ChM_teory2

Ɂɚɞɚɱɚ ɿɧɬɟɪɩɨɥɸɜɚɧɧɹ ɮɭɧɤɰɿʀ ɦɚɽ ɧɟ ɽɞɢɧɢɣ ɪɨɡɜ’ɹɡɨɤ. ȼɢɛɟɪɟɦɨ

ɫɢɫɬɟɦɭ ɥɿɧɿɣɧɨ ɧɟɡɚɥɟɠɧɢɯ ɮɭɧɤɰɿɣ {Mk

x }kn

0 , k x

C[a,b] ɿ ɩɨɛɭɞɭɽɦɨ

ɥɿɧɿɣɧɭ ɤɨɦɛɿɧɚɰɿɸ

) x )n x

¦ckMk x ,

(1)

n

k 0

n

¦ckMk xi

yi ,i

0,n,

(2)

k

0

ɪɨɡɜ’ɹɡɤɨɦ ɹɤɨʀ ɽ c&

c0 ,...,cn

. əɤɳɨ

x0

x0

0

...

n

D x0 ,..,xn

...

...

...

0 ,

M0 xn

...

Mn

xn

ɬɨ ɫɢɫɬɟɦɚ (2) ɦɚɽ ɽɞɢɧɢɣ ɪɨɡɜ’ɹɡɨɤ.

ɋɢɫɬɟɦɚ

ɮɭɧɤɰɿɣ

^

k x `

ɧɚɡɢɜɚɽɬɶɫɹ ɫɢɫɬɟɦɨɸ

ɑɟɛɢɲɨɜɚ,

k

0,n

ɹɤɳɨ, D x0 ,..,xn

0, {xi}in

0 , xi [a,b],i

i

0,n,

xi

xj

j .

ɉɪɢɤɥɚɞɢ ɫɢɫɬɟɦ ɑɟɛɢɲɨɜɚ.

1.

Mk x

xk

- ɚɥɝɟɛɪɚʀɱɧɚ ɫɢɫɬɟɦɚ.

ȼɢɡɧɚɱɧɢɤ D x0 ,.., xn

0 ɽ ɜɢɡɧɚɱɧɢɤɨɦ ȼɚɧɞɟɪɦɨɧɞɚ:

D x0 ,.., xn

x

0

...

xn

1

xk

xm

1

0

... ... ...

...

0.

k x

Lk x

1

x

n

...

xn 1

0dk

mdn

n

2.

-

ɨɪɬɨɝɨɧɚɥɶɧɿ ɛɚɝɚɬɨɱɥɟɧɢ Ʌɟɠɚɧɞɪɚ;

k

x

Tk

x -

ɨɪɬɨɝɨɧɚɥɶɧɿ ɛɚɝɚɬɨɱɥɟɧɢ ɑɟɛɢɲɨɜɚ.

3.

Mk x : 1,cos x,sin x,...,cosnx,sin nx .

)n x

Tn x

n

bk sin kx

Ɍɨɞɿ

a0 ¦ ak coskx

-

ɬɪɢɝɨɧɨɦɟɬɪɢɱɧɢɣ

ɛɚɝɚɬɨɱɥɟɧ.

k

1

6.2. ȱɧɬɟɪɩɨɥɹɰɿɣɧɚ ɮɨɪɦɭɥɚ Ʌɚɝɪɚɧɠɚ [ɋȽ, 127-129],[ȻɀɄ,38-42]

əɤɳɨ Mk x

)n x

Pn x

n

xk ,

ɬɨ

¦ck xk .

Ɂɚɞɚɱɚ

ɿɧɬɟɪɩɨɥɸɜɚɧɧɹ

ɮɭɧɤɰɿʀ f (x) ɚɥɝɟɛɪɚʀɱɧɢɦ,

k

0

ɛɚɝɚɬɨɱɥɟɧɨɦ

ɩɨɥɹɝɚɽ

ɜ

ɡɧɚɯɨɞɠɟɧɧɿ

ɤɨɟɮɿɰɿɽɧɬɿɜ ck , k

ɞɥɹ ɹɤɢɯ ɜɢɤɨɧɭɽɬɶɫɹ ɭɦɨɜɚ

f xi

xi

0,n

i

0,n .

ɉɪɟɞɫɬɚɜɢɦɨ ɿɧɬɟɪɩɨɥɹɰɿɣɧɢɣ ɛɚɝɚɬɨɱɥɟɧ ɭ ɜɢɝɥɹɞɿ

Pn x

Ln x

n

¦ f xk )(kn) x

(1)

k 0

Ɍɭɬ Ln x - ɿɧɬɟɪɩɨɥɹɰɿɣɧɚ ɩɨɥɿɧɨɦ;

)(kn)

x

-

ɩɨɥɿɧɨɦɢ n -ɝɨ ɫɬɟɩɟɧɹ, ɹɤɿ

ɧɚɡɢɜɚɸɬɶ ɦɧɨɠɧɢɤɚɦɢ Ʌɚɝɪɚɧɠɚ. Ɂ ɭɦɨɜɢ

Ln

xi

f xi

ɜɢɩɥɢɜɚɽ,

ɳɨ

ɦɧɨɠɧɢɤ Ʌɚɝɪɚɧɠɚ ɩɨɜɢɧɟɧ ɡɚɞɨɜɨɥɶɧɹɬɢ ɭɦɨɜɢ

)(kn)

xi

Gik .

(2)

Ɍɚɤ ɹɤ )(kn) x - ɛɚɝɚɬɨɱɥɟɧ ɫɬɟɩɟɧɹ n, ɬɨ ɜɿɧ ɦɚɽ ɜɢɝɥɹɞ

xn ,

)(kn) x

Ak

x

x0 ... x

xk

1

x

xk 1

... x

ɞɟ Ak –ɱɢɫɥɨ . Ɂɧɚɣɞɟɦɨ ɣɨɝɨ ɡ ɭɦɨɜɢ )(kn)

xk

1:

xn .

Ak

xk

x0 ... xk

xk

1 xk

xk 1 ... xk

1

Ɍɚɤɢɦ ɱɢɧɨɦ ɛɚɝɚɬɨɱɥɟɧ )(kn)

x ɦɚɸɬɶ ɜɢɝɥɹɞ:

xk

xk 1 ... xk

xn

(3)

)k

x

xk

x0 ... xk

xk

1

(n)

x

x0

... x

xk

1

x

xk 1

... x

xn

i 0

x xk Ȧcn xk

ɉɨɡɧɚɱɢɜɲɢ Zn x

n

, ɦɚɽɦɨ )(kn) x

x

x

xi

x

Ȧn

. Ɉɫɬɚɬɨɱɧɨ

ɮɨɪɦɭɥɚ Ʌɚɝɪɚɧɠɚ ɦɚɽ ɜɢɝɥɹɞ:

Ȧn

Ln x

n

¦ f xk

'

xk

(4)

k 0

x xk Ȧn

rn x

Ɂɚɦɿɧɸɸɱɢ

ɮɭɧɤɰɿɸ

f

x

ɧɚ

Ln

x

ɦɢ ɞɨɩɭɫɤɚɽɦɨ

ɩɨɯɢɛɤɭ

f x

Ln x . ɐɟ ɡɚɥɢɲɤɨɜɢɣ ɱɥɟɧ ɿɧɬɟɪɩɨɥɸɜɚɧɧɹ.

Ɂ

ɨɡɧɚɱɟɧɧɹ

ɜɢɩɥɢɜɚɽ,

ɳɨ, rn xi

0, xi

a,b@. Ɉɰɿɧɢɦɨ ɩɨɯɢɛɤɭ ɭ

ɤɨɠɧɿɣ ɬɨɱɰɿ x >a,b@. ȼɜɟɞɟɦɨ ɞɨɩɨɦɿɠɧɭ ɮɭɧɤɰɿɸ:

g t

f t

t

K n t , t

a,b@, g xi 0, i

.

Ln

0,n

Ɂɧɚɣɞɟɦɨ ɬɚɤɟK , ɳɨɛ g x

0, ɜ ɞɟɹɤɿɣ ɬɨɱɰɿ x

a,b@,

. Ʌɟɝɤɨ

x xi , i

0,n

ɛɚɱɢɬɢ, ɳɨ

Zn

x

K

f

x

Ln x

.

g t 0 ɜ

ɉɪɢɩɭɫɬɢɦɨ ɳɨ

f x Cn 1>a,b@, ɬɨɞɿ

g t Cn 1 a,b@.

Ɏɭɧɤɰɿɹ

(n+2)-ɯ ɬɨɱɤɚɯ, ɚ ɫɚɦɟ t

x ,

t

xi ,

i

0,n

. Ɂ ɬɟɨɪɟɦɢ Ɋɨɥɥɹ ɜɢɩɥɢɜɚɽ, ɳɨ

ɿɫɧɭɽ (n+1)-ɚ ɬɨɱɤɚ, ɞɟ

g

ti

0 ,

. ɉɪɨɞɨɜɠɭɸɱɢ ɰɟɣ

ɩɪɨɰɟɫ,

i

0,n

ɨɬɪɢɦɚɽɦɨ, ɳɨ ɿɫɧɭɽ ɯɨɱɚ ɛ ɨɞɧɚ

a,b

ɬɚɤɚ, ɳɨ

g(n 1) [

0.

Ɍɚɤ ɹɤ

g(n 1)

t

f (n 1) t

0 K

n

1 !, ɬɨ

, ɳɨ

ɛɭɞɟɦɨ ɞɨɫɥɿɞɠɭɜɚɬɢ ɰɟ ɩɢɬɚɧɧɹ.

Ȼɚɝɚɬɨɱɥɟɧɨɦ ɑɟɛɢɲɨɜɚ (n-ɬɨɝɨ ɫɬɟɩɟɧɹ ,

1–ɝɨ

ɪɨɞɭ)

ɧɚɡɢɜɚɽɬɶɫɹ

ɩɨɥɿɧɨɦ, ɹɤɢɣ ɡɚɞɚɽɬɶɫɹ ɬɚɤɢɦɢ ɪɟɤɭɪɟɧɬɧɢɦɢ ɫɩɿɜɜɿɞɧɨɲɟɧɧɹɦɢ:

Tn 1

x

2xTn

x

Tn 1 x

0

(1)

T0

x

1, T1

x

x .

(2)

Ɂɧɚɣɞɟɦɨ ɹɜɧɢɣ ɜɢɝɥɹɞ ɛɚɝɚɬɨɱɥɟɧɚ ɑɟɛɢɲɨɜɚ.

Ȼɭɞɟɦɨ ɲɭɤɚɬɢ

ɪɨɡɜ‘ɹɡɨɤ ɪɿɜɧɹɧɧɹ (1) ɭ ɜɢɝɥɹɞɿ Tn x

qn ,

ɞɟ q

q x

. ɉɿɞɫɬɚɜɢɜɲɢ ɜ (1),

ɨɬɪɢɦɭɽɦɨ

ɯɚɪɚɤɬɟɪɢɫɬɢɱɧɟ

ɪɿɜɧɹɧɧɹ

q2

2xq

1

0 .

Ɍɨɞɿ

ɩɪɢ

x t1 q1,2

x

x2

1, ɚ ɩɪɢ

x

1 q1,2

cos

isin

,

arccos x .

Ɋɨɡɝɥɹɧɟɦɨ ɨɛɢɞɜɚ ɜɢɩɚɞɤɢ ɞɟɬɚɥɶɧɿɲɟ:

ɚ) ɩɪɢ

x

d 1 Tn x

Acos n

Bsin n

. Ɂ (2) ɜɢɩɥɢɜɚɽ A=1, B=0 ɿ ɬɨɦɭ

Tn x

cos narccosx

(3)

ɛ) ɩɪɢ

x

1

Tn x

ª x

1 n x

1 n º .

1

x2

x2

(4)

2

«

»

¬

¼

Ɂɧɚɣɞɟɦɨ ɧɭɥɿ ɬɚ ɟɤɫɬɪɟɦɭɦɢ ɛɚɝɚɬɨɱɥɟɧɚ ɑɟɛɢɲɨɜɚ:

x

>

1,1@, cos narccosx

2k

1

Tn

0 , x

0, arccosx

, k

0,n

1.

2n

Ɉɬɠɟ ɧɭɥɿ ɛɚɝɚɬɨɱɥɟɧɚ ɑɟɛɢɲɨɜɚ:

[

2k

1

xk

cos

1,1], k

0,n

1.

2n

1,1@:

Ʌɨɤɚɥɶɧɿ ɟɤɫɬɪɟɦɭɦɢ ɛɚɝɚɬɨɱɥɟɧɚ ɑɟɛɢɲɨɜɚ ɧɚ x

xkc

k

cos

, k

0,n.

2n