ChM_teory2

7.2. ɉɪɨ ɨɛɱɢɫɥɸɜɚɥɶɧɭ ɩɨɯɢɛɤɭ ɱɢɫɟɥɶɧɨɝɨ ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ [ɋȽ, 186-188], [ȻɀɄ, 80-81]

ɇɟɯɚɣ ɡɧɚɱɟɧɧɹ ɮɭɧɤɰɿʀ ɨɛɱɢɫɥɟɧɧɿ ɡ ɞɟɹɤɨɸ ɩɨɯɢɛɤɨɸ. ɉɨɫɬɚɽ

ɩɢɬɚɧɧɹ ɩɪɨ ɜɩɥɢɜ ɰɢɯ ɩɨɯɢɛɨɤ ɧɚ ɡɧɚɱɟɧɧɹ ɩɨɯɿɞɧɢɯ ɨɛɱɢɫɥɟɧɢɯ ɡɚ ɮɨɪɦɭɥɚɦɢ ɱɢɫɟɥɶɧɨɝɨ ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ.

ɉɟɪɟɞ ɰɢɦ ɡɪɨɛɢɦɨ ɡɚɭɜɚɠɟɧɧɹ ɩɪɨ ɜɩɥɢɜ ɡɛɭɪɟɧɧɹ ɮɭɧɤɰɿʀ ɧɚ

ɡɧɚɱɟɧɧɹ ɡɜɢɱɚɣɧɢɯ ɩɨɯɿɞɧɢɯ.

ɇɟɯɚɣ f x C1[a,b] ɿ ʀʀ ɡɛɭɪɟɧɧɹ ɦɚɽ ɜɢɝɥɹɞ:

f

x

f

x

1

sin Zx .

~

n

~

1

~

ɉɪɢ n o f ɦɚɽɦɨ

f x

f x

o

0 ɡɜɿɞɫɢ

f

x f x . Ɍɚɤɢɦ

C[a,b]

n

Z

nof

~

2

ɱɢɧɨɦ ɰɟ ɦɚɥɿ ɡɛɭɪɟɧɧɹ. Ɇɚɽɦɨ

f c x

f c

x

cos

x . ɇɟɯɚɣ

n

, ɬɨɞɿ

f c

f c

n

n of .

~

C[a,b]

nof

n

ɐɟɣ ɩɪɢɤɥɚɞ ɿɥɸɫɬɪɭɽ ɧɟɫɬɿɣɤɿɫɬɶ

ɨɩɟɪɚɬɨɪɚ

ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ. ȯ

ɫɩɨɞɿɜɚɧɧɹ, ɳɨ ɰɹ ɧɟɫɬɿɣɤɿɫɬɶ ɦɚɽ ɦɿɫɰɟ ɿ ɞɥɹ ɱɢɫɟɥɶɧɨɝɨ ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ.

	~
ɇɟɯɚɣ		fi i , fi f xi , i 0,n				. Ɋɨɡɝɥɹɧɟɦɨ ɜɩɥɢɜ
	fi				i
ɩɨɯɢɛɨɤ i ɧɚ ɤɨɧɤɪɟɬɧɢɯ ɮɨɪɦɭɥɚɯ ɱɢɫɟɥɶɧɨɝɨ ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ.

ɉɪɢɤɥɚɞ 1 Ɉɰɿɧɿɦɨ ɜɩɥɢɜ ɡɛɭɪɟɧɶ ɧɚ ɩɨɯɢɛɤɭ ɨɛɱɢɫɥɟɧɧɹ ɩɟɪɲɨʀ ɩɨɯɿɞɧɨʀ

n 1,k

1.

fic

fic

Gi

Gi 1 ,

~

~

fi

fi 1

fi

fi 1

h

h

fic

fi

fi 1

d fic

fi

Gi

Gi

h

(1)

fi 1

1 d M2h 2G of ,

~

~

ho0

h

h

h

2

h

Ɍɚɤɢɦ ɱɢɧɨɦ, ɹɤ ɿ ɞɥɹ ɚɧɚɥɿɬɢɱɧɨɝɨ ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ, ɦɚɽɦɨ

ɧɟɤɨɪɟɤɬɧɿɫɬɶ: ɩɪɢ ɦɚɥɢɯ ɡɛɭɪɟɧɧɹɯ

i

ɦɨɠɭɬɶ ɛɭɬɢ ɹɤ ɡɚɜɝɨɞɧɨ ɜɟɥɢɤɿ

ɩɨɯɢɛɤɢ, ɹɤɳɨ

o f ɩɪɢ h o 0.

h

Ɇɿɧɿɦɿɡɭɽɦɨ ɜɩɥɢɜ ɰɢɯ ɡɛɭɪɟɧɶ. ɉɨɡɧɚɱɢɦɨ

M h

M2h

2

.

2

h

Ɍɨɞɿ ɦɿɧɿɦɭɦ ɰɿɽʀ ɮɭɧɤɰɿʀ ɞɨɫɹɝɚɽɬɶɫɹ ɞɥɹ ɬɚɤɢɯ h :

Mc h

M2

2

0 , h0

2

G .

h

2

2

M2

ɉɪɢ ɬɚɤɨɦɭ ɡɧɚɱɟɧɧɿ h ɨɰɿɧɤɚ ɩɨɯɢɛɤɢ (1) ɬɚɤɚ:

§

1

·

M h0 2 M2G O¨G

2

¸ o 0 .

¨

¸Go0

61

ɉɪɢɤɥɚɞ 2 ɉɨɞɢɜɢɦɨɫɹ ɧɚ ɜɩɥɢɜ ɡɛɭɪɟɧɶ ɧɚ ɩɨɯɢɛɤɭ ɨɛɱɢɫɥɟɧɧɹ ɩɟɪɲɨʀ

ɩɨɯɿɞɧɨʀ ɩɪɢ ɜɢɤɨɪɢɫɬɚɧɧɿ ɰɟɧɬɪɚɥɶɧɨʀ ɪɿɡɧɢɰɟɜɨʀ ɩɨɯɿɞɧɨʀ:

fic

~

~

d

fic

fi 1

fi 1

Gi 1 Gi 1

d

3h

2

G M h .

fi 1

fi 1

M

2h

6

2h

2h

h

G ,

h0

3G

Ɂ ɪɿɜɧɹɧɧɹ Mc h

M3h

0 ɦɚɽɦɨ: h03

3

. Ɉɬɠɟ,

3

h

2

M3

M3

§

2 ·

M h0

M

3 3

9G

2

G

1 3

M

3G

2

3 03G

2

3

M

3G

2

3

¨

3 ¸

6

2

3G

2

3

3

2

3

O¨G

¸.

M3

3

©

¹

M3

o 0 ɩɨɯɢɛɤɢ ɮɨɪɦɭɥɢ ɱɢɫɟɥɶɧɨɝɨ

Ɍɚɤɢɦ ɱɢɧɨɦ ɲɜɢɞɤɿɫɬɶ ɡɛɿɠɧɨɫɬɿ ɩɪɢ

ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ ɰɟɧɬɪɚɥɶɧɨɸ ɩɨɯɿɞɧɨɸ ɜɢɳɚ ɧɿɠ ɞɥɹ ɮɨɪɦɭɥɢ ɡ ɩɪɢɤɥɚɞɭ 1 (ɩɨɯɿɞɧɚ ɜɩɟɪɟɞ ɚɛɨ ɧɚɡɚɞ).

Ɂɚɞɚɱɚ 24 Ⱦɨɫɥɿɞɢɬɢ ɩɨɯɢɛɤɭ ɱɢɫɟɥɶɧɨɝɨ ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ ɞɥɹ n 2,k 2, ɜɢɛɪɚɬɢ ɨɩɬɢɦɚɥɶɧɢɣ ɤɪɨɤ h0 , ɞɚɬɢ ɨɰɿɧɤɭ h0 .

8. Ⱥɩɪɨɤɫ ɦɭ ɚɧɧɹ ɮɭɧɤ ɿ

8.1. ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɿ ɚɩɪɨɤɫɢɦɚɰɿʀ [ɅɆɋ, 8-13], [ȻɀɄ, 160-161]. ɇɚɛɥɢɠɟɧɧɹ ɮɭɧɤɰɿɣ ɡɚɫɬɨɫɨɜɭɸɬɶ ɭ ɜɢɩɚɞɤɚɯ, ɹɤɳɨ

-ɮɭɧɤɰɿɹ ɫɤɥɚɞɧɚ (ɬɪɚɧɫɰɟɧɞɟɧɬɧɚ ɚɛɨ ɽ ɪɨɡɜ‘ɹɡɤɨɦ ɫɤɥɚɞɧɨʀ ɡɚɞɚɱɿ) ɿ ʀʀ ɡɚɦɿɧɸɸɬɶ ɮɭɧɤɰɿɽɸ, ɹɤɚ ɥɟɝɤɨ ɨɛɱɢɫɥɸɽɬɶɫɹ ( ɧɚɣɱɚɫɬɿɲɟ, ɩɨɥɿɧɨɦɨɦ);

-ɧɟɨɛɯɿɞɧɨ ɩɨɛɭɞɭɜɚɬɢ ɮɭɧɤɰɿɸ ɧɟɩɟɪɟɪɜɧɨɝɨ ɚɪɝɭɦɟɧɬɭ ɞɥɹ ɮɭɧɤɰɿʀ, ɹɤɚ ɡɚɞɚɧɚ ɫɜɨʀɦɢ ɡɧɚɱɟɧɧɹɦɢ (ɬɚɛɥɢɱɧɚ);

-ɬɚɛɥɢɱɧɚ ɮɭɧɤɰɿɹ ɧɚɛɥɢɠɚɽɬɶɫɹ ɬɚɛɥɢɱɧɨɸ ɠ ɮɭɧɤɰɿɽɸ (ɡɝɥɚɞɠɭɜɚɧɧɹ).

ȱɧɬɟɪɩɨɥɸɜɚɧɧɹ ɧɟ ɤɪɚɳɢɣ ɫɩɨɫɿɛ ɧɚɛɥɢɠɟɧɧɹ ɮɭɧɤɰɿɣ ɱɟɪɟɡ ɪɨɡɛɿɠɧɿɫɬɶ ɰɶɨɝɨ ɩɪɨɰɟɫɭ ɞɥɹ ɩɨɥɿɧɨɦɿɜ. Ɍɢɦ ɛɿɥɶɲɟ ɞɨɰɿɥɶɧɿɫɬɶ ɡɚɫɬɨɫɭɜɚɧɧɹ ɿɧɬɟɪɩɨɥɸɜɚɧɧɹ ɫɭɦɧɿɜɧɚ, ɹɤɳɨ ɮɭɧɤɰɿɹ ɬɚɛɥɢɱɧɚ, ɚ ʀʀ ɡɧɚɱɟɧɧɹ

ɧɟɬɨɱɧɿ. ɉɨɬɪɿɛɧɨ ɛɭɞɭɜɚɬɢ ɚɩɪɨɤɫɢɦɭɸɱɭ ɮɭɧɤɰɿɸ ɡ ɿɧɲɢɯ ɦɿɪɤɭɜɚɧɶ.
ɇɚɣɛɿɥɶɲ ɡɚɝɚɥɶɧɢɣ ɩɪɢɧɰɢɩ: ɧɚɛɥɢɡɢɬɢ f x ɮɭɧɤɰɿɽɸ ) x ɬɚɤ, ɳɨɛ
ɞɨɫɹɝɚɥɚɫɹ ɞɟɹɤɚ ɡɚɞɚɧɚ ɬɨɱɧɿɫɬɶ		:
	f	x	) x				.

ɥɟ ɪɨɡɜ’ɹɡɨɤ ɜ ɬɚɤɿɣ ɩɨɫɬɚɧɨɜɰɿ ɦɨɠɟ ɧɟ ɿɫɧɭɜɚɬɢ ɚɛɨ ɛɭɬɢ ɧɟ ɽɞɢɧɢɦ. Ɂɚɝɚɥɶɧɚ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɿ ɧɚɛɥɢɠɟɧɧɹ ɬɚɤɚ. ɇɟɯɚɣ ɦɚɽɦɨ ɟɥɟɦɟɧɬ f

ɥɿɧɿɣɧɨɝɨ ɧɨɪɦɨɜɚɧɨɝɨ ɩɪɨɫɬɨɪɭ R .	ɉɨɛɭɞɭɽɦɨ ɩɿɞɩɪɨɫɬɿɪ		Mn , ɜ ɹɤɨɦɭ
ɟɥɟɦɟɧɬɢ ɽ ɥɿɧɿɣɧɨɸ ɤɨɦɛɿɧɚɰɿɽɸ		R
n
) ¦ciMi Mn			(1)
i 0
ɩɨ ɟɥɟɦɟɧɬɚɯ ɥɿɧɿɣɧɨ ɧɟɡɚɥɟɠɧɨʀ ɫɢɫɬɟɦɢ
{Mi}i	0 , i	R	(2)

62

ȼɿɞɯɢɥɟɧɧɹ ) M n ɜɿɞ

f R ɽ ɱɢɫɥɨ

' f ,)

f

)

.

ɉɨɡɧɚɱɢɦɨ

)

' f .

inf

f

) Mn

ȿɥɟɦɟɧɬ)0 ɬɚɤɢɣ, ɳɨ' f ,)0

f

)0

inf

f

)

' f ,

(3)

) Mn

ɧɚɡɢɜɚɽɬɶɫɹ ɟɥɟɦɟɧɬɨɦ ɧɚɣɤɪɚɳɨɝɨ ɧɚɛɥɢɠɟɧɧɹ (ȿɇɇ).

əɫɧɨ, ɳɨ ɭɦɨɜɭ ɬɨɱɧɨɫɬɿ ɬɪɟɛɚ ɩɟɪɟɜɿɪɹɬɢ ɧɚ ɰɶɨɦɭ ɟɥɟɦɟɧɬɿ. ɍ

ɜɢɩɚɞɤɭ ʀʀ ɧɟɜɢɤɨɧɚɧɧɹ ɬɪɟɛɚ ɡɛɿɥɶɲɭɜɚɬɢ ɤɿɥɶɤɿɫɬɶ ɟɥɟɦɟɧɬɿɜ n ɜ (1).

Ɍɟɨɪɟɦɚ 1 Ⱦɥɹ ɛɭɞɶ-ɹɤɨɝɨ ɥɿɧɿɣɧɨɝɨ ɧɨɪɦɨɜɚɧɨɝɨ ɩɪɨɫɬɨɪɭ R ɿɫɧɭɽ

ɟɥɟɦɟɧɬ ɧɚɣɤɪɚɳɨɝɨ ɧɚɛɥɢɠɟɧɧɹ

0

Mn .

n

ȼɜɟɞɟɦɨ F(c&)

F c0 ,c1,...,cn

f

)

f

¦ciIi

.

ɐɟ

ɧɟɩɟɪɟɪɜɧɚ

ɮɭɧɤɰɿɹ ɚɪɝɭɦɟɧɬɿɜ c&

c0,c1,...,cn . Ⱦɥɹ

i 0

ɟɥɟɦɟɧɬɿɜ, ɹɤɿ ɡɚɞɨɜɨɥɶɧɹɸɬɶ ɭɦɨɜɿ

F c&

)

2

f

f R1

) Mn ,

' f .

(4)

ɦɚɽɦɨ

)

t

)

f

f

2

f

f

f

)

0

) :

)

d

Mn

Ɂɧɚɱɢɬɶ ȿɇɇ

. Ɂɚ

ɬɟɨɪɟɦɨɸ Ʉɚɧɬɨɪɚ

0 , ɞɟ

2

f

U

F c& ɞɨɫɹɝɚɽ ɦɿɧɿɦɭɦɭ.

)0

d

f

)

.

ɉɪɢɱɨɦɭ

f

ȿɥɟɦɟɧɬɿɜ ɧɚɣɤɪɚɳɨɝɨ ɧɚɛɥɢɠɟɧɧɹ ɜ ɥɿɧɿɣɧɨɦɭ ɧɨɪɦɨɜɚɧɨɦɭ ɩɪɨɫɬɨɪɿ

ɦɨɠɟ ɛɭɬɢ ɿ ɞɟɤɿɥɶɤɚ.

ɉɪɨɫɬɿɪ R ɧɚɡɢɜɚɽɬɶɫɹ ɫɬɪɨɝɨ ɧɨɪɦɨɜɚɧɢɦ, ɹɤɳɨ ɡ ɭɦɨɜɢ

f g

f

g

,

f

0,

g

0

ɜɢɩɥɢɜɚɽ, ɳɨ

0 ɬɚɤɟ, ɳɨ

g

f

(5)

Ɍɟɨɪɟɦɚ 2 əɤɳɨ ɩɪɨɫɬɿɪ R ɫɬɪɨɝɨ

ɧɨɪɦɨɜɚɧɢɣ, ɬɨ

ɟɥɟɦɟɧɬ ɧɚɣɤɪɚɳɨɝɨ

ɧɚɛɥɢɠɟɧɧɹ )0 ɽɞɢɧɢɣ.

)01

)02 –

Ⱦɨɜɟɞɟɧɧɹ

ɜɿɞ

ɫɭɩɪɨɬɢɜɧɨɝɨ. ɇɟɯɚɣ

ɿɫɧɭɸɬɶ

ɞɜɚ

ɟɥɟɦɟɧɬɢ

ɧɚɣɤɪɚɳɨɝɨ ɧɚɛɥɢɠɟɧɧɹ. ȼɿɡɶɦɟɦɨ

0,1@, ɬɨɞɿ

D f

)02

' f d

f

D )o1

1

D

)02

D f

)01

1

d

d D

)

01

1 D

f

)

02

D '

f

1 D ' f

' f

f

Ɍɨɛɬɨ ɜɫɿ “d

” ɦɨɠɧɚ ɡɚɦɿɧɢɬɢ ɧɚ “=”.Ɉɬɪɢɦɚɽɦɨ

D f

)01

1

D f

)

02

D

f

)

01

1

D

f

)02

D

)01

D

f

)02

Ɂɚ ɩɪɢɩɭɳɟɧɧɹɦ

ɬɚɤɟ, ɳɨ

f

1

. ȼɢɛɟɪɟɦɨ

D

1

. Ɍɨɞɿ

f

)

01

f

)

02

. Ɉɫɤɿɥɶɤɢ

f

)01

f

)02

' f , ɬɨ

2

63

ɨɫɬɚɧɧɹ ɪɿɜɧɿɫɬɶ ɦɚɽ ɦɿɫɰɟ ɬɿɥɶɤɢ ɞɥɹ										1. Ɂɜɿɞɫɢ
f	)01	f	)02 )01			)02			.
	Ɉɬɠɟ, ɦɢ ɨɬɪɢɦɚɥɢ ɩɪɨɬɢɪɿɱɱɹ ɡ ɩɪɢɩɭɳɟɧɧɹɦ, ɳɨ ɿ ɞɨɜɨɞɢɬɶ ɿɫɧɭɜɚɧɧɹ
ɽɞɢɧɨɝɨ ɟɥɟɦɟɧɬɚ ɧɚɣɤɪɚɳɨɝɨ ɧɚɛɥɢɠɟɧɧɹ.
Ɍɟɨɪɟɦɚ 3 Ƚɿɥɶɛɟɪɬɿɜ ɩɪɨɫɬɿɪ H – ɫɬɪɨɝɨ ɧɨɪɦɨɜɚɧɢɣ.
	H :	u,v u,v H					u		u,u	. ɇɟɯɚɣ

f g

f

g

,

x, y H

g

2

f

f

2

2

f

g

g

2 .

Ɂ ɿɧɲɨɝɨ ɛɨɤɭ

g 2

f g, f

g

f 2 2 f , g g 2

f

Ɂɜɿɞɫɢ f g f , g . Ⱦɥɹ ɞɨɜɿɥɶɧɨɝɨ ɝɿɥɶɛɟɪɬɨɜɨɝɨ ɩɪɨɫɬɨɪɭ

Ɍɚɤɢɦ ɱɢɧɨɦ ɧɚ ɟɥɟɦɟɧɬɚɯ (6) ɧɟɪɿɜɧɿɫɬɶ Ʉɨɲɿ – ɩɟɪɟɬɜɨɪɸɽɬɶɫɹ ɜ ɪɿɜɧɿɫɬɶ. Ɋɨɡɝɥɹɧɟɦɨ

f

g

2

f

2 2 f , g 2

g

2

f

2

f

g

2

g

2

(6)

f , g d f g .

Ȼɭɧɹɤɨɜɫɶɤɨɝɨ
f	g 2 ,

Ɍɨɞɿ ɞɥɹ

f

ɦɚɽɦɨ

f

g

0. Ɂɜɿɞɫɢ

: f

g , ɬɨɛɬɨ ɇ – ɫɬɪɨɝɨ

g

ɧɨɪɦɨɜɚɧɢɣ.

!)0 Mn .

ɇɚɫɥɿɞɨɤ R

H

ɉɪɢɤɥɚɞɢ ɫɬɪɨɝɨ ɧɨɪɦɨɜɚɧɢɯ ɩɪɨɫɬɨɪɿɜ:

³

L2 (a,b) ɡ ɧɨɪɦɨɸ u

b

u2dx .

1)

a

· p

§b

¨ ³

1

p

¸

2)

Lp (a,b)ɡ ɧɨɪɦɨɸ

u

¨

u pdx¸

, p 1.

© a

¹

ɉɪɨɫɬɿɪ C[a,b] ɧɟ ɽ ɫɬɪɨɝɨ ɧɨɪɦɨɜɚɧɢɦ, ɚɥɟ ɜ ɧɶɨɦɭ ɿɫɧɭɽ ɽɞɢɧɢɣ ɟɥɟɦɟɧɬ ɧɚɣɤɪɚɳɨɝɨ ɧɚɛɥɢɠɟɧɧɹ (ɩɪɨ ɰɟɣ ɮɚɤɬ ɜ ɧɚɫɬɭɩɧɨɦɭ ɩɭɧɤɬɿ).

8.2. ɇɚɣɤɪɚɳɟ ɪɿɜɧɨɦɿɪɧɟ ɧɚɛɥɢɠɟɧɧɹ [ȻɀɄ, 180-186], [ɅɆɋ, 66-82]

ɇɚɣɤɪɚɳɟ ɪɿɜɧɨɦɿɪɧɟ ɧɚɛɥɢɠɟɧɧɹ – ɰɟ ɧɚɛɥɢɠɟɧɧɹ ɜ ɩɪɨɫɬɨɪɿ R C[a,b], ɞɟ

	f		C[a,b]	max		f (x)		– ɪɿɜɧɨɦɿɪɧɚ ɦɟɬɪɢɤɚ.
				x [a,b]
Ɍɟɨɪɟɦɚ 1 (					ɏɚɚɪɚ ) Ⱦɥɹ ɬɨɝɨ, ɳɨɛ f C[a,b] ɿɫɧɭɜɚɜ ɽɞɢɧɢɣ ɟɥɟɦɟɧɬ

ɧɚɣɤɪɚɳɨɝɨ ɪɿɜɧɨɦɿɪɧɨɝɨ ɧɚɛɥɢɠɟɧɧɹ ɧɟɨɛɯɿɞɧɨ ɿ ɞɨɫɬɚɬɧɶɨ, ɳɨɛ ɫɢɫɬɟɦɚ {Mi }if 0 ɛɭɥɚ ɫɢɫɬɟɦɨɸ ɑɟɛɢɲɨɜɚ.

64

ɋɢɫɬɟɦɚ

{Mi }if

0

ɧɚɡɢɜɚɽɬɶɫɹ

ɫɢɫɬɟɦɨɸ

ɑɟɛɢɲɨɜɚ,

ɹɤɳɨ

ɟɥɟɦɟɧɬ

)n (x)

n

n

¦ciMi (x)

ɦɚɽ ɧɟ ɛɿɥɶɲɟ

n

ɧɭɥɿɜ,

ɩɪɢɱɨɦɭ

¦ci2

0 . ɇɚɩɪɢɤɥɚɞ,

i 0

xi `if

i

1

ɫɢɫɬɟɦɨɸ ɑɟɛɢɲɨɜɚ ɽ ɩɨɥɿɧɨɦɿɚɥɶɧɚ ɫɢɫɬɟɦɚ

0 .

ɉɨɡɧɚɱɢɦɨ Qn0 (x) – ɛɚɝɚɬɨɱɥɟɧ ɧɚɣɤɪɚɳɨɝɨ ɪɿɜɧɨɦɿɪɧɨɝɨ ɧɚɛɥɢɠɟɧɧɹ

(ɞɚɥɿ

–

ȻɇɊɇ.).

Ƀɨɝɨ

ɜɿɞɯɢɥɟɧɧɹ

ɜɿɞ

f :

'( f )

Qn0 (x)

f (x)

ɋ

inf

Qn (x)

f (x)

.

Qn (x)

Ɍɟɨɪɟɦɚ 2 ( ɑɟɛɢɲɨɜɚ ) Qn0 (x) – ȻɇɊɇ ɧɟɩɟɪɟɪɜɧɨʀ ɮɭɧɤɰɿʀ

f (x) ɬɨɞɿ ɬɚ

ɬɿɥɶɤɢ

ɬɨɞɿ,

ɹɤɳɨ

ɧɚ

ɜɿɞɪɿɡɤɭ

[a,b]

ɿɫɧɭɽ

ɯɨɱɚ

ɛ

(n 2)-ɚ ɬɨɱɤɢ

a d x0...d xm

d b, m t n

1 ɬɚɤɿ, ɳɨ

D(

'( f ),

f (xi ) Qn0 (xi )

1)i

(1)

ɞɟ i

,

0,m

1.

Ɍɨɱɤɢ

{xi }im

0 ,

ɹɤɿ

ɡɚɞɨɜɨɥɶɧɹɸɬɶ

ɭɦɨɜɚɦ

ɬɟɨɪɟɦɢ

ɑɟɛɢɲɨɜɚ,

ɧɚɡɢɜɚɸɬɶɫɹ ɬɨɱɤɚɦɢ ɱɟɛɢɲɨɜɫɶɤɨɝɨ ɚɥɶɬɟɪɧɚɧɫɭ.

Ɍɟɨɪɟɦɚ 3 Qn0 (x) – ȻɇɊɇ ɞɥɹ ɧɟɩɟɪɟɪɜɧɨʀ ɮɭɧɤɰɿʀ ɽɞɢɧɢɣ.

ɉɪɢɩɭɫɬɢɦɨ, ɿɫɧɭɸɬɶ ɞɜɚ ȻɇɊɇ ɫɬɟɩɟɧɹ n : Qn(1) (x)

Qn(2) (x):

'( f )

f

Qn(1)

C

f

Qn(2)

C

.

Ɂɜɿɞɫɢ ɜɢɩɥɢɜɚɽ, ɳɨ

Qn(2)

d

'( f ),

f

Qn(1)

f

Qn(1)

f

Qn(2)

2

Qn(1)

2

2

ɬɨɛɬɨ ɛɚɝɚɬɨɱɥɟɧ

Qn(1) (x)

(x

)

ɬɚɤɨɠ ɽ ȻɇɊɇ. ɇɟɯɚɣ x

,x ,...,x

–

0

m

2

1

ɜɿɞɩɨɜɿɞɧɿ ɣɨɦɭ ɬɨɱɤɢ ɱɟɛɢɲɨɜɫɶɤɨɝɨ ɚɥɶɬɟɪɧɚɧɫɭ.

ɐɟ ɨɡɧɚɱɚɽ, ɳɨ

Qn(1) (xi ) Qn(2) (xi )

'( f ) ,

f (xi )

2

ɚɛɨ

>Qn(1) (xi )

f (xi )@ Qn(2) (xi )

f (xi )@

2'( f )

(2)

Ɍɚɤ ɹɤ

f (xi )

d '( f ),

1,2, ɬɨ (2)

ɦɨɠɥɢɜɟ ɥɢɲɟ ɭ ɬɨɦɭ

Qn(k) (xi )

k

ɜɢɩɚɞɤɭ, ɤɨɥɢ

0,n 1.

Qn(1) (xi ) f (xi ) Qn(2) (xi ) f (xi ), i

Ɂɜɿɞɤɢ

ɜɢɩɥɢɜɚɽ,

ɳɨ

Q(1)

(x)

Q(2) (x),

ɚ

ɰɟ

ɫɭɩɟɪɟɱɢɬɶ

ɩɨɱɚɬɤɨɜɨɦɭ

n

n

ɩɪɢɩɭɳɟɧɧɸ.

8.3. ɉɪɢɤɥɚɞɢ ɩɨɛɭɞɨɜɢ ȻɇɊɇ [ȻɀɄ, 186-181], [ȼɨɥɤɨɜ, 81-91]

65

ɋɤɿɧɱɟɧɨɝɨ ɚɥɝɨɪɢɬɦɭ ɩɨɛɭɞɨɜɢ ȻɇɊɇ ɞɥɹ ɞɨɜɿɥɶɧɨʀ ɮɭɧɤɰɿʀ ɧɟ ɿɫɧɭɽ. ȯ ɿɬɟɪɚɰɿɣɧɢɣ [ɅɆɋ, 73-79]. ɥɟ ɜ ɞɟɹɤɢɯ ɜɢɩɚɞɤɚɯ ɦɨɠɧɚ ɩɨɛɭɞɭɜɚɬɢ ȻɇɊɇ ɡɚ ɬɟɨɪɟɦɨɸ ɑɟɛɢɲɨɜɚ.

10. ɉɨɬɪɿɛɧɨ ɧɚɛɥɢɡɢɬɢ ɛɚɝɚɬɨɱɥɟɧɨɦ ɧɭɥɶɨɜɨɝɨ ɫɬɟɩɟɧɹ.

M

Q0(x

m

ɇɟɯɚɣ M

Ɋɢɫ. 11

f (x1 ) . ɬɨɞɿ Q0 (x) – ȻɇɊɇ ɦɚɽ

max f (x)

f (x0 ),

m

min f (x)

[a,b]

[a,b]

ɜɢɝɥɹɞ (ɞɢɜ. ɪɢɫ. 11):

Q0

(x)

M

m

,

M m

2

M m

ɞɟ f (x

0

)

M m

, f (x )

M m

m

M m

,

2

2

1

2

2

2

'( f )

M m

, ɚ x0, x1 – ɬɨɱɤɢ ɱɟɛɢɲɨɜɫɶɤɨɝɨ ɚɥɶɬɟɪɧɚɧɫɭ.

2

C[a,b] ɧɚɛɥɢɠɚɽɬɶɫɹ ɛɚɝɚɬɨɱɥɟɧɨɦ ɩɟɪɲɨɝɨ

20. Ɉɩɭɤɥɚ ɮɭɧɤɰɿɹ f (x)

ɫɬɟɩɟɧɹ

Q1(x) c0 c1x .

Ɉɫɤɿɥɶɤɢ

f (x) ɨɩɭɤɥɚ, ɬɨ ɪɿɡɧɢɰɹ

f (x)

(c0 c1x)

ɦɨɠɟ ɦɚɬɢ ɥɢɲɟ ɨɞɧɭ

ɜɧɭɬɪɿɲɧɸ ɬɨɱɤɭ ɟɤɫɬɪɟɦɭɦɭ. Ɍɨɦɭ ɬɨɱɤɢ a,b ɽ ɬɨɱɤɚɦɢ ɱɟɛɢɲɨɜɫɶɤɨɝɨ

ɚɥɶɬɟɪɧɚɧɫɭ. ɇɟɯɚɣ ɬɟɨɪɟɦɨɸ ɑɟɛɢɲɨɜɚ,

ɬɪɟɬɹ – ɬɨɱɤɚ ɱɟɛɢɲɨɜɫɶɤɨɝɨ ɚɥɶɬɟɪɧɚɧɫɭ. Ɂɝɿɞɧɨ ɡ ɦɚɽɦɨ ɫɢɫɬɟɦɭ:

°	c1a D'( f )
­ f (a) c0
® f ([) c0	c1[ D'( f )
°	c1b D'( f )
¯ f (b) c0

Ɂɜɿɞɫɢ f (b)	f (a) c (b	a) ɬɚ c	f (b)	f (a)	.
	1	1	b	a
ɐɸ ɫɢɫɬɟɦɭ ɬɪɟɛɚ ɡɚɦɤɧɭɬɢ, ɜɢɤɨɪɢɫɬɚɜɲɢ ɳɟ ɨɞɧɟ ɪɿɜɧɹɧɧɹ ɡ ɭɦɨɜɢ:
ɬɨɱɤɚ	ɽ ɬɨɱɤɨɸ	ɟɤɫɬɪɟɦɭɦɭ	ɪɿɡɧɢɰɿ		f (x) (c0 c1x). Ɍɨɦɭ ɞɥɹ
ɞɢɮɟɪɟɧɰɿɣɨɜɚɧɨʀ ɮɭɧɤɰɿʀ f (x) ɞɥɹ ɜɢɡɧɚɱɟɧɧɹ					ɦɚɽɦɨ ɪɿɜɧɹɧɧɹ ( ɞɨɬɢɱɧɚ ɿ

ɫɿɱɧɚ ɩɚɪɚɥɟɥɶɧɿ ):

66

f c([) c1	f (b)	f (a)	.
	b
		a

Ɋɢɫ. 12 Ƚɟɨɦɟɬɪɢɱɧɨ ɰɹ ɩɪɨɰɟɞɭɪɚ ɜɢɝɥɹɞɚɽ ɧɚɫɬɭɩɧɢɦ ɱɢɧɨɦ (ɞɢɜ. ɪɢɫ. 12).

ɉɪɨɜɨɞɢɦɨ ɫɿɱɧɭ ɱɟɪɟɡ ɬɨɱɤɢ (a, f (a)),(b, f (b)). Ⱦɥɹ ɧɟʀ ɬɚɧɝɟɧɫ ɤɭɬɚ ɞɨɪɿɜɧɸɽ c1. ɉɪɨɜɨɞɢɦɨ ɩɚɪɚɥɟɥɶɧɭ ʀɣ ɞɨɬɢɱɧɭ ɞɨ ɤɪɢɜɨʀ y f (x), ɚ ɩɨɬɿɦ ɩɪɹɦɭ, ɪɿɜɧɨɜɿɞɞɚɥɟɧɭ ɜɿɞ ɫɿɱɧɨʀ ɬɚ ɞɨɬɢɱɧɨʀ, ɹɤɚ ɿ ɛɭɞɟ ɝɪɚɮɿɤɨɦ Q1 (x) . ɉɪɢ

ɰɶɨɦɭ x0

a ,

x1

,

x2

b.

30. ɉɨɬɪɿɛɧɨ ɧɚɛɥɢɡɢɬɢ f (x)

xn

1,

x

[

1,1]

ɛɚɝɚɬɨɱɥɟɧɨɦ ɫɬɟɩɟɧɹ n

Qn0 (x). ȼɜɟɞɟɦɨ

n 1 (x)

xn 1

xn 1

a1xn ...

'( f )

P

Qn (x)

xn 1

n 1

n 1(x)

inf

Qn0 (x)

inf

P

0

T

Qn (x)

C

Pn 1

C

xn 1

Qn0 (x)

n 1 (x)

Qn0 (x)

xn 1

n 1 (x) .

T

T

Ɂɚɞɚɱɚ

25

Ⱦɥɹ

ɩɪɢɤɥɚɞɭ

3

ɜɤɚɡɚɬɢ

ɬɨɱɤɢ

ɱɟɛɢɲɨɜɫɶɤɨɝɨ

ɚɥɶɬɟɪɧɚɧɫɭ

{xi },i

0,n

1.

a0 ... an

40. ɉɨɬɪɿɛɧɨ ɧɚɛɥɢɡɢɬɢ

f (x)

Pn 1(x)

1xn 1,

an 1

0,

x [a,b] ȻɇɊɇ ɫɬɟɩɟɧɹ n . Ɂɚɩɢɲɿɦɨ ɣɨɝɨ ɭ ɜɢɝɥɹɞɿ:

Qn0 (x)

Pn 1(x) an 1Tn[a1,b],

- ɧɨɪɦɨɜɚɧɢɣ ɛɚɝɚɬɨɱɥɟɧ ɑɟɛɢɲɨɜɚ ɧɚ ɩɪɨɦɿɠɤɭ x [a,b].

ɞɟ Tn[a1,b](x)

Ⱦɿɣɫɧɨ ɰɟ ȻɇɊɇ: ɜɢɪɚɡ ɭ ɩɪɚɜɿɣ ɱɚɫɬɢɧɿ ɽ ɛɚɝɚɬɨɱɥɟɧɨɦ ɫɬɟɩɟɧɹ n ,

ɨɫɤɿɥɶɤɢ

ɤɨɟɮɿɰɿɽɧɬ

ɩɪɢ

xn

1

ɞɨɪɿɜɧɸɽ

ɧɭɥɸ,

ɚ

ɣɨɝɨ ɧɭɥɿ

b

a

b

a

2k

1

ɽ

ɬɨɱɤɚɦɢ

ɱɟɛɢɲɟɜɫɶɤɨɝɨ

xk

tk , tk

cos

,k

0,n

2

2

2(n

1)

ɚɥɶɬɟɪɧɚɫɭ ɞɥɹ Q

0(x).

n

Ɂɚɞɚɱɚ

26

ɉɨɤɚɡɚɬɢ,

ɳɨ

ɞɥɹ

f (x) ɩɚɪɧɨʀ

(ɧɟɩɚɪɧɨʀ) ɮɭɧɤɰɿʀ

ȻɇɊɇ

ɰɟ

ɛɚɝɚɬɨɱɥɟɧ ɩɨ ɩɚɪɧɢɯ (ɧɟɩɚɪɧɢɯ) ɫɬɟɩɟɧɹɯ x .

50. Ɍɟɥɟɫɤɨɩɿɱɧɢɣ ɦɟɬɨɞ. Ⱦɭɠɟ ɱɚɫɬɨ ȻɇɊɇ ɬɨɱɧɨ ɡɧɚɣɬɢ ɧɟ ɜɞɚɽɬɶɫɹ. ȼ ɬɚɤɢɯ ɜɢɩɚɞɤɚɯ ɲɭɤɚɽɬɶɫɹ ɛɚɝɚɬɨɱɥɟɧ, ɛɥɢɡɶɤɢɣ ɞɨ ɧɶɨɝɨ. Ȼɚɠɚɧɨ ɳɨɛ ɰɟɣ

67

ɛɚɝɚɬɨɱɥɟɧ ɛɭɜ ɧɟɜɢɫɨɤɨɝɨ ɫɬɟɩɟɧɹ (ɦɟɧɲɟ ɚɪɢɮɦɟɬɢɱɧɢɯ ɨɩɟɪɚɰɿɣ ɧɚ ɣɨɝɨ

			n
	ɨɛɱɢɫɥɟɧɧɹ)	ɋɩɨɱɚɬɤɭ ɛɭɞɭɸɬɶ ɬɚɤɢɣ ɛɚɝɚɬɨɱɥɟɧ Pn (x)	¦a j x j ,	ɳɨɛ
			j 0
	ɜɿɞɯɢɥɟɧɧɹ	ɜɿɞ f (x) ɛɭɥɚ ɞɨɫɬɚɬɧɶɨ ɦɚɥɨɸ. (ɧɚɩɪɢɤɥɚɞ	ɦɟɧɲɨɸ ɡɚ			).
				2

Ɇɨɠɧɚ ɰɟ ɡɪɨɛɢɬɢ, ɧɚɩɪɢɤɥɚɞ, ɡɚ ɮɨɪɦɭɥɨɸ Ɍɟɣɥɨɪɚ. ɉɨɬɿɦ ɧɚɛɥɢɠɚɸɬɶ ɛɚɝɚɬɨɱɥɟɧ Pn (x) ɛɚɝɚɬɨɱɥɟɧɨɦ ɧɚɣɤɪɚɳɨɝɨ ɪɿɜɧɨɦɿɪɧɨɝɨ ɧɚɛɥɢɠɟɧɧɹ

Pn 1(x) (ɡɚ ɚɥɝɨɪɢɬɦɨɦ ɩ. 4; ɞɥɹ ɩɪɨɫɬɨɬɢ x [ 1,1]): d Pn 1(x) Pn (x) anTn (x)21 n .

Ɉɫɤɿɥɶɤɢ Tn (x) 1 ɧɚ ɜɿɞɪɿɡɤɭ [ 1,1], ɬɨ

Pn 1(x) Pn (x) d an 21 n .

Ⱦɚɥɿ ɧɚɛɥɢɠɚɸɬɶ ɛɚɝɚɬɨɱɥɟɧ Pn 1(x) ɛɚɝɚɬɨɱɥɟɧɨɦ ɧɚɣɤɪɚɳɨɝɨ ɪɿɜɧɨɦɿɪɧɨɝɨ ɧɚɛɥɢɠɟɧɧɹ Pn 2 (x) ɿ ɬ. ɞ. ɉɨɧɢɠɟɧɧɹ ɫɬɟɩɟɧɹ ɩɪɨɞɨɜɠɭɽɬɶɫɹ ɞɨ ɬɢɯ ɩɿɪ,

ɩɨɤɢ ɫɭɦɚɪɧɚ ɩɨɯɢɛɤɚ ɜɿɞ ɬɚɤɢɯ ɩɨɫɥɿɞɨɜɧɢɯ ɚɩɪɨɤɫɢɦɚɰɿɣ ɡɚɥɢɲɚɽɬɶɫɹ ɦɟɧɲɨɸ ɡɚ ɡɚɞɚɧɟ ɦɚɥɟ ɱɢɫɥɨ .

8.4. ɇɚɣɤɪɚɳɟ ɫɟɪɟɞɧɶɨɤɜɚɞɪɚɬɢɱɧɟ ɧɚɛɥɢɠɟɧɧɹ [ȻɀɄ, 156-166], [ɅɆɋ, 5358]

ɇɚɛɥɢɡɢɦɨ ɮɭɧɤɰɿɸ f (x) H	ɡ ɝɿɥɶɛɟɪɬɨɜɨɝɨ ɩɪɨɫɬɨɪɭ H			ɮɭɧɤɰɿɹɦɢ
ɡ ɫɤɿɧɱɟɧɧɨ-ɜɢɦɿɪɧɨɝɨ ɩɿɞɩɪɨɫɬɨɪɭ	M n	,	ɩɪɨɫɬɨɪɭ H . Ɍɭɬ H	ɝɿɥɶɛɟɪɬɿɜ
ɩɪɨɫɬɿɪ ɿɡ ɫɤɚɥɹɪɧɢɦ ɞɨɛɭɬɤɨɦ	u,v		ɧɨɪɦɚ ɿ ɜɿɞɫɬɚɧɶ	ɞɥɹ ɹɤɨɝɨ

ɜɢɡɧɚɱɚɸɬɶɫɹ ɮɨɪɦɭɥɚɦɢ:

u,u , ' u,v

u

u v

.

ɉɨɛɭɞɭɽɦɨ

Mn

H ,

n

ɞɟ ^Mi `if

u

¦ciMi

(1)

i 0

0 - ɥɿɧɿɣɧɨ ɧɟɡɚɥɟɠɧɚ ɫɢɫɬɟɦɚ ɟɥɟɦɟɧɬɿɜ ɡ H .

ȿɥɟɦɟɧɬ

ɧɚɣɤɪɚɳɨɝɨ

ɫɟɪɟɞɧɶɨɤɜɚɞɪɚɬɢɱɧɨɝɨ

ɧɚɛɥɢɠɟɧɧɹ

(ɜ

ɩɨɞɚɥɶɲɨɦɭ ȿɇɋɄɇ) )0

ɬɚɤɢɣ, ɳɨ

)0

)0

)0, f

) M n

) .

f

f

inf

f

f

H ,

Ɍɟɨɪɟɦɚ 1

ɇɟɯɚɣ

0 M n

–

ɟɥɟɦɟɧɬ

ɧɚɣɤɪɚɳɨɝɨ ɫɟɪɟɞɧɶɨ-

ɤɜɚɞɪɚɬɢɱɧɨɝɨ ɧɚɛɥɢɠɟɧɧɹ

)0

)

M M n

f

inf

f

.

Ɍɨɞɿ

f

0,

0,

Mn

(2)

ɇɟɯɚɣ (2) ɧɟ ɜɢɤɨɧɭɽɬɶɫɹ, ɬɨɛɬɨ

1 :

f

)

0 ,)1

0,

1

Mn ,

)1

1.

68

ɉɨɛɭɞɭɽɦɨ )2

0

)1

,

f

)2

2

f

)2 , f

)2

f

)0

2

D2

f

)0

2 .

Ɉɬɠɟ,

ɟɥɟɦɟɧɬ )

2 ɤɪɚɳɢɣ ɡɚ ɟɥɟɦɟɧɬ ɧɚɣɤɪɚɳɨɝɨ ɫɟɪɟɞɧɶɨɤɜɚɞɪɚɬɢɱɧɨɝɨ

ɧɚɛɥɢɠɟɧɧɹ )

0 .

ɰɟ ɫɭɩɟɪɟɱɧɿɫɬɶ.

ɇɚɫɥɿɞɨɤ

v ,

ɞɟ )0 Mn , ɚ

f

)0

v

Mn (

ɩɨɩɪɚɜɤɚ

v

ɡ ɨɪɬɨɝɨɧɚɥɶɧɨɝɨ

ɞɨɩɨɜɧɟɧɧɹ ɞɨ M n ).

Ɂɧɚɣɬɢ ȿɇɋɄɇ

)0

n

¦ciMi

(3)

ɨɡɧɚɱɚɽ ɡɧɚɣɬɢ ɤɨɟɮɿɰɿɽɧɬɢ ci .

i

0

Ⱦɥɹ ɜɢɤɨɧɚɧɧɹ (2) ɞɨɫɬɚɬɧɶɨ, ɳɨɛ

f

k

0 ,

0, k

0,n.

(4)

ɉɿɞɫɬɚɜɢɦɨ (3) ɭ ɮɨɪɦɭɥɭ (4):

·

§

n

¨ f

¦ciMi

,Mk ¸ 0.

©

i

0

¹

Ɍɚɤɢɦ ɱɢɧɨɦ ɦɚɽɦɨ ɋɅ

Ɋ ɞɥɹ ci :

n

¦ci (Mi ,Mk )

f ,Mk ,k

0,n

(5)

i

0

Ɂ ɬɟɨɪɟɦɢ 1 ɜɢɬɿɤɚɽ ɥɢɲɟ ɞɨɫɬɚɬɧɿɫɬɶ ɭɦɨɜ (5) ɞɥɹ ɡɧɚɯɨɞɠɟɧɧɹ

ɤɨɟɮɿɰɿɽɧɬɿɜ

ci .

Ɋɨɡɝɥɹɧɟɦɨ

ɡɚɞɚɱɭ

f

)0

inf

f

)

,

ɹɤ ɡɚɞɚɱɭ

ɦɿɧɿɦɿɡɚɰɿʀ ɮɭɧɤɰɿʀ ɛɚɝɚɬɶɨɯ ɡɦɿɧɧɢɯ:

M M n

n

2

F(a0,..,an )

f

)

2

f

¦aiMi

.

i

0

ɍɦɨɜɢ ɦɿɧɿɦɭɦɭ ɰɿɽʀ ɮɭɧɤɰɿʀ ɩɪɢɜɨɞɹɬɶ ɞɨ (5).

Ɂɚɞɚɱɚ 27 ɉɨɤɚɡɚɬɢ, ɳɨ ɞɥɹ ɤɨɟɮɿɰɿɽɧɬɿɜ ci

ɟɥɟɦɟɧɬɚ ɧɚɣɤɪɚɳɨɝɨ ɫɟɪɟɞɧɶɨ

ɤɜɚɞɪɚɬɢɱɧɨɝɨ ɧɚɛɥɢɠɟɧɧɹ ɭɦɨɜɢ (5) ɽ ɧɟɨɛɯɿɞɧɢɦɢ ɬɚ ɞɨɫɬɚɬɧɿɦɢ.

Ɇɚɬɪɢɰɹ ɋɅ Ɋ (5) ɫɤɥɚɞɚɽɬɶɫɹ

ɡ ɟɥɟɦɟɧɬɿɜ

gik

i , k

,

ɬɨɛɬɨ ɰɟ

ɦɚɬɪɢɰɹ

Ƚɪɚɦɦɚ:

G

gik

in,k

0 .

Ɉɫɤɿɥɶɤɢ

ɰɟ

ɦɚɬɪɢɰɹ

Ƚɪɚɦɦɚ

ɥɿɧɿɣɧɨ

ɧɟɡɚɥɟɠɧɨʀ ɫɢɫɬɟɦɢ, ɬɨ detG

0, ɳɨ ɳɟ ɪɚɡ ɞɨɜɨɞɢɬɶ ɿɫɧɭɜɚɧɧɹ ɬɚ ɽɞɢɧɿɫɬɶ

ȿɇɋɄɇ.

Ɉɫɤɿɥɶɤɢ

GT

G , ɬɨ ɞɥɹ ɪɨɡɜ’ɹɡɤɭ ɰɿɽʀ ɫɢɫɬɟɦɢ ɜɢɤɨɪɢɫɬɨɜɭɸɬɶ

ɦɟɬɨɞ ɤɜɚɞɪɚɬɧɢɯ ɤɨɪɟɧɿɜ.

əɤɳɨ ɜɡɹɬɢ x [0,1] ɬɚ Mi

xi , i

L2 (0,1) , ɬɨ

0,n , H

³

1

1

gik

0

xi xk dx

i k 1

, i,k 0,n.

ɐɟ ɦɚɬɪɢɰɹ Ƚɿɥɛɟɪɬɚ, ɹɤɚ ɽ ɩɨɝɚɧɨ ɨɛɭɦɨɜɥɟɧɨɸ: condG

107 ,

n

6. ɉɪɚɜɿ

ɱɚɫɬɢɧɢ

69

v - ɡɚɥɢɲɨɤ (ɩɨɯɢɛɤɚ). Ɍɚɤɢɦ ɱɢɧɨɦ ȿɇɋɄɇ ɽ ɜɿɞɪɿɡɤɨɦ

i 0

¦( fn ,Mi )Mi ,

¦n ( f ,Mi )Mi ,.

i 0

fk f ,Mk ³1	f (xi )xk dx,
0

ɹɤ ɩɪɚɜɢɥɨ, ɨɛɱɢɫɥɸɸɬɶɫɹ ɧɚɛɥɢɠɟɧɨ, ɬɨɦɭ ɩɨɯɢɛɤɢ ɨɛɱɢɫɥɟɧɧɹ ci ɦɨɠɭɬɶ ɛɭɬɢ ɜɟɥɢɤɢɦɢ.

ɓɨ ɪɨɛɢɬɢ? əɤɳɨ ɜɢɛɢɪɚɬɢ ɫɢɫɬɟɦɭ {Mi }if			0	ɨɪɬɨɧɨɪɦɨɜɚɧɨɸ, ɬɨɛɬɨ
	1,	i		k
(Mi ,Mk ) Gik	­
	®			,
	¯0,	i		k

ɬɨ ɫɢɫɬɟɦɚ (5) ɦɚɽ ɹɜɧɢɣ ɪɨɡɜ‘ɹɡɨɤ:

) (6)

0

əɤɳɨ ^ i ` – ɩɨɜɧɚ ɨɪɬɨɧɨɪɦɨɜɚɧɚ ɫɢɫɬɟɦɚ, ɬɨ ɞɨɜɿɥɶɧɭ ɮɭɧɤɰɿɸ ɦɨɠɧɚ ɩɪɟɞɫɬɚɜɢɬɢ ɭ ɜɢɝɥɹɞɿ ɪɹɞɭ Ɏɭɪ’ɽ:

	f )		f
ɿ		0 ¦ciMi
		i n 1
ɪɹɞɭ Ɏɭɪ’ɽ. Ⱦɚɥɿ
			f	)0				2

f 2

( f

f 2

)0

f

)0 , f

2)0 2

2 ¦f ci2 i 0

(7)

)0 )				f				2	2( f ,)0 )												)0				2
2(v,				)0 )									)0				2

¦		¦		n
0 ɡɚ ɬɟɨɪɟɦɨɸ1		f		o 0 .
n	ci2		ci2
i 0		i n 1			of

ɤɜɚɞɪɚɬ ɩɨɯɢɛɤɢ

Ɉɫɬɚɧɧɽ ɜɢɬɿɤɚɽ ɡ ɜɿɞɩɨɜɿɞɧɨʀ ɬɟɨɪɟɦɢ ɦɚɬɟɦɚɬɢɱɧɨɝɨ ɚɧɚɥɿɡɭ. Ɍɚɤɢɦ ɱɢɧɨɦ,

ɹɤɳɨ {Mi }if 0

- ɩɨɜɧɚ ɨɪɬɨɧɨɪɦɨɜɚɧɚ ɫɢɫɬɟɦɚ, ɬɨ

f

ci o0 ɬɚ

0

nof

o f .

¦

nof

Ɂɧɚɱɢɬɶ ɜɿɪɧɚ

i n 1

)0(n) ɩɨ

Ɍɟɨɪɟɦɚ 2 ȼ ɝɿɥɶɛɟɪɬɨɜɨɦɭ ɩɪɨɫɬɨɪɿ H

ɩɨɫɥɿɞɨɜɧɿɫɬɶ ȿɇɋɄɇ

ɩɨɜɧɿɣ ɨɪɬɨɧɨɪɦɨɜɚɧɿɣ ɫɢɫɬɟɦɿ {Mi}if 0 ɡɛɿɝɚɽɬɶɫɹ ɞɨ f .

Ɂɚɭɜɚɠɟɧɧɹ 1. ȼɿɞɯɢɥɟɧɧɹ ɦɨɠɧɚ ɨɛɱɢɫɥɢɬɢ ɡɚ ɮɨɪɦɭɥɨɸ:

n

'2 ( f )

f

)0

2

f

2

2( f ,)0 )

)0

2

f

2

)0

2

f

2

¦ci2 .

{Mi }if

i 0

əɤɳɨ

0

–

ɨɪɬɨɝɨɧɚɥɶɧɚ ɫɢɫɬɟɦɚ,

ɚɥɟ

ɧɟ

ɧɨɪɦɨɜɚɧɚ, ɬɨɛɬɨ

(Mi ,Mk )

Gik

Mi

2 , ɬɨ

70