬Ó‡Ú âã ç-×-îÛÜÕ0×-ÔzÝ!ë2×-åÌۉòêۉ0Ú-Ý ï
ÐÑ¿ÒÓԗÕÓÖ Ú-×-ÒÞ å#×-ûzèfèfÓ÷Ó Þ%Ú-Ñ c (h) ×-Ù%Ú-ÓÖÛÜÕÝ-ÞÑóå%ßÌçۉÚ
ß ÞÑóèf× ÖrßêÕ¬Þۍ٠-× Ô­Ú Û
Þ
0
ãzã
ã
â
â
â â ú
Ô­Ú-îÛÜó Be = W
fpm Ô p ∈ (1, ∞) Ó m > n/p ï Ð×
ò‚Ý-Ö ã Ö Þ× â ÖnÑ î-Ó#ë+Û


f m (Q)k = 
kf |W
p
ÐéԗÕßÌÒÛ ÷ ÕÑódÒ Ú-ÞÑó
ã ã
ã
m
Z
Q
dx
X
k
1/p
fk (1 + |2πk|2k )m/2 e2πikx |p 
.
û­Ú-ÓÞ× nÖ Ñ Ôz×-î-ÙÛêë+Û
à_Ú¬Ôfç-×‰Õ Ú Ûêë2Ó÷Ó×-ÞÞÑ¿ÖnÓ
â
ãzã â

f m (Q)k = 
kf |W
p
Z
Q
1/p
dx|(1 − 4)m/2 f (x)|p 
,
ßÌÙ×
Ú ‰ç Õìî ÓÖnÓÔ¶ìgøcÞÛ
Ù Ó Ö øÂî Û
ç-×
ډÛÜó¿¾ ï ÿ ï±À ×-õÞ%ù¬ß -× îÛ ï f Û
å|Ó Û ÞÝ øÂÙ׉Õ2Û Û Ö
â1ã Ôf÷ Õä Ñ¿ÖnÓ
â ÓÖ zã â á
â
â
â ä ã
í ã
Ñ
Ö
Ù
Ó
p
T
ï
Á
0
h=
N
ã
ãzã â p = p − 1
N
f m (Q))∗ k = k1 − c0 (h) + c0 (h)l1 |(W
f m (Q))∗ k =
klh0 |(W
p
p
‡‰ˆ‹ŠˆsŒ hnoh!&hrTc{
\!Â


=



=
Z
=
Z
dx|1 − c0 − c0 (h)
β6=0
Q
Z
β6=0


 m
±
h 
ºÂ×-åÌۉù
X e2πiβx/h
|2πβ|
Z
Q
2πiβx/h
e
0
|p
2
m/2
+ |2πβ| )
|2πβk
X
+ hm
m
2πiβx/h
(h2
X

e2πiβx/h
≶
Z
0
|p 

=
1/p
e
0
|p 
2
m/2
+ |2πβ| )
β6=0
!1/p
/h|2 )m/2
k=1
β6=0
β6=0
(h2
n
P
(1 +
Q
Q
+hm [1 − c0 (h)]
e2πβx/h
dx|1 − c0 (h) − hm c0 (h)
dx|1 − c0 (h) − hm
X
X
1/p
=
[(h2 + |2πβ|2 )m/2 − |2πβ|m ]
+
|2πβ|m (h2 + |2πβ|2 )m/2
dx|1 − c0 (h) − hm
X e2πiβx/h
β6=0
Q
|2πβ|m
1/p0
|p 
0
±
p0 1/p0
1
Z
2πiβx/h
X
m
e
m
2
2
2 2 −1 
dx +
dt
h
(th
+
|2πβ|
)

m 2
2 m/2
2
β6=0 |2πβ| (h + |2πβ| )
0

+hm · |1 − c0 (h)| 
Z
dx|
Q
X
β6=0
e2πiβx/h
(h2 + |2πβ|2 )
1/p0 
0

|p   ≡ I + II.
m/2
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î-Þzã Ñ ã ú
1
2
í â
ã
ú ãzâ
âã
âí ã
â
ã
m
Ù Ó γ = β/h 6= 0 Ó
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â
(t + |2πγ|2 ) 2 −1
ϕ1 (γ) =
m ,
(1 + |2πγ|2 ) 2
ϕ2 (γ) =
Û
î-ÞÑ %Þ ßêÕàNÙ Óq×-ԭډÛÜÕÝ-ÞÑó
â #ì ë2×-î ã ô!ß Ý îâ Ù ×-Ô­Ú Û
ÞÔ­Ú-î
â
â ã â â
ã
A1 :
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X
X
fγ e2πiγx →
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|2πγ|m
m
(1 + |2πγ|2 ) 2
γ ∈ Zn
Ú
L p0 , ï ã ï
X
X
ø*ìî‰Õìà_Ú-Ô¶ì Û
î-Þ×-Ö Þ×Ù×
ãzâ
×-٠ۉÚ-× Ñ â
ãzâ â
II ≤
ë
í ã
bm (x) =
h
2π
P e2πiβx ï
m
β6=0 |β|
m h
Ó
fγ ϕ1 (γ)e2πiγx ,
fγ ϕ2 (γ)e2πiγx ,
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ßÌà_öÓ î Ù -× Ô­Ú Û
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ÞÓÒ ÞÑ Û
î-Þ×-Ö Þ× Ù×
ãzâ
î-Ñ ã ۉù ÞÓ îã åîÛêë â ۉÚ-â Þ ÑóÔzå#×-ç-åÌÛÜpó‡×-÷ ÞÓîÛ Ú-Ô¶ìdډízÛ
â å  ã â
â ã ã
â
ã
ã
0
h
i

m 2
h kA1 k + (1 − c0 (h))kA2 k 
2
Z
Q
1/p0
dx|bm (x/h)|p 
0
h
Ó t ï º ×-û­Ú-×-Örß
,
t
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bm (Q)
Z
Q
ý
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ãzâ
ã
ãzízâ
1/p0
dx|bm (x/h)|p 
0

=
Z


Z

Q/h
0
Q
1/p0
II = o(hm ) ï

=
X e2πiβx/h
dy|1 − c0 (h) − hm
Q/h

=
€Ú‰Û
ågø
ºÂ×-ԗÕ
ã
Z
dx|1 − c0 (h) −
Q

òêÛ
Ö Þ Ñ
ã
f m (Q))∗ k = 
klh0 |(W
p
1 − c0 (h) =
h
2π
f m (Q))∗ k =
klh0 |(W
p
Z
|2πβ|m
h
2π
m
dx|1 − c0 (h) −
R0 (h) + o(hm )
h
2π


Z
Q
1/p0
|p 
0
=
=
1/p0
1/p0
bm (x)|p 
0
m
m
ã
X e2πiβy 0 
|p 
m
|2πβ|
β6=0
Q
m
0
dy|bm (y)|p 
≡ kbm |Lp (Q)k
β6=0
Z
1/p0
»×-ÒÞ×ډÛ
ådù
dx|1 − c0 (h) − hm
Q

Z

=  hn
dy|bm (y)|p 
Þ êò Û
î-ÓÔzÓ%Ú ×
Ú h ï r» Û
åÓ֙×-ç ‰Û ò‚×-Ö¿ø
ã
â
\7Ã
=
.
1/p0
h
0
bm (x)|p 
2π
+ o(hm ).
Ù׉ÕßÌÒÛ Ö
ã
1/p0
dx|R0 (h) − bm (x)|p 
0
+ o(hm ).
Ò -î Ó#ë2Þ×øÖnÓÞÓÖnÓ%òêÛ
÷Ó%ì (W
f m )∗ ý Þ× â ÖnÑ l0 Ù× c0 ûzåî-ÓîÛÜÕ ã Þ%Ú-ÞÛÖnÓÞÓÖnÓ%òêÛ
÷ÓÓÙ× R0 ÓÞ%Ú ã—ú
ã Z
p
h
ÛÜÕ2Û dx|R − b (x)|p . €òñÞ òêÛ
î-ÓÔzÓÖn×-Ô­Ú-Ó b (x) ×
Ú h Ô—Õ ëß ÚdÞ òêÛ
î-ÓÔzÓÖn×-Ô­Ú-Ýd×
Ú h Ó R ,
0
m
m
0
ízâ
ã
ã ã ã
Q
Z
û­Ú-ÓÖáÙ×-Ô­Ú-×êìÞÞÑ¿ÖáÙ× h ÒÓԗÕ×-Ö R
R = arg min dx|R − b (x)|p . Ä

0
0
0
m
R
0
Û
Q
f m (Q))∗ k
klh0 |(W
p
lh0 (x)
=
h
2π
m
kR0 − bm (x)|Lp0 k · (1 + o(hm )),
m
X
h
m
= χQ (x) − 1 −
R0 + o(h ) hn
δ(x − hk).
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hk∈Q
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