Buxoro davlat universiteti qo’lyozma huquqida udk abdullayev Behzod Rajabovich

Teoreama 3.1.2. (3.12) tenglik shuni tasdiqlaydiki,

N
_   n ,k  


^Ñ  ^Yk ,
_ (  ) _

^U ^  ^ _{ }
 1 _Y
^ 

1
k 1
^k m  ^{n  k  2} m k ,

U _ _
funksiya (3.1) ko’rinishdagi kubatur formula uchun ekstremal funksiya

bo’ladi va
U _ _  L^m _ S _ , bu yerda
^Yk ,
^{( ) - sferik garmonikalar k - tartibli}

2
ko’rinishdagi va garmonikalar soni.
 (n, k ) - k – tartibli chiziqli bog’liq bo’lmagan sferik

Endi n+1=3 hol uchun quyidagi formulani qarab chiqamiz.

2  
 
0 0
^f ^{ ,}  ^{sin }


d  d 


N
 _ C _ f
 1


( ^{(  )} , ^{(  )} ) . (3.13)

Bubda


^Yk ,
( ) 
( 2 k  1)( k  | |)!_{e  i}
2 ( k  | |)!
^ P ^{| |} (cos  )
ko’rinishda bo’ladi, bu yerda

k

k
P ^{| |} (cos ) ^{- Lejandr funksiyasi. (3.5) ga asosan quyidagicha bo’ladi}


 k

_{ } . (3.14)_{m m}
2

k ( k  1)

k 1   k

2
Endi
L^{(m )} ( S )
fazoda xatolik funksionali normasini baholaymiz.

3.2.

_L( m ) _{ S }

Sobolev fazosida vaznli kubatur formulaning xatolik

2
funksionali normasini hisoblash va ekstremal funksiyani aniqlash.
Bu bo’limda biz vaznli kubatur formulani qarab chuqamiz. Vaznli kubatur formula quyidagi ko’rinishda bo’ladi:

_ p ( ) f ( ) d
S
N


 _ C f ( ^{(  )} )
 1
(3.15)

2
L^m ( S ) - fazoda sfera sirtida, by yerda S  n  o’lchovli birlik sfera,

  (₁ , ₂ ,..., _n ),
  1 ,
p ( )
- S sferada integrallanuvchi funksiya,

ya’ni
_ p ( )d  
S

N
va _ C _ 
 1
n


2 ²



n
Г ( ) 2


^pˆ 0 ,0

bunda

^pˆ k ,

 _ P ( )Y_{k ,}

S

( )d  ,

bu yerda Y_{k ,}
^ 

• 
  ^{sferik garmonikalar} k ^{- tartibli ko’rinishdagi 1    ( n , k ) .}
  

 ( n , k )  ( 2 k  n  2) ^{( k  n  3)!}
( k  2)!k !

• 
  k – tartibli chiziqli bog’liq bo’lmagan sferik
  

garmonikalar soni. Y_{k ,} ( )
funksiyani S sferada orthogonal deb hisoblaymiz.

(3.15) ko’rinishdagi vaznli kubatur formulaning xatolik funksionali quyidagicha bo’ladi:

( ) 
p ( )
N
( )  _ C
 (
  ^{(  )} ) , (3.16)

N S 
 1

bu yerda
 ( )

• 
  Dirakning delta funksiyasi, C _ va
  

_ (  )

• 
  (3.15) ko’rinishdagi
  

vaznli kubatur formulaning koeffisiyetlari va tugun nuqtalari.
Teorema 3.2.1. (3.15) ko’rinishdagi vaznli kubatur formulaning xatolik

2
funksionalini normasi
L^m _ S _
fazoda quyidagiga teng

_{ } N
1
_ ² _ 2

^ _   n ,k  _ ^pˆ k . ^{  C}  ^Yk ,
^ _ 

_{ } ^   1  ^ _,
^{ } m ^
_ ^{k 1 1 k} m  ^{k  n  2}  _
 
 

bunda
^pˆ k ,
 _ p _ _Y_{k ,}
S
_ _ d .

Ushbu teoremani to’liq isbotini keltiramiz.
Isboti. Ma’lumki agar funksiymiz


^f ^  ^{ L}m  ^S 
ga tegishli bo’lsa, unda

2
quyidagi qator absolyut tekis yaqinlashuvchi bo’ladi.

2
bu yerda yetarli.
Y_{k } _

f _ _  _ Y_{k } _ ,
k  0

• 
  k - tartibli sferik garmonikalar, bunda 2m  n
  

shart bajarilishi

Shunday qilib funksiyani
f _ _  L^m
sferik garmonikalar bo’yicha absolyut

tekis yaqinlashuvchi qatorga yoysak

_{ } ^  ^{n ,k} 

f _ _  _ Y_{k } _  _{ }
^ak , ^Yk ,
^  ^{, (3.17)}

k  0 k  0 1

bu yerda Y_{k ,}
^ 

• 
  ^{sferik garmonikalar k - tartibli ko’rinishdagi;}
  

^ak ,
 _ Y_{k ,}
S
_ _ f _ _ d ;
 _ n, k _

• 
  k – tartibli chiziqli bog’liq bo’lmagan sferik
  

garmonikalar soni:

^  ^{n , k}  ^{  k  n  3 !}  ^{n  2 k  2}  ^{.
k !} ^{n  2} ^!

(3.15) ni chap qismiga (3.17) keltirib qo’ysak quyidagini topamiz :

 _ _ , f _ _ 
p _ _ 
N
( )  _ C 
_   ^{  } _ , _ Y
_ _ 

N S 
 1
k


k 1

 p ( )


_S ( ), _ Y_k
k 1
N
_ _    _ C _
 1


k
_   ^{(  )} _, _ Y
k 1
_ _ 

_ ^  ^{n ,k}  _{N } ^  ^{n ,k} 

^ _ ^p ^ _{ }
^ak , ^Yk ,
^  ^{d   } _ ^C
_ _
  ^{  } _, _{ }
^ak , ^Yk ,
^  ^

_S k 1 1
 1
k 1 1

_ ^  ^{n ,k}  _ ^  ^{n .k}  _N

k ,
^  
k 1 1
a_{k ,} p _ _Y_{k ,
} _ d
^  
k 1 1
^ak ,
^{ C} 
 1
  _
  ^{  } _,Y
_ _ 

_ ^  ^{n ,k}  _{ N }

^  
a p^ˆ

• 
  _ C Y
  

_  ^  _
. (3.18)

k 1 1
k , _ k ,


 1
 k , _



Agar (3.18) ni o’ng tomonidagi

a_{k ,} ni
m _m


^{k 2}  ^{k  n  2}  ²

ga ko’paytirib,

yig’indini shu ko’paytuvchiga bo’lsak va Koshi tengsizligini qo’llasak, (3.15) ga asosan quyidagini olamiz

_ ^  ^{n ,k}  ^m



k ,

_m ^pˆ k ,


N

• 
  ^{ C}  ^Yk ,
  

_  ^  _

^  
^{a k 2}  ^{k  n  2}  ²
_ 1 _
m _m

k 1 1
k ² _ k  n  2 _ ²

 _
¹ _ p^ˆ
N

• 
  _ C Y
  

_  ^  _
1


_ 2 _{ 2}


_  ^{ ( n ,k )}
m ^{ 2}
_  ^{ ( n ,k )  k ,}
 k , _{ }

^{ } 
^a 2 ^k m  ^{k  n  2}  _
^{ } 
  1  _{ }

k ,
 ^{k 1 1}
 _ ^{k 1 1
k} m  ^{k  n  2} m _

 
 

 _
 p^ˆ
N

• 
  _ C Y
  

_  ^  _
1


_ 2 _{ 2}


_{ } ^  ^{n ,k}   ^{k ,  k ,}  _

^{ f L}m  ^S 
^{ } 
  1  _
. (3.19)

2
_ k 1 1
^k m  ^{k  n  2} m _

 
 
(3.19) dan quyidagi kelib chiqadi

 _
 p^ˆ
N

• 
  _ C Y
  

_  ^  _
1


_ 2 _{ 2}


_{ } ^  ^{n ,k}   ^{k ,  k ,}  _

^{  } 
  1  _
. (3.20)

_ k 1 1
^k m  ^{k  n  2} m _

 
 
Quyidagi funksiyani qarab chiqamiz
_ ^  ^{n ,k} 

U _ _  _{ }
^bk , ^Yk ,
_ _ , (3.21)

bu yerda
k 1 1

^pˆ k ,
N

• 
  ^{ C}  ^Yk ,
  

_ (  ) _

^bk ,
^^{ 1} . (3.22)
^k m  ^{n  k  2} m

Sferik funksiyalar uchun quyidagi baho o’rinli [1]

max
Y_{k } _
 C _ n _ k
n

2
 m  1 2
f _ _
^Lm  ^S  ^,

2
(3.22) dan kelib chiqadiki koeffisiyetlari U _ _  L^m _ S _
tegishli bo’ladi

Bu funksiya uchun (3.22) kubatur formula xatoligini hisoblasak quyidagi tenklikni olamiz:

_{N } ^  ^{n ,k}  ^pˆ k ,
N

• 
  ^{ C}  ^Yk ,
  

_  ^  _

  p _ _ 
( )  _ C  _   ^{ } _, _{ }
 1
Y _ _  

S 
 1
k 1 1
^k m  ^{k  n  2} m k ,

_ ^  ^{n ,k}  ^pˆ k ,
N

• 
  ^{ C}  ^Yk ,
  

_  ^  _


^  
 1 _{ }
p _ _ 
( ), Y


k 1 1
^k m  ^{k  n  2} m ^
S k ,

_ ^  ^{n ,k}  ^pˆ k ,
N

• 
  ^{ C}  ^Yk ,
  

_  ^  _

^  
 1 _{ }
p _ _Y


k 1 1
^k m  ^{k  n  2} m _ ^ k ,

S

_   n ,k  _
^pˆ k ,

^   ^{
 U L}m  ^S 
2
. (3.23)

2
k 1 1
(3.20) va (3.23) dan quyidagi kelib chiqadi

2

N
_Lm  _{ S }
 U L^m _ S _ ,

2
Bu yerda U _ _
funksiya (3.15) vaznli kubatur formula uchun ekstremal funksiya

bo’ladi, ya’ni
xatolik funksionali uchun
U _ _

• 
  Ris funksiyasi bo’ladi,
  

shunday qilib quyidagi teorema isbotlandi.