Buxoro davlat universiteti qo’lyozma huquqida udk abdullayev Behzod Rajabovich

Teorema 3.1.1. (3.1) ko’rinishdagi kubatur formulaning xatolik

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

_{ } N
 _ ^{C Y}
_ (  ) _



1
_ ² _ 2


_{ }   n ,k  _  k , _{ }

^{  } 
  1  _
. (3.5)

_ k 1 1
^k m  ^{k  n  2} m _

 
 

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 ,
^  ^{, (3.6)}

k 1 k 1 1

^{bu yerda} Y_{k ,}
^ 
^{- sferik garmonikalar} k ^{- tartibli ko’rinishdagi.}

Xatolik funksionali (3.4) va (3.6) dan foytalanib quyidagini topamiz

 _{N } _ , f _ _     _s
N
( )  _ C _
 1




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

| 

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

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

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

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

_S k 1 1
 1
k 1 1

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

^ _{ } ^{a Y} ^  ^{d } _{ } ^a _ ^{C  }
_   ^{  } _, Y
^  ^{ }

k 1 1
k , _ k ,
S
k 1 1


k ,  k ,
 1

_ ^  ^{n ,k}  _N

^{   a}k , ^{ C}  ^Yk ,
_ ^{ } _ . (3.7)

k 1 1  1

Agar (3.7) 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, (3.2) ga asosan quyidagini olamiz

_ ^  ^{n ,k}  ^m


k ,

m _ ^N

_ m _m

^  
^{a k 2}  ^{k  n  2}  ^2
 _^{ C}  ^Yk ,
_  ^  _
^{k 2}  ^{k  n  2}  ^{2 }

k 1 1
  1 

_{ } N
¹ _{ } C Y
_  ^  _



1


_ 2 _{ 2}


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

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

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

 
 

_{ } N
_{ } C Y
_  ^  _



1


_ 2 _{ 2}


_{ } ^  ^{n ,k}   ^{ k ,}  _

^{ f L}m  ^S 
^{ } 
  1  _
. (3.8)

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

 
 

(3.8) dan quyidagi kelib chiqadi

_{ } N
_{ } C Y
_  ^  _



1


_ 2 _{ 2}


_{ } ^  ^{n ,k}   ^{ k ,}  _

^{  } 
  1  _
. (3.9)

_ k 1 1
^k m  ^{k  n  2} m _

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

U _ _  _{ }
^bk , ^Yk ,
_ _ , (3.10)

bunda
k 1 1

N
^{ C}  ^Yk ,
_ (  ) _

^bk ,
 ^{ 1} . (3.11)
^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
(3.11) dan kelib chiqadiki (3.10) ning koeffisiyetlari U _ _  L^m _ S _
Bu funksiya uchun (3.8) kubatur formula xatoligini hisoblasak quyidagi tenklikni olamiz:

N
_{N } ^  ^{n ,k}   ^C  ^Yk ,
_  ^  _

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

s 

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

N


_ ^  ^{n ,k}   ^C  ^Yk ,
_  ^  _


^  
 1 _{  }
( ), Y

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

N
_ ^  ^{n ,k}   ^C  ^Yk ,
_  ^  _


N
 ^ 

^  
 1 _{  Y
} _ d 
 _ C Y _
^ ^

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

 1
 k ,





_   n ,k  _


N
^{ C}  ^Yk ,
2


_  ^  _^

^  
  1  _
^{U L}m  ^S 
2
. (3.12)

k 1 1
^k m  ^{k  n  2} m 2

(3.9) va (3.11) dan quyidagi kelib chiqadi.

2

N
_Lm  _{ S }
 U L^m _ S _ ,

2
bu yerda
U _ _
funksiya (3.1) ko’rinishdagi kubatur formula uchun ekstremal

2
funksiya bo’ladi. Bu teoremaga asosan (3.1) ko’rinishdagi kubatur formula xatolik

funksionali uchun
L^m _ S _
fazoda quyidagi baho o’rinli bo’ladi:

 _
^ _ ^  ^{n ,k}  


N
^{ C}  ^Yk ,


_ ^  _

1

_ 2 _{ 2}
^{ } _ 
1
^  ^{n ,k}  _{ 2}

 _ _ , f
_ _   ^
  1  ^ _
a ² k ^m _ k  n  2 _m .

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