numericna analiza

8/19/2019 Numericna analiza

1/412


2/412


3/412


4/412


5/412


6/412


7/412


8/412


9/412


10/412


11/412


12/412


13/412

f ∈ X

f ∞ := f ∞,[a,b ] := supa≤x≤b |f (x)|

f ∞ := f ∞,xxx := max1≤i≤N |f (x i )|, xxx = ( x i)N i=1 .

f

X

, f, g := ba f (x)g(x)ρ(x)dx, f, g ∈L 2ρ([a, b]), ρ > 0,

f, g :=N

i=1

f (x i )g(x i)ρ(x i ), ρ(x i ) > 0,

f, g + g, f = 2 f, g = 2 g, f α∈

R

αf,g = f,αg = α f, g .

f := f 2 := f, f = b

af (x)2ρ(x)dx,

f := f 2,xxx := f, f = N

i=1f (x i )2ρ(x i)

̃f

≈f

∈X

X f̃ S ⊂X S X

C ([a, b]) S


14/412

• P n { ≤n } • T n {

≤n } • R n,m { ≤n

≤m

}

• S 1,xxx { xxx = ( x i ) }

• S k,xxx { ≤k xxx = ( x i )

k −1 }• P k,xxx,ν ν ν { ≤k

xxx = ( x i ) ν ν ν = ( ν i )

}

̃f ∈ S f f ∈ X

f − f̃ = inf s∈S f −s =: dist ( f, S ) .

f S f − f̃ f S

dist ∞ (f, S ) ̃f

f ∈ X f ∈X

λ i : X → R f

• f xi λi f := f (x i )

• r f x

i

λ i f := f ( )(x i ) = 0 , 1, . . . , r −1 • f λif :=

1β i −α i β iα i f (x)dx

• f λi f := ba f (x)s i (x)dx si • λif := N j =1 f (x j )s i (x j ) ̃f ∈ S S

̃

f ∈ S

λi ̃f f

λ i f̃ = λ i f, za vse i.


15/412

̃f f f ∈ X

f 1 ≤ c < ∞

̃f

∈S f

∈X

f − f̃ ≤c dist ( f, S ) f

S n S = S n

(S n ) n

f̃ f

̃f ∈S S L( )

≤n P n = L x i

ni=0 ,

P n = L({ i}ni=0 ), i (x) :=n

j =0 j = i

x −x jx i −x j

, x0 < x 1 < · · ·< x n ,

i (x j ) = δ i,j

P n = L({(x −x0)(x −x1) · · ·(x −x i−1)}ni=0 ), x0 ≤x1 ≤ · · · ≤xn .

x 0 x 1 x 2 x 3 x 4

x

1

x 0 x 1 x 2 x 3 x 4

x

1

n = 4


16/412

δ i,j i j 0 T n

a0 +n

k=1

(ak coskx + bk sin kx) .

S 1,xxx

x0 < x 1 < · · ·< x n

1, (x −x i)+ , i = 0 , 1, . . . , n −1. n∈

N

zn+ := max ( z, 0)n

. 0

z0+ := 0, z < 0,1, z > 0,

z = 0

S 1,xxx

H i (x) :=

x

−x i

−1

x i −x i−1 , xi−1 < x < x i , 0 < i,1, x = xi ,x i+1 −xx i+1 −x i

, xi < x < x i+1 , i < n,

0, sicer,

, i = 0 , 1, . . . , n ,

x 1 x 2 x 3 x 4

x

1

x 1 x 2 x 3 x 4

x

1


17/412

H i supp H i H i [x i−1, x i+1 ]

f ∈ S sss := ( s i) S ααα := ( α i ) f = i α is i δf = i δα i s i δαδαδα := ( δα i )

S

f =i

α i s i , f + δf =i

(α i + δα i )s i

c = c(sss)

δf f ≤ c(sss)

δαδαδαααα

δf f

= δf δαδαδα ·

αααf ·

δαδαδαααα

c(sss) = M m

,

M := supδαδαδα =0

δf δαδαδα

= supδαδαδα =0

i δα is iδαδαδα

= supδαδαδα =1 i

δα is i ,

m := inf ααα =0

f ααα

= inf ααα =1

iα i s i .

c(sss) ≥ 1

= ∞ M = sup

ααα ∞ =1 iα iH i ∞ ≤ supααα ∞ =1 i |

α i|H i ∞ ≤i

H i ∞ = 1 ,

i H i [x i−1, x i]

m = inf ααα ∞ =1 i

α iH i∞ ≥

inf ααα ∞ =1

max j | i

α iH i (x j )

|=

= inf ααα ∞ =1

max j |

iα i δ i,j | = inf ααα ∞ =1 max j |α j | = 1 .


18/412


19/412

n

Bn p2(x) =n

i=0

in

2 ni

x i (1 −x)n−i =n

i=1

in

2 ni

x i (1 −x)n−i =

=n

i=1

in

(n −1)!(i −1)! (n −i)!

x i (1 −x)n−i =

= xn−1

i=0

i + 1n

n −1i

x i (1 −x)n−1−i =

= xn

((n −1) Bn−1 p1(x) + Bn−1 p0(x)) =

=

n

−1

n x2

+

1

n x = x2

+

1

n x(1 −x) = x2

+ O1

n .

Bn f f x∈[0, 1]

f (x) = 1 · f (x) 1 Bn p0

|f (x) −Bn f (x)| =n

i=0f (x) − f

in

ni

x i (1 −x)n−i ≤

≤

n

i=0f (x)

− f

i

n

n

ix i (1

−x)n−i .

i in x f

I 1 := i | 0 ≤ i ≤n,in −x <

14√ n , I 2 := {0, 1, . . . , n }\I 1.

i∈ I 1

i∈I 1f (x)

− f

i

n

n

ix i (1

−x)n−i

≤≤ω f ;

14√ n

i∈I 1ni

x i (1 −x)n−i ≤

≤ω f ; 1

4√ n n

i=0

ni

x i (1 −x)n−i = ω f ; 14√ n .

ω (f ; h) [a, b]

ω (f ; h) := max|x−y|≤hx, y ∈[a, b]

|f (x) −f (y)|,


20/412

Ω x ;h

Ω x;h

Ω x;h

0.002 0.004 0.006 0.008 0.010h

0.02

0.04

0.06

0.08

0.10

Ω

ω (√ x; h) ω(ex ; h) ω (x; h) [0, 1] h

h ↓0 i∈ I 2

in −x ≥

14√ n =⇒

(i − nx)2n√ n ≥ 1,

i∈I 2f (x) − f

in

ni

x i (1 −x)n−i ≤

≤2 f ∞i∈I 2

ni

x i (1 −x)n−i ≤

≤2 f ∞i∈I 2

(i − nx)2n√ n

ni

x i (1 −x)n−i ≤

≤2 f ∞√ nn

i=0

in

2

−2x in + x2 n

ix i (1 −x)n−i =

= 2 f ∞√ nn −1

n x2 +

1n

x −2x2 + x2 == 2 f ∞

x(1 −x)√ n ≤ 12√ n f ∞.

f −Bn f ∞ ≤ω f ; 14√ n + 12√ n f ∞


21/412

n f (x) = sin5 x f (x) = |4x −2| −1

[0, 1] c nα α

f −Bn f ∞ ≈ max0≤i≤N |f (x i ) −Bn f (x i )| =: en = c nα + O (nα ) .

α n, m = n + 2

α ≈ln enem

ln nm .

O 1n O 1√ n

0.2 0.4 0.6 0.8 1.0 x

1.0

0.5

0.5

1.0

0.2 0.4 0.6 0.8 1.0 x

1.0

0.5

0.5

1.0

f (x) = sin5 x f (x) = |4x −2| −1

f ∈C 2([a, b]) f −Bn f ∞ = O

1n2

.


22/412

n α

n α

x = cos θ

T n = a0 +n

k=1

(ak coskθ + bk sin kθ) .

Bn : C ([0, 1]) →P n •

• f ≥0 =⇒Bn f ≥0 • Bn 1 = 1 , Bn x = x, Bn x2 →x2 n → ∞

C ([a, b])

(Ln ) Ln : C ([a, b]) →C ([a, b])

f

−Ln f ∞

→0, n

→ ∞,

f = p0 p1 p2 pi (x) := xi f ∈C ([a, b]) Ln f ≥ g f −g ≥0 Ln (f −g) ≥0 Ln f ≥Ln g

Ln |f | ≥ |Ln f |,

|f | ≥f, |f | ≥ −f, L n |f | ≥Ln f, L n |f | ≥ −Ln f.

∆ n pi := Ln pi − pi , i = 0 , 1, 2, 0 n

f ∈C ([a, b]) ε > 0 f


23/412

[a, b] ε > 0 δ > 0

|x −y| < δ =⇒ |f (x) −f (y)| < ε, x, y ∈[a, b]. ε > 0 δ

c := 2f ∞δ 2

.

x, y ∈[a, b], |x −y| ≥δ |f (x) −f (y)| ≤2 f ∞ ≤2 f ∞

x −yδ

2

= c (x −y)2 . x, y ∈[a, b]

|f (x) −f (y)| ≤ε + c(x −y)2. y x ∈ [a, b]

f pi

|f −f (y) p0| ≤ε p0 + c p2 −2yp1 + y2 p0 .

|Ln f −f (y)Ln p0| ≤Ln |f −f (y) p0| ≤≤εLn p0 + c(Ln p2 −2yLn p1 + y2Ln p0).

x∈[a, b] x = y

|Ln f (y) −f (y)Ln p0(y)| ≤≤ εLn p0(y) + c Ln p2(y) −2yLn p1(y) + y2Ln p0(y) == ε (1 + ∆ n p0(y)) +

+ c ∆ n p2(y) + y2 −2y (y + ∆ n p1(y)) + y2(1 + ∆ n p0(y) == ε (1 + ∆ n p0(y)) + c ∆ n p2(y) −2y∆ n p1(y) + y2∆ n p0(y) ≤≤ ε (1 + ∆ n p0 ∞) + c ( ∆ n p2 ∞ + 2 p1 ∞ ∆ n p1 ∞ + p2 ∞ ∆ n p0 ∞) .

∆ n pi ∞ →0 n Ln f −fL n p0 ∞ ≤2ε

Ln f −f ∞ ≤ Ln f −fL n p0 ∞ + fL n p0 −fp 0 ∞ ≤2ε + f ∞ ∆ n p0 ∞. n f ∞ ∆ n p0 ∞ < ε,

Ln f −f ∞ < 3 ε. Bn

S 1,xxx xxx = ( x i )ni=0

a := x0 < x 1 < · · ·< x n =: b.


24/412

∆ x i := xi+1 −x i > 0

∆ x := max0≤i≤n−1 ∆ x i . n

I 1,xxx : C ([0, 1]) →S 1,xxx : f →n

i=0

f (x i)H i ,

H i I 1,xxx H i I 1,xxx

x ∈ [a, b]

i

x i ≤ x ≤ xi+1 I 1,xxx f f x i xi+1 f I 1,xxx f

I 1,xxx p2 − p2, p2(x) := x2 i x i ≤x ≤x i+1 H i H i+1

|I 1,xxx p2(x) − p2(x)| = x2i H i (x) + x2i+1 H i+1 (x) −x2 == x

2i

x i+1

−x

∆ x i + x2i+1

x

−x i

∆ x i −x2

=

=1

∆ x ix2i x i+1 −x2i+1 x i + x(x2i+1 −x2i ) −x2 =

= −x2 + x(x i + x i+1 ) −x i x i+1 == ( x −x i)(x i+1 −x) ≤≤ (

x i + x i+12 −x i)(x i+1 −

xi + x i+12

) = ∆ x2i

4 ≤ ∆ x2

4 .

i

∆ x (S 1,xxx )∆ x→0

C ([0, 1]) H i

S

bn,i (x) :=n

ix i (1

−x)n−i .

[0, 1] n → ∞


25/412

bn,i (bn,i )ni=0 P n [0, 1]

k Bn f f

f [0, 1] c > 0

|f (x) −f (y)| ≤c|x −y|, x, y ∈[0, 1].

|f (x) −Bn f (x)| ≤ c2√ n , x∈[0, 1],

Bn f ≤n

Bn f ≥g Bn f ≥Bn g f ∈ C 1([0, 1]) (Bn f )

f n → ∞

f −Bn f ∞ f (x) = √ x, x√ x, x 2√ x

f

f −Bn f ∞ = O 1n2

.

f ∈ C ([a, b])

f c =⇒ ω(f ; h) ≤c h f =⇒ ω(f ; h) ≤ f ∞h lim

h↓0ω (f ; h) = 0

h < k =⇒ω (f ; h) ≤ω (f ; k)

ω (f ; h + k) ≤ω (f ; h) + ω (f ; k) ω(f, h ) = O (h) f

f (x) := √ xα 0 ≤ α ≤ 1 f ∈ C ([0, 1]) ω (f ; h) h

f [0, 1]

Bn f ≥Bn +1 f.


26/412

Bn f (0, 1) f (0, 1)

P (u) = nk=0 bkuk (0, ∞) (bk )n

k=0

f ∈ C ([0, 1]) n ∈ N I 1f

f (0) , f 1n

, . . . , f n −1

n, f (1) .

p∈P 1 Bn f

[0, 1] I 1f

f

∈ C ([0, 1]) n

∈ N

Bn f [0, 1] in

, f in

, i = 0 , 1, . . . , n .


27/412

X

S ⊂ X f ∈ X S f ∗ ∈S

f −f ∗ ≤f −s , za vsak s∈S.

f ∗

X = R 3, S = R 2 (f 1, f 2, f 3) 2 = f 21 + f 22 + f 23 .

f ∗ f R 2 f −f ∗

R 2

f f f

f

R2

R3

R 3

R 2

X xxx = ( x i)N i=1 f, g = N i=1 f (x i )g(x i) 2 S P 1 f ∗ ∈ P 1 f ∈ X

f −f ∗22 =N

i=1(f (x i ) −f ∗(x i)) 2 .

f ∗ f


28/412

x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7

x

X = L 2 ([a, b]) S = P n p∗

b

a

(f (x)

− p∗(x)) 2 dx

≤ b

a

(f (x)

− p(x))2 dx, p

∈

P n ,

f

X = C ([a, b]) ∞ p∗

maxa≤x≤b |f (x) − p∗(x)| ≤ maxa≤x≤b |f (x) − p(x)|,

p∈P n ,

f

n = 5

f (x) = ex [−1, 1]

f − p∗2 = 0 .0000391087, f − p∗∞ = 0 .000107613,

f − p∗2 = 0 .0000450258, f − p∗∞ = 0 .0000454396.

2 0.0000391087 < 0.0000450258

∞


29/412

1.0 0.5 0.5 1.0 x

0.00004

0.00002

0.00002

0.000040.00006

0.00008

0.0001

napaka

ex [−1, 1]

S

X S ⊂ X f ∈ X f ∗

∈S

f ∈X K := K (f ; f ) f f inf s∈S

f −s ≤ f −0 = f , f 0∈S

s ∈ S K E := S ∩K E S

g(s) := f −s S

|g(s + h) −g(s)| = | f −(s + h) −f −s | ≤ h . g E S

f ∗ ∈S f −f ∗ = inf s∈ S f −s = dist( f, S ).


30/412

X K ⊂ X u, v ∈K

λu + (1 −λ)v , λ∈[0, 1], K K

K

X ε > 0 δ > 0 f, g ∈X f = g = 1

12

(f + g) > 1 −δ =⇒ f −g < ε.

X

f + g 2 + f −g 2 = 2( f 2 + g 2), f, g ∈X. f,g, f = g = 1 δ > 0

12 (f + g) > 1 −δ

f −g 2 = 4 −4 12 (f + g)2

≤ 4 1 −(1 −δ )2= 4 1 −1 + 2 δ −δ 2 < 8δ.

ε > 0 δ = δ (ε) = 18 ε2

S ⊂X X f ∈ X

f ∗

∈S

f −f ∗ = inf s∈ S f −s .

f ∈X f ∗ ∈S S := S −f S f ∈ X

f ∗

d := inf s∈S

s .

d = 0 S (sn ) , s n ∈S

0 = limn→∞

sn = limn→∞sn ,


31/412

limn→∞

sn = 0 =⇒0∈S ,

S f

f ∗ = f d > 0 d S := 1d S ,

inf s∈S

s = 1 , s ≥1, s∈S .

(sn ) , sn ∈S , limn→∞sn = 1 . ε > 0 δ

s̃n := 1

snsn .

sn −s̃n = sn − 1sn

sn = 1 − 1sn sn →0, ko n → ∞,

n0 n ≥n0 sn −s̃n < δ m, n ≥n0 S 12 (sm + sn )∈

S

12 (sm + sn )

≥ 1

12

(s̃m + s̃n ) = 1

2 (sm + sn ) −(sm −s̃m ) −(sn −s̃n )

≥ 1

2sm + sn −

12

sm −s̃m − 12

sn −s̃n> 1−

12

δ − 12

δ = 1 −δ. s̃m −s̃n < ε (s̃n )

X

s̃ = limn→∞s̃n , s̃∈X.

sn −s̃ ≤sn −s̃n + s̃n −s̃ →0, ko n → ∞. s̃ (sn ) S s̃ ∈ S

S X f ∈X f ∗ ∈S

S X


32/412

S ⊂ X f ∗ ∈ S f̃ ∈ S f ∈X

f ∗−f = f̃ −f ≤s −f , s∈S. λ 0 ≤λ ≤1 λf ∗ + (1 −λ) f̃ f ∗

f̃

(λf ∗ + (1 −λ) f̃ ) −f = λ(f ∗−f ) + (1 −λ)( f̃ −f ) ≤≤ λ f ∗−f + (1 −λ) f̃ −f = f ∗−f .

λf ∗ + (1 −λ) f̃ f

X f, g ∈X, g = 0

f + g = f + g , f = λ g λ

X S ⊂X S f ∈X

f ∗ ∈S f̃ ∈S f ∈S f −f ∗= f − f̃ ≤ f −s , s∈S.

f ∈S f 0 = f −f = f −f ∗= f − f̃

f = f ∗

= f̃ f /∈ S

12 f ∗ + f̃

12 (f −f ∗) 12 f − f̃

f −f ∗ = f − 12

f ∗ + f̃ =12

f −f ∗ + 12

f − f̃ ≤

≤12

f −f ∗ +12

f − f̃ = f −f ∗. X λ

12

f −f ∗ = λ2

f − f̃ .


33/412

λ = 1 λ = 1

f = 11 −λ

f ∗− λ1 −λ

f̃ ∈S. f /

∈

S

x, y = x1y1 + x2y2 x, y = 0 .8x1y1 + 1 .3x2y2 x, y = 1 .2x1y1 + 0 .9x2y2 x ∞ ≤ 1

x 1 ≤ 1

1 1

1

1

1 1

1

1

R 2

∞ C ([0, 1])

f (x) := 32 x, 0 ≤x < 13 ,12 ,

13 ≤x ≤1,

g(x) := f (1 −x) f + g ∞ = max0≤x≤1 |f (x) + g(x)| = max13 ≤x≤23 |

f (x) + g(x)| = 1 = f ∞ + g ∞, f g

X


34/412

S

X = R 2 ∞ f = 12 , 1

S = {α (1, 0) |α∈R }.

mins∈ S

f −s = minα∈R max12 −α , |1 −0|

= min

|12

−α

|≤1

max12 −α , 1 = 1

S

f

f

1 1

1

1S

f f

1 1

1

1

{12 , 1}

dist( f, S ) = 1

α(1, 0)

12 −α ≤ 1

f

S = {α (1, 1) |α∈R }.

mins∈ S

f −s = minα∈R max12 −α , |1 −α |

= minα∈R34 −α +

14 =

14

34 (1, 1) f


35/412

X S ⊂ X f ∈X S

(sk )k∈

N , sk

∈S,

f ∈X lim

k→∞f −sk = inf s∈S f −s .

(sk )k∈N f ∈ X s∗ ∈ S

s∗ f ∈ X S

X S ⊂ X f

∈X f ∗

∈S

S ⊂ X X f ∈X

f ∗ ∈S

R n 1

X = R 3

f = (1 , 3, 2)∈X

L({(1, 0, 0), (0, 1, 0)})

X

X 0

X = xxx = ( x1, x2, . . . ) | limn→∞xn = 0 , xxx ∞ = maxn∈N |xn |.

A = xxx∈X |∞

n =12−

nxn = 0

X

xxx∈X \ A xxx∈A


36/412

∞ f ∈ X = C ([a, b])

S ⊂ C ([a, b])

f ∈ C ([a, b]) f ∗ ∈ S C ([a, b])

S

S S = P n S

X = C ([a, b]) S = P n f ∈ C ([a, b]) p∗ ∈P n p∗

E ⊂[a, b] E

E ⊂ [a, b]

f

E

f ∞,E = maxx∈E |f (x)| f

E P n

M n (E ; f ) := min p∈

P nf − p ∞,E = min p∈P n maxx∈E |f (x) − p(x)| =: dist ∞,E (f,

P n ) .

p∗ ∈P n f

r := f

− p∗

n + 1

E = {x i |a ≤x0 < x 1 < .. . < x n ≤b}.

p∗ =n

i=0f (x i ) i ,

( i)ni=0

P n

p∗(x j ) =n

i=0f (x i ) i (x j ) =

n

i=0f (x i )δ i,j = f (x j ), j = 0 , 1, . . . , n .


37/412

p∗ f E M n (E ; f ) = 0 p∗, ˜ p ∈ P n

f xi ˜ p(x i) = f (x i) = p∗(x i) i = 0 , 1, . . . , n ˜ p(x i ) − p∗(x i ) = 0 i = 0 , 1, . . . , n ˜ p − p∗ n n + 1 ˜ p = p∗ p∗

E

E = {x i |a ≤x0 < x 1 < . . . < x n < x n +1 ≤b},

f (x) = ex n = 1

E = {a = x0 = 0 , x1 = 12

, b = x2 = 1} α β

M 1(E ; f )

r(x) = f (x) − (αx + β )

0.2 0.4 0.6 0.8 1.0 x

0.4

0.2

0.2

0.4

residual

r (x) = ex −(αx + β ) α ∈ (−∞, 1] α∈[1, e] α∈[e, ∞) α ≤ 1 α ≥ e

β

r (0) = −r (1) , β = 12

(−α + e + 1) . β α ≤1 r r (0) =

12

(1 + α −e) ≤1 − 12

e < 0


38/412

(α, β ) = 1, 12 e α ≥e r (α, β ) = e, 12

r (0) = 12 1 < α < e r (0) α β α max |r ( 12 )|, |r (1) |

α

r12

= −r (1) , α = 23 −2β + e + √ e .

α max( |r (0) |, |r (1)|) |r ( 12 )| = |r (1) | β

β = 14

3 + 2√ e −e α = e −1. r α β

r (0) = −r12

= r (1) = 14

√ e −12 .

E M n (E ; f )

r = f − p p ∈ P n x∈E m := max

x∈E |f (x)

− p(x)

|= max

0≤i≤n +1 |f (x

i)−

p(xi)|.

r xi

r (x i ) = f (x i) − p(x i ) = ( −1)i u i m. u i |u i| ≤1

p m

p(x) =n

i=0a i x i .

u0m + a0 + a1x0 + · · · + an xn0 = f (x0)−u1m + a0 + a1x1 + · · · + an xn1 = f (x1)

· · ·

(−1)n +1 un +1 m + a0 + a1xn +1 + · · · + an xnn +1 = f (xn +1 ).

u i

u0 1 x0 . . . xn0

−u1 1 x1 . . . xn1

(−1)n +1 un +1 1 xn +1 · · · xnn +1

m

a0

an

=

f (x0)

f (x1)

f (xn +1 )

.


39/412

A ∈ R n +2 ,n +2 A(−)i,j A

i j

D i := det A(−)i+1 ,1, i = 0 , 1, . . . , n + 1 .

D i

D i = V (x0, x1, . . . , x i−1, x i+1 , x i+2 , . . . , x n +1 ) .

V (y0, y1, . . . , , y n ) =

1 y0 . . . yn01 y1 . . . yn1

1 yn · · · ynn=

n

i=0

i−1

j =0(yi −y j )

x j ∈ E

Di > 0

i

m

m = |m| =D0f (x0) −D1f (x1) + · · ·+ ( −1)n +1 Dn +1 f (xn +1 )

u0D0 + u1D1 + · · ·+ un +1 Dn +1 ≥

≥

n +1

i=0(−1)iD if (x i )n +1

i=0 |u i |D i≥

n +1

i=0(−1)i D i f (x i )

n +1

i=0D i

.

m u i 1 −1

A det A = ±n +1

i=0D i = 0

f E u i = ε = ±1 ε i

m m = |m| = M n (E ; f ) E

f (x i )

− p∗(x i) = ε(

−1)iM n (E ; f ), i = 0 , 1, . . . , n .

f ∈ C ([a, b]) E = {x i |x0 < x 1 < . . . < x n +1 }⊂[a, b] f E

ε m = M n (E ; f ) > 0

E

M n (E ; f )

f ∈ C ([a, b]) E = {x i | x0 < x 1 < . . . < x n +1 } ⊂ [a, b] q ∈ P n r = f −q


40/412

E |r (x i )| min

0≤i≤n +1 |r (x i)| < M n (E ; f ) < max0≤i≤n +1 |r (x i)| .

M n (E ; f ) = min p∈

P nmaxx∈E

|f (x) − p(x)| == min

p∈P n

maxx∈E

|(f (x) −q (x)) −( p(x) −q (x)) ˜ p(x) |=

= min˜ p∈

P nmaxx∈E

|(f (x) −q (x)) − ˜ p(x)| = M n (E ; f −q ). M n (E ; f −q )

M n (E ; f −q ) =n +1

i=0 (−1)iD i (f (x i) −q (x i ))

n +1

i=0D i

=

n +1

i=0 D i |r (x i )|n +1

i=0D i

<

< max0≤i≤n +1 |r (x i )|.

|r (x i )| M n (E ; f ) < max

0≤i≤n +1 |r (x i )|

E

|E | = n + 2

f ∈C ([a, b]) P n G⊂[a, b]

|G| = m ≥n + 2 , m < ∞, E ⊂G |E | = n + 2

M n (E ; f ) ≥M n (E ; f ) E ⊂G, |E | = n + 2 . f G

f E

G E

M n (E ; f ) = M n (G; f ).

E ⊂G maxx∈E |

f (x)

− p(x)

| ≤maxx∈G |

f (x)

− p(x)

|, p

∈

P n

.

p G M n (E ; f ) ≤ M n (G; f )


41/412

p∗ f E r

M n (E ; f ) = maxx∈E

|f (x) − p∗(x)| ≥maxx∈G |f (x) − p∗(x)|(≥M n (G; f )!) y ∈G

y /∈E

|f (y) − p∗(y)| = |r (y)| > M n (E ; f ) = |r (x i)|.

i 0 ≤ i ≤n xi < y < x i+1 a ≤y < x 0 y xi xn +1 < y ≤b y xi

xi y xi 1r xi r y

r xi 1 x

xi y xi 1r xi

r y r xi 1

x

x i < y < x i+1 j = i

j = i + 1

y x 0 x 1r y r x 0

r x 1 x

y

x 0 x1

r y

r x 0

r x1 x

a ≤y < x 0 j = 0 j = n + 1

j r E E = E \ {x j }∪ {y}

j ∈ {i, i + 1} j ∈ {0, n + 1} E

M n (E ; f ) = |r (x i )| = minx∈E |r (x)| < M n (E ; f ), E


42/412

G p∗ ˜ p f G

M n (E ; f ) = maxx∈E

|f (x) − p∗(x)| ≤maxx∈E |f (x) − ˜ p(x)| ≤maxx∈G |f (x) − ˜ p(x)| =

= M n (G; f ) ˜ p f E

˜ p = p∗

G →[a, b] f ∈ C ([a, b]) P n

f [a, b] f E ⊂[a, b] E |E | = n + 2

M n (E ; f ) ≥M n (E ; f ) E ⊂[a, b] |E | = n + 2

M n ([a, b]; f ) = M n (E ; f ) .

E

G E

E = {x i |a ≤x0 < x 1 < . . . < x n +1 ≤b} n + 2

x i Ξ : E → Ξ(E ) = xxx = ( x i )n +1i=0

K = {(x i )n +1i=0 | a ≤x0 ≤x1 ≤. . . ≤xn +1 ≤b}⊂R n +2 xxx → E := Ξ −1(xxx)

K E

m(xxx) := M n (Ξ−1(xxx); f ) =

n +1

i=0(−1)i D i f (x i)

n +1

i=0D i

,

D i

K R n +2 m K m(xxx) := 0 ,

xi = xi+1

i, 0 ≤ i ≤n. m K

K m xi


44/412

p∗(x i) − p(x i) = ( f (x i ) − p(x i )) −(f (x i ) − p∗(x i)) sgn(f (x i)− p(x i))( p∗(x i ) − p(x i)) =

= |f (x i ) − p(x i )| −sgn(f (x i ) − p(x i ))( f (x i) − p∗(x i )) ≥≥m − f − p∗∞ > 0

sgn(f (x i )− p(x i ))( p∗(x i+1 ) − p(x i+1 )) == sgn( f (x i ) − p(x i ))( f (x i+1 ) − p(x i+1 ))−

−sgn(f (x i) − p(x i ))( f (x i+1 ) − p∗(x i+1 )) ≤≤ −m + f − p∗∞ < 0 .

xi xi 1

f p

f p

p p

x xi xi 1

f p

f p

p p

x

f − p f − p∗

p∗− p ≤n x i xi+1 p∗− p∈P n

(x i , x i+1 ), i = 0 , 1, . . . , n p∗− p ≡0 f (x) = xn p∗ ∈P n−1

[−1, 1] x = cos θ θ ∈ [0, π ] θ = arccos x [0, π ] → [−1, 1] T n

T n (x) := cos( nθ) = cos( n arccos x) .

T n ≤n x cos(nθ) =

1

2e−inθ + einθ = 1

2(e−iθ )n + ( eiθ )n

e±iθ = cos θ ±i sin θ = x ±i 1 −x2 = x ± x2 −1


45/412

cos(nθ) = 12

x − x2 −1 n + x + x2 −1 n ==

12

n

j =0

n j (−1)

n

− j

x j

(x2

−1)n − j

2 +

n

j =0

n j x

j(x

2

−1)n − j

2 .

12 1 + ( −1)n− j = 0 n − j 1

cos(nθ) =n

j =0n− j sod

n j

x j (x2 −1)n − j

2 ∈P n .

T n

12

n

j =0

n j

+ 12

n

j =0

n j

(−1)n− j = 12

((1 + 1) n + (1 −1)n ) = 2 n−1 = 0 .

T n n

T 0

T 1

T 2

T 3T 4

1.0 0.5 0.5 1.0 x

1.0

0.5

0.5

1.0

T i

T n


46/412

xk = cos θk = cos kπ

n , k = 0 , 1, . . . , n ,

T n (xk ) = cos( n arccos xk ) = cos( kπ ) = (

−1)k , k = 0 , 1, . . . , n .

|T n | ≤ 1 T n ∈ C ([−1, 1]) T n ∞ = 1 n + 1 xk

2−n +1 T n (x) = xn −. . . f (x) = xn P n−1 p∗ = f −2−n +1 T n ∈ P n−1

f n 2−n +1 T n [−1, 1]

dist ∞ (xn , P n−1) = 2 −

n +1

P n

yi ∈[−1, 1] ω(x) = ( x −y1) · · ·(x −yn ) ω ∞ [−1, 1]

ω = 2−n +1 T n T n

cos k + 12

πn

, k = 0 , 1, . . . , n −1.

[a, b]

f ∈ C ([a, b]) p ∈ P n r = f − p E = {x i | a ≤ x0 < x 1 M n ([a, b]; f ). p∗− p = ( f − p) −(f − p∗)

[a, b] E ⊂ [a, b] u i = ε =

±1

n +2 E E M n (E ; f )


47/412

p∗ ∈ P n f ∈C ([a, b]) ε > 0 E 0 :=

{x i |

a

≤x0 < x 1 < . . . < x n +1

≤b

}

k := 0

p∗k f E k

y ∈[a, b], |f (y) − p∗k (y)| = f − p∗k ∞ |f (y) − p∗k (y)| −M n (E k ; f ) < ε

x j ∈E k y r k

E k+1 := E k

\ {x j

} ∪ {y

} j

k

ε

f ∈C ([a, b]) ( p∗k )k≥0 p∗

f [a, b] c > 0 0 < θ < 1

0 ≤ f − p∗k ∞−M n ([a, b];f ) ≤c θk .

f − p∗k ∞ →f − p∗∞ p∗k − p∗∞ →0, k → ∞. M n (E k ; f )

E k+1 = {yi | y0 < y1 < . . . < y n +1 } := E k \ {x j } ∪ {y} rk := f − p∗k

M n (E k+1 ; f ) = M n (E k+1 ; r k ) =

n +1

i=0 D i |r k (yi )|n +1

i=0D i

=n +1

i=0θ(k)i |r k (yi)| ≥

≥ min0≤i≤n +1 |r k (yi )|n +1

i=0θ(k)i = min0≤i≤n +1 |r k (yi )| = M n (E k ; f ) ,

θ(k)i := D in +1

j =0D j

D i

E k+1 n +1

i=0θ(k)i = 1 0 < θ

(k)i < 1


48/412

θ 0 < θ < 1 θ(k)i 0 θ(k)i > 1 −θ > 0 i k

M n (E k+1 ; f ) −M n (E k ; f ) =n +1

i=0θ

(k)i |r k (yi )| −

n +1

i=0θ

(k)i M n (E k ; f ) =

=n +1

i=0θ(k)i (|r k (yi)| −M n (E k ; f )) ≥

≥ (1 −θ) ( |r k (y)| −M n (E k ; f )) ≥ ≥ (1 −θ) (M n ([a, b]; f ) −M n (E k ; f )) ≥0.

0 ≤M n ([a, b]; f ) −M n (E k+1 ; f ) == ( M n ([a, b]; f ) −M n (E k ; f )) −(M n (E k+1 ; f ) −M n (E k ; f )) ≤≤ (M n ([a, b]; f ) −M n (E k ; f ))(1 −(1 −θ)) == θ (M n ([a, b]; f ) −M n (E k ; f )) .

k

0 ≤M n ([a, b]; f ) −M n (E k ; f ) ≤θk (M n ([a, b]; f ) −M n (E 0; f )) . M n (E k ; f ) → M n ([a, b];f ) k → ∞ p∗k → p∗

k → ∞

|r k (y)| −M n ([a, b]; f ) ≤ |r k (y)| −M n (E k ; f ) ≤≤

11 −θ

(M n (E k+1 ; f ) −M n (E k ; f )) ≤≤

11 −θ

(M n ([a, b]; f ) −M n (E k ; f )) ≤≤

θk

1 −θ (M n ([a, b]; f ) −M n (E 0; f )) .

c

c = 11 −θ

(M n ([a, b]; f ) −M n (E 0; f )) > 0. θ θ

θ(k)i i 0 D i Di

i xi xi+1 ∈ E k (E k )

E = {x i |x0 ≤x1 ≤. . . ≤x = x +1 ≤. . . ≤xn +1 }, p f

E f E


49/412

r := f − p r [a, b] ε > 0 δ > 0 |r (x) −r (y)| < ε

|x −y| < δ, x, y ∈[a, b]. ε 0 < ε < M n (E 0; f ) δ

ε E (E k ) k E k = {x i | x0 < x 1 < . . . < x n +1 }

|x i −x i| < δ i E k |r (x i) −r (x i)| < ε |r (x i )| < ε + |r (x i )| p f (x i) = p (x i ) r (x i ) = 0 |r (x i )| < ε < M n (E 0; f ) i

M n (E 0; f ) > ε > maxi |r (x i )| = maxi |f (x i ) − p (x i )| ≥

≥ min p∈P n maxi |f (x i ) − p (x i)| = M n (E k ; f ) .

0.5 1.0 1.5 2.0 x

0.2

0.1

0.1

0.2

r k

r k p∗ ∈P 4 e2x [0, 2]

r k p∗ ∈P 4

e2x [0, 2]

M n (E k ; f ) E k+1 |r k (y)| M n (E k ; f )

E k r k |r k|

f ∈C ([a, b]) ε > 0


50/412

k E k

×10 ×10

×10

×10 ×10 ×10 ×10 ×10 ×10 ×10

|f (y) − p∗k (y)|−M n (E k ; f ) E k

p∗ ∈ P n f ∈C ([a, b]) ε > 0 E 0 := {x i |a ≤x0 < x 1 < . . . < x n ≤b} k := 0

p∗k f E k

y ∈[a, b] |r k (y)| = r k ∞ |r k (y)| −M n (E k ; f ) < ε z0 := a zn +2 := b

r k (zi ) = 0 , zi ∈(x i−1, x i ) i = 1 , 2, . . . , n + 1 yi ∈(zi , zi+1 ) sgn(r k (x i )) r k (yi) = i = 0 , 1, . . . , n + 1

E k+1 := {yi |y0 < y1 < . . . < y n +1 } k

n + 1 f ∈ C 1([a, b])

r k (zi , zi+1 ) r k rk

O(n)


51/412

xi yi xi 1

r xi

r yi

r xi 1 zi 1 zi zi 1 zi 2

r

x

yi

P n

S

f f f = ( f i )ni=0 [a, b]⊂

R [a, b]

V (xxx; f f f ) := V ((x j )n j =0 ;f f f ) := V (x0, x1, . . . , x n ; f f f ) := det ( f i(x j ))ni,j =0

x j ∈[a, b] p =

n

i=0α i f i

f f f := ( f 0, f 1, . . . , f n ) x j ∈I j ⊂[a, b]

f f f = (1 ,x ,x 2) [a, b]

V (x0, x1, x2; f f f ) =1 1 1x0 x1 x2x20 x21 x22

= ( x2 −x1)(x2 −x0)(x1 −x0) = 0

x i x f f f = (1 , x2) [−1, 1]

V (x0, x1; f f f ) = 1 1x20 x21= ( x1 −x0)(x1 + x0)

V (x0, x1; f f f ) = 0 x1 = −x0 = x0


52/412

|E | = n +1 (f i(x j )) ni,j =0

f f f := ( f 0, f 1, . . . , f n ) [a, b] ⊂ R xxx := ( x i )ni=0 , x0 < x 1 < . . . < x n +1 xi ∈ [a, b]

V (xxx; f f f )

xxx = ( x i )ni=0 yyy = ( yi)ni=0 V (xxx; f f f )V (yyy; f f f ) < 0 f i [a, b]

xxx yyy α∈(0, 1)

V (αxxx + (1 −α)yyy; f f f ) = 0 f i αxxx + (1

−α)yyy

V 0 i j i > j αx i + (1 −α)yi = αx j + (1 −α)y j α(x i −x j ) = (1 −α)(y j −yi) α > 0

x i −x j > 0 (1−α) > 0 y j −yi < 0

{1, cos θ , . . . , cos nθ, sin θ , . . . , sin nθ} n ∈ N [0, 2π) [−π, π )

−π ≤

θ0 < θ

1 <

· · ·< θ

2n < π.

cos kθ = eikθ + e−ikθ

2 =

ωk + ω−k2

, sin kθ = eikθ −e−ikθ

2i =

ωk −ω−k2i

{ω−n , ω−n +1 , . . . , 1, . . . , ω n−1, ωn}

ω j = eiθ j ∈ {z |z ∈C ; |z| = 1}

p(x) = αx + β f (x) = ex [0, 1]

f

∈C 1([a, b])

E 0 = {a, a+ b2 , b}.


53/412

f (x) = sin(3 x) [0, 2π]

P 3

cosh x [

−1, 1]

[−1, 1] |x|

E 0 = {−23

, −13

, 13

, 23}.

dist ∞ (xn , P n−1) = 2 −

n +1 , [−1, 1] . f (x) = x [0, 2]

S = L{ex , e2x}

• 1, x2, x3 • 1, xx , e2x •

12 + x

, 13 + x

, 14 + x

[0, 1]

• 1, x2, x3 • {|x|, |1 −x|} • {x + 1 , ex}

[−1, 1]

F = 1

Q(x) , xQ(x) , . . . ,

xn

Q(x) ,

Q [a, b] F [a, b] Q(x) = x + 1

L 1

Q(x),

xQ(x)

f (x) := ( x − 1)2 [0, 2]

{f 0, f 1, . . . , f n} X

ni=0 a i f i n X


54/412

{eα 1 x , eα 2 x , . . . , e α n x}, 0 < α 1 < α 2 < · · ·< α n , [a, b]

f (x) = a0 +n

k=1

(ak cos kx + bk sin kx) ,

g(x) = c0 +n

k=1

(ck cos kx + dk sin kx) ,

ak , bk , ck , dk ∈ R f (x i) = g(x i ) 2n + 1 [0, 2π) f ≡g

p ≤n max

−1≤x≤1| p(x)| ≤1, T n 2n−1

{H 0, H 1, . . . , H n} x0 < x 1 < · · · < x n

i xi H i


55/412

X

, S ⊂X

X f ∗ f ∈X

S ⊂X f ∗ ∈S f

∈X f

−f ∗

⊥

S

f ∗ ∈ S f − f ∗⊥ S s ∈ S f ∗ − s ∈ S f −f ∗

f −s 2 = f −f ∗ + f ∗−s 2 = f −f ∗2 + f ∗−s 2 ≥ f −f ∗2. f ∗

f ∗ ∈S f ∈X s∈S λ > 0 f ∗−λs f

0 ≤ f −f ∗ + λs2

− f −f ∗2

== f −f ∗2 + 2 λ f −f ∗, s + λ2 s 2 − f −f ∗2 == λ(2 f −f ∗, s + λ s 2).

λ f −f ∗, s λ s 2 f −f ∗, s ≥ 0 s

−s f −f ∗, s ≤0 s f −f ∗⊥ S

f ∗ (s i ) S

f ∗ f ∗ = j α j s j f − j α j s j⊥ S (s i) S

f − j

α j s j , s i = 0 , i.

ααα := ( α j )

Gααα = bbb, G := ( s j , s i ) , bbb := ( f, s i ) . G

G


56/412

X = C ([0, 1]) , f, g := 10 f (x)g(x)dx, S = P n , si = xi ,

G = 1

i + j −1n

i,j =0

n S (s i) s j , s i = δ i,j G = I f ∗

f ∗ = j

f, s j s j .

f, s j

f −f ∗, f ∗= 0 ,

f −f ∗2 = f −f ∗, f −f ∗= f −f ∗, f == f, f −f ∗, f = f 2 −

jf, s j s j , f =

= f 2 − j

f, s j s j , f = f 2 − j

f, s j 2 ≥0,

[0, 2π] X = L 2 ([0, 2π]) 1 S T n

1√ 2π ,

1√ π cos x , . . . ,

1√ π cos nx,

1√ π sin x , . . . ,

1√ π sin nx

[0, 2π]

2π0 dx = 2π


57/412

2π0 cos kx cos jx dx = π δ k,j , j > 0 ali k > 0,

2π

0 sin kx sin jx dx = π δ k,j ,

2π0 cos kx sin jx dx = 0 . T n (x) = cos ( n arccos x) [−1, 1]

[0, π ] [−1, 1] x = x(θ) := cos θ

π

0 cos kθ cos jθ dθ = π2 δ k,j , k > 0 ali j > 0

π0 dθ = π x = cos θ, dx = −sin θ dθ = − 1 −x2dθ k = 0 j = 0

T k , T j = 2π 1−1 T k (x)T j (x) 1√ 1 −x2 dx

= 2π

π

0 cos kθ cos jθ dθ = δ k,j .

X = L 2ρ([−1, 1]) ρ(x) :=

2π

1√ 1 −x2

1√ 2 T 0, T 1, T 2, . . . ,

[−1, 1] ρ(x) = (1 −x) p (1 + x)q , p > −1, q > −1.

p = q p = 0 q = 0 P n

A∈R m,n , m > n ,


58/412

bbb∈R m

xxx∗ ∈R n Axxx∗−bbb 2 ≤ Axxx −bbb 2 , xxx∈R n .

2

xxx,yyy =m

k=1

xkyk , xxx = ( xk )mk=1 , yyy = ( yk )mk=1 .

n m > n

S = R(A) := L{AAA j }n j =1⊂

Rm

, AAA j := ( a ij )mi=1 .

AAA j dim S = n

AAAi ,AAA j = AAAT i AAA j =m

k=1

aki akj , i, j = 1 , 2, . . . , n ,

G = AT A.

AAAi , bbb = AAAT i bbb =m

k=1

aki bk , i = 1 , 2, . . . , n .

AT Axxx∗ = AT bbb.

AT A AT A

AT A G = AT A A

A = Q R,

Q ∈ R m,n R QT Q = I

∈ R n,n

rang A = n

Rxxx = QT bbb.


59/412

cos(n + 1) θ = 2 cos θ cosnθ −cos(n −1)θ, x := cos θ

T n +1 (x) = 2 xT n (x) −T n−1(x), T n (x) = cos( n arccos x)

(Q0, Q1, . . . , Q n ) , Q i = 1 , Q i , Q j = δ i,j , Qi i Qn +1

n + 1 Qn +1 n + 1 L({Q i}ni=0 )

Qn Qn +1 −xQ n ≤n

(Q i)ni=0

Qn +1 (x) −x Q n (x) = −αn Qn (x) −β n Qn−1(x) +n−2

i=0γ n,i Q i (x).

, Q j

0 = γ n,j , j = 0 , 1, . . . , n −2, j

Qn +1 , Q j = 0 , −xQ n , Q j = Qn , −xQ j = 0 , Qi , Q j = δ i,j . −xQ n , Q j = Qn , −xQ j

Q i

Q j ∈P j ⇒x Q j ∈P j +1 . γ n,i

Qn +1 (x) = ( x −αn ) Qn (x) −β n Qn−1(x), n = 0 , 1, . . . ,

Q−1(x) := 0 ,

Q0 = Q0, Q0 = 1 .

, Qn ,

, Qn−1

αn , β n

αn = xQ n , Qn , β n = xQ n , Qn−1 .


60/412

Qn +1

Qn +1 = 1Qn +1

Qn +1 .

β n

β n = xQ n , Qn−1 = 1Qn

Qn , xQ n−1

= 1Qn

Qn , Qn − Qn −xQ n−1 = Qn , Qn −xQ n−1 < n αn

n 2n

On2

Qn +1 (x) = x −αn

β n +1Qn (x) −

β nβ n +1

Qn−1(x), n = 0 , 1, . . . ,

2n + 2 α i , β i Q0

f, g = 1−1 f (x)g(x)dx

f, g = 2m + 1

m

i=0f (x i )g(x i ), xi = 2

im −1.

Q0 = 1

√ 2

(α i , β i )

P n

(α i , β i )

p(x) = ni=0

a iQ i(x) x (Q i )ni=0

(α i , β i ) Q0 Q0 = 1


61/412

n α n β n αn β n

1.00929 ×10

1

√ 30

2√ 15 −1.21115 ×10

3√ 35 8.07435 ×10

43√ 7 −1.81673 ×10

53√ 11 8.07435 ×10

6√ 143 −6.35855 ×10

m = 10

δ n (x) := an

δ n−1 (x) := x −αn−1

β nδ n (x) + an−1

k = 2 , 3, . . . , n

δ n−k (x) := x −α n−k

β n−k+1δ n−k+1 (x) −

β n−k+1β n−k+2

δ n−k+2 (x) + an−k

p(x) := δ 0 (x) Q0(x)

(α i , β i )

f (x) = ex sin4x [−1, 1] n = 2 , 3, . . . , 6


62/412

1.0 0.5 0.5 1.0 x

0.5

0.5

1.0 0.5 0.5 1.0 x

0.5

0.5

n = 2 n = 6

(Qn )n≥0 Qn

n

Qn (x) = cn xn + . . . , cn = 0 .

n

i=0Qi (x)Qi (t) =

cncn +1

Qn +1 (x)Qn (t) −Qn +1 (t)Qn (x)x −t

.

n i

ci

Qi

ci+1 x i+1 + · · ·= x −α i

β i+1cix i + . . . −

β iβ i+1

ci−1xi−1 + . . . .

x i+1

β i+1 = cici+1

, i = 0 , 1, . . . , c−1 = β 0 c0 = 0 .

x Q i (x)

x Q i(x) = cici+1

Qi+1 (x) + α i Qi (x) + ci−1

ciQ i−1(x).

Qi (t)

xQ i(x)Q i(t) = cici+1

Q i+1 (x)Q i (t) + α i Qi (x)Qi (t) + ci−1

ciQ i−1(x)Qi (t) ,

x

t

t Q i (t)Q i (x) = cici+1

Qi+1 (t)Qi (x) + α iQ i (t)Q i(x) + ci−1

ciQ i−1(t)Q i(x) .


63/412


64/412

(s i )ni=1 (u i )ni=1 S

k = 1 , 2, . . . , n sk := 1u k uk

j = k + 1 , k + 2 , . . . , n

u j := u j − u j , sk sk

S ⊂X n X (s i )ni=1

S si , i < n sn s̄n , s̄n = 1 sn −s̄n ∈S,

sn −s̄n =n

i=1sn −s̄n , s i s i ,

sn = s̄n −n−1

i=1s̄n , s i s i + sn −s̄n , s n sn ,

s i ⊥sn , i < n s̄n 1 = s̄n 2 = s̄n , s̄n = sn −sn + s̄n , s n −sn + s̄n

= sn 2 −2 sn −s̄n , s n + sn −s̄n 2= 1 −2 sn −s̄n , s n + sn −s̄n 2

sn −s̄n , s n = O sn −s̄n 2


65/412

sn −s̄n O sn −s̄n 2 sn

s̄n −n−1

i=1

s̄n , s i s i ,

s̄n , s i = s̄n −sn , s i = O( sn −s̄n ) . s i

s i O(i) On2 S

m = 1000

( Q i , Q j )6i,j =0

π2

i Q i , Q6 Q i , Q6 2.62 ×10 −5.33 ×10 −2.38 ×10 2.75 ×10 2.21

×10

−6.91

×10

−1.41 ×10 −6.74 ×10 6.80 ×10 −3.00 ×10 −3.46 ×10 −2.31 ×10

x i ∈[a, b] f, g = ba f (x)g(x)ρ(x)dx, 2 = , , ρ > 0 ,

x i = a + ih, i = 0 , 1, . . . , N, h = b−a

N ,

f, g h = hN

i=0f (x i )g(x i)ρ(x i),

2,h =

,

h .


66/412

(Q i)i≥0 (Q i,h )i≥0 Qi , Q i,h ∈P i

, , h Qi,h

−Q i ∞

→0, h

→0.

f, g h f, g i = N O(h)

| f, g h − f, g | →0, f 2,h →f 2, h →0, f ,g ∈C ([a, b]) .

Q0,h = 11 2,h →

11 2

= Q0, h →0. Q−1,h = 0 = Q−1

Q i,h →Q i , i = −1, 0. j ≤i

Q i+1 ,h α i,h := xQ i,h , Q i,h h

α i,h →xQ i , Q i = α i , h →0.

| xQ i,h , Q i,h h − xQ i , Q i | ≤≤ |

xQi,h

, Qi,h h −

xQi, Q

i h |+

|xQ

i, Q

i h −xQ

i, Q

i |.

| xQ i,h , Q i,h h − xQ i , Q i h | ≤ | xQ i,h , Q i,h −Q i h |+ | Qi,h −Q i , xQ i h | ≤≤ Qi,h −Q i 2,h ( xQ i,h 2,h + xQ i 2,h ) ≤≤ Qi,h −Q i ∞ 1 2,h ( xQ i,h 2,h + xQ i 2,h ) .

1 2,h

→1 2 <

∞

xQ i,h 2,h ≤ xQ i,h −xQ i 2,h + xQ i 2,h ≤≤ Qi,h −Qi ∞ x 2,h + xQ i 2,h .

β i,h = Q i,h 2,h →β i = Q i 2, h →0.

Q i+1 ,h (x)

→ Q i+1 (x) = ( x

−α i) Q i(x)

−β i Q i

−1(x), h

→0.

Q i+1 ,h (x) i + 1


67/412

f ∈ C ([a, b]) p∗h p∗ ∈ P n

p∗h =n

i=0f, Q i,h h Qi,h , p∗ =

n

i=0f, Q i Qi .

p∗h − p∗∞ →0, h →0.

| f, Q i,h h − f, Q i | ≤ |f, Q i,h h − f, Q i h |+ | f, Q i h − f, Q i |

| f, Q i,h −Qi h | ≤ f ∞ Qi,h −Q i ∞ 1 22,h , f, Q i,h h

f, Q i

f, Q i,h h Q i,h − f, Q i Q i ∞ ≤≤ f, Q i,h h Q i,h − f, Q i,h h Q i ∞ + f, Q i,h h Qi − f, Q i Q i ∞ ≤≤ | f, Q i,h h | Qi,h −Q i ∞ + | f, Q i,h h − f, Q i |Qi ∞ →0, h →0,

P n n

(s i ) 2 (s i )

(Q i)i≥0 Qi ∈P i

L 2ρ([a, b]) f 2 = ba f (x)2ρ(x)dx f ∈C ([a, b]) f −

n

i=0α i Qi

2→0, n → ∞, α i := f, Q i .

p∗n ∈ P n f [a, b] f −

n

i=0α iQ i

2≤ f − p∗n 2 .

f − p∗n 22 = b

a(f (x) − p∗n (x))2 ρ(x)dx ≤ f − p∗n 2∞

b

aρ(x)dx →0,

n


68/412

(Q i)i≥0 C ([a, b])⊂L2ρ([a, b]) L 2ρ([a, b])

(Qi )i≥0 Qi ∈P i

L 2ρ([a, b]) (Q i )i≥

0

f ∈L 2ρ([a, b])

f, Q i = ba f (x)Qi (x)ρ(x)dx = 0 , i = 0 , 1, . . . f = 0

F (x) := xa f (t)ρ(t)dt. F ∈C ([a, b]) F (a) = 0

F (b) = ba f (t)ρ(t)dt = 1Q0(a) f, Q 0 = 0 .

0 = f, x n = ba tn f (t)ρ(t)dt

dF (t )

= tn F (t)b

a −n ba tn−1F (t)dt, F ⊥x

n n F F = 0 f ρ 0

ρ f = 0

f ∈L 2ρ([a, b])

[

−1, 1]

f 1(x) := x + 1 , x ≤0x −1, x > 0

f 2(x) := 1 − |x|. f 1 1 0

1 −1 2 O 1√ n

f 2 O 1n α

f [−1, 1] ρ = 1


69/412

2 ∞n α α

2 ∞ f 1

2 ∞n α α

2

∞ f 2Q i P i

Q i (x) = i + 12 P i(x) = i + 12i! 2i didx i x2 −1 i , i = 0 , 1, . . . f ∈ C 2([−1, 1]) p∗n ∈ P n

f

2

p∗n =n

i=0α i Qi , α i = 1−1 f (x)Q i (x)dx.


70/412

> 0 n

f − p∗n ∞ ≤ √ n .

P i (x) = 1i! 2i

didx i

x2 −1i ,

2i+1 (i + 1)! P i+1 (x) = di+2

dx i+2x2 −1

i+1 =

= di

dx i d2

dx2x2 −1

i+1 =

= 2( i + 1) di

dx i ddx x

2

−1i

x =

= 2( i + 1) di

dx ii x2 −1

i−1 2 x2 −1 + 1 + x2 −1i =

= 2( i + 1) di

dx i(2i + 1) x2 −1

i + 2 i x2 −1i−1 =

= 2 i+1 (i + 1)! (2i + 1) P i(x) + P i−1(x) Q i

1 i + 32 Q i+1 (x) − 1 i − 12 Q i−1(x) = 2i + 1 i + 12 Q i (x). f ∈C 2([−1, 1]) f f

α i := 1−1 f (x)Q i (x)dx, α i := 1

−1f (x)Q i(x)dx

i α i±1

α i±1 = 1

−1f (x)Q i±1(x)dx = f (x)Q i±1(x)

1

−1 − 1

−1f (x)Q i±1(x)dx.

f (±1) = f (±1) = 0 p3 ∈ P 3 f −1 1

f f := f − p3 α i±1 = − 1−1 f (x)Q i±1(x)dx, α i±1 = −

1

−1f (x)Q i±1(x)dx.

f (x) [

−1, 1]

− 1

i + 32 α i+1 + 1

i − 12 α i−1 = 2i + 1

i + 12 α i


71/412

P 0

P 1

P 2

P 3

P 4P 5

1.0 0.5 0.5 1.0 x

1.0

0.5

0.5

1.0

P i

P n

α i = −α i+1

i12

+ O1i

+ α i−1

i12

+ O1i

.

α i α i f

f

α i = Oα ii

= Oα ii2

= O1i2

.

|P i(x)| ≤1, x∈[−1, 1],

|Q i (x)

| ≤ i +

1

2, x

∈

[

−1, 1].

> 0 n i ≥n i2|α i | < 2√ 2 m ≥n x∈[−1, 1] | p∗m (x) − p∗n (x)|


72/412

| p∗m (x) − p∗n (x)| =m

i=0α iQ i (x) −

n

i=0α iQ i (x) =

=m

i= n +1α i Q i(x) ≤

m

i= n +11i2 i

2|α i| |Q i(x)| ≤

≤ 2√ 2

m

i= n +1

1i2 i + 12 ≤ 2 mi= n +1 1i 32 ≤

≤ 2 ∞n x−32 dx = √ n .

p∗i

p∗ := limi→∞ p∗i . ( p∗i ) f

0 = limi→∞

f − p∗i 2 = f − limi→∞ p∗i 2 = f − p∗2. f p∗ f = p∗

m → ∞

(x1, y1), (x2, y2), . . . , (xm , ym ) p∗(x) = αx + β,

(1, 2), (2, 3), (3, 5), (4, 8)

[

−1, 1] f (x) = ex p∗

∈ P 1

f g, h = 1−1 g(x)h(x)dx

(x i , yi ), i = 1 , 2, . . . , n L({1, ex})

f 1, f 2, . . . , f n det G = 0 , G = ( f i , f j )ni,j =1

[a, b]

f, g = ba f (t)g(t)dt.


73/412

f ∗ ∈ P n f ∈C ([a, b]) f = 0 . f −f ∗ n + 1

[a, b].

f, g = f (1)g(1) + 2 f (2)g(2) + 2 f (3)g(3) + 2 f (4)g(4) + f (5)g(5) .

f (x) = 2 cos 2(πx4

)

(−1, 12), (0, 7), (1, 6), (2, 9)

f, g = 1−1 f (t)g(t)dt. x i [−1, 1]

(s i)ni=1 si ∈ C ([a, b]) ( s j , s i )ni,j =1


74/412


75/412


76/412

f

f (x i) xi s1, s2, . . . , s n n

s = ni=1 α is i α i λ1, λ 2, . . . , λ n L({s1, s 2, . . . , s n})

rrr = ( r i)ni=1 α i s λis = r i i

α i rrr

s1, s 2, . . . , s n λ1, λ 2, . . . , λ n

(λ i s j )ni,j =1

a = x0 < x 1 < x 2 < . . . < x n = b, [a, b]⊂R .

s i

s i(x) = i(x) =n

j =0 j = i

x −x jx i −x j

, i = 0 , 1, . . . , n .

λ i xi λi f := f (x i ) λ i j = δ i,j

rrr i ααα i si = i

ni=0 ααα i i (x)

≤n x xi rrr i

ααα i = rrr i xi x i

x0 = 0 , xi = xi−1 + 1 , i = 0 , 1, . . . , n x0 := 0 , xi = xi−1 + rrr i −rrr i−1 2, i = 1 , 2, . . . , n

x0 := 0 , xi = xi−

1 +

rrr i

−rrr i

−1 2, i = 1 , 2, . . . , n

s i S = L({s1, s2, . . . , s n})


77/412

r 0

r 1

r 2

r 3

r 4

r 5

r 6

r 7

r 8

r 9

x i

S = P 2n +1 a = x0 < x 1 < x 2 < .. . < x n = b x i

λ2if = f (x i ) = r i , i = 0 , 1, . . . , n ,λ2i+1 f = f (x i) = r i , i = 0 , 1, . . . , n .


78/412

P 2n +1

p∈P 2n +1 λi p = 0

i λ2i p = p(x i ) = 0 q

p(x) = q (x)

n

j =0(x −x j ) ,

λ2i+1 p = p (x i) = 0

p (x i ) = q (x i) ·0 + q (x i )n

j =0 j = i

(x i −x j ) = 0 .

q (x i) = 0 , i = 0 , 1, . . . , n q n

S = P n a = x0 < x 1 < x 2 < . . . < x m = b [a, b] xi µi

λ i,j f = f ( j ) (x i ) = r i,j , i = 0 , 1, . . . , m ; j = 0 , 1, . . . , µ i −1, µi ∈ N mi=0 µi = n + 1

p∈P n , p ≡0

p(x) = q (x)m

i=0(x

−x i)µ i .

mi=0 µi = n + 1 p ∈ P n q ≡ 0

p ≡ 0

S

λ i f = f (x i) xi

xxx = ( x i )ni=0 [a, b]

a = x0 < x 1 < x 2 < · · ·< x n = b.

τ 0 < τ 1 < · · ·< τ n ,

x0 ≤τ 0 < x 1, xi−1 < τ i < x i+1 , i = 1 , 2, . . . , n −1, xn−1 < τ n ≤xn ,


79/412

λ i f = f (τ i ) S 1,xxx n = 1

det H 0(τ 0) H 1(τ 0)H 0(τ 1) H 1(τ 1)= det

x1 −τ 0∆ x0

τ 0 −x0∆ x0x1 −τ 1

∆ x0

τ 1 −x0∆ x0

= ∆ τ 0∆ x0

(H j (τ i ))1i,j =0 n > 1 0 < < n τ

x −1 < τ ≤x x < τ < x +1 H j (τ i) = 0 , i = 0 , 1, . . . , , j = + 1 , + 2 , . . . , n .

(H j (τ i )) i,j =0 , (H j (τ i ))ni,j = +1

u

x 2

x 1

x 1

x 2

x 2

x 1

x1 x2

E ⊂ R m m ≥ 2 E u s1, s 2, . . . , s n , n ≥ 2 E


80/412

u n xi ∈ R m s i

det( s j (x i ))ni,j =1 = 0 .

u x1 x2

x2 x1 x2 x2

x1 si det( s j (x i))


81/412

P n a = x0 < x 1 < x 2 < . . . < x n = b r i

f ∈C

([a, b])

x i

p =n

i=0f (x i ) i , i (x) =

n

j =0 j = i

x −x jx i −x j

,

i

I n : C

([a, b]) →P

n : f →I n f :=n

i=0 f (x i ) i

q ≤n I n q = q I n

• p(x) x On2

O(n)

n

j =0 j = i

1x i −x j

x i

•

• x i


82/412

i x −x j j = i

ω ω(x) := ( x

−x

0)(x

−x

1)

· · ·(x

−x

n)

i ω x −x i ω x = xi

ω (x i ) =n

j =0 j = i

(x i −x j ) .

i(x) = ω(x)

(x −x i )ω (x i ) .

p(x) =n

i=0f (x i)

ω(x)(x −x i )ω (x i )

= ω(x)n

i=0f (x i )

1(x −x i)ω (x i )

.

1 ≤n

1 = I n 1 = ω(x)n

i=0

1 · 1

(x

−x i )ω (x i)

.

p 1 = I n 1

p(x) =n

i=0f (x i )wi(x) ,

wi (x) :=

1(x −x i )ω (x i)n

j =0

1(x −x j )ω (x j )

.

wi 1 n

i=0wi (x) ≡1

f

x i pk ∈ P k f x0, x1, . . . , x k pk−1


83/412

k (x −x0)(x −x1) · · ·(x −xk−1)

pk (x j ) = pk−1(x j ) = f (x j ), j = 0 , 1, . . . , k −1,

pk (xk ) = f (xk )

x0, x1, . . . , x k f [x0, . . . , x k ]f [x0, . . . , x k ]f

x 1 x 2 x 3 x 4 x 5

x

12

x 1 x 2 x 3 x 4 x 5

x

12

(x − x0)(x − x1) · · ·(x −x i−

1) [x0, . . . , x i]f p0, p1, . . . , p 5 f (x) =sin5x sin x + 12 [0, 3]

p ∈ P k ≤k f xi , x i+1 , . . . , x i+ k p

[x i , . . . , x i+ k ]f k f x i , x i+1 , . . . , x i+ k

p

≤k

k p < k x j

f ∈ C ([a, b]) p ∈ P 1 f x i , x i+1 ∈[a, b]

p(x) = f (x i) + f (x i+1 ) −f (x i )

x i+1 −x i(x −x i ).

[x i , x i+1 ]f = f (x i+1 ) −f (x i )

x i+1 −x i.


84/412

f (x) = xr , r

≤k, r

∈N 0

[x i , . . . , x i+ k ]f = 0, 0 ≤ r < k,1, r = k.

I k r ≤ k

x i , x i+1 , . . . , x i+ k

[x i , . . . , x i+ k ]f =i+ k

j = if (x j ) [x i , . . . , x i+ k ] j =

i+ k

j = if (x j )i+ k

r = ir = j

(x j −x r ).

x j j k

j 1

i+ k

r = ir = j

(x j −

xr)

x j p f

xxx := ( x j )i+ k j = i , a ≤x i ≤x i+1 ≤ · · · ≤x i+ k ≤b, f

f f |xxx := ( f j )

i+ k j = i

f j := f (r ) (x j ) , r := max {m | j −m ≥ i, x j−m = x j }.

f

g

xxx

f |xxx = g|xxx .


85/412

f ∈ C k ([a, b]) xi+ j = xi ∈ [a, b], j = 1 , 2, . . . , k f xi (k + 1)

p(x) =k

j =0

f ( j ) (x i) j !

(x −x i ) j ,

[x i , x i , . . . , x i

j +1]f :=

1 j !

f ( j )(x i ), j = 0 , 1, . . . , k .

p(x) =

n

i=0 (x −x0)(x −x1) · · ·(x −x i−1)[x0, . . . , x i ]f.

x i

[x i , . . . , x i+ k ] : f →[x i , . . . , x i+ k ]f , x j

[x i , . . . , x i+ k ] x j , j = i, i + 1 , . . . , i + k [. . . ]

≤ k f

[x i , . . . , x i+ k ]

[x i , . . . , x i+ k ](αf + βg) = α[x i , . . . , x i+ k ]f + β [x i , . . . , x i+ k ]g .

p ∈ P k xi , x i+1 , . . . , x i+ k f q ∈ P k g αp + βq αf + βg

f f = g·h

[x i , . . . , x i+ k ]f =i+ k

= i

[x i , . . . , x ]g [x , . . . , x i+ k ]h .

p1(x) :=i+ k

r = i(x −x i )(x −x i+1 ) · · ·(x −x r −1)[x i , . . . , x r ]g


86/412

x i , x i+1 , . . . , x i+ k g

p2(x) :=i+ k

= i

(x −x i+ k )(x −x i+ k−1) · · ·(x −x +1 )[x , . . . , x i+ k]h x i+ k , x i+ k−1, . . . , x i h p := p1 · p2

x i , x i+1 , . . . , x i+ k f

p(x) = p1(x) p2(x) =i+ k

r = i

i+ k

= i

ωr, (x) [x i , . . . , x r ]g [x , . . . , x i+ k ]h.

ωr,

ωr, (x) := ( x −x i)(x −x i+1 ) · · ·(x −xr −1)(x −x +1 )(x −x +2 ) · · ·(x −x i+ k ) , k + r

− x < r

ωr, (x j ) = 0 j = i, i + 1 , . . . , i + k < r x i , x i+1 , . . . , x i+ k p f

q

q (x) :=i+ k

r = i

i+ k

= r

ωr, (x) [x i , . . . , x r ]g [x , . . . , x i+ k ]h,

x i , x i+1 , . . . , x i+ k f q ≤ k

[x i , . . . , x i+ k ]f.

= r k ωr,r q

i+ k

= i

[x i , . . . , x ]g [x , . . . , x i+ k ]h .

f

f (x) = ( x −t)k−1+ k ≥ 2 (x − t)k−1+ = ( x − t)(x − t)k−2+ [x i , . . . , x i+ k ]f

[x i ]( −t) = xi −t, [x i , x i+1 ]( −t) = 1 ( −t)

[xi, . . . , x

i+ k](

−t)k−1

+ =( x

i −t)[x

i, . . . , x

i+ k](

−t)k−2

+ +

+ [x i+1 , . . . , x i+ k ]( −t)k−2+ .


87/412

f k k+1

f ∈ C k f (x) := √ x {0, δ }, δ > 0 [0, δ ]f = √ 0 −√ δ 0 −δ

= 1√ δ . δ → 0 f 0 0 δ

f

f ∈C k ([a, b]) x j

∈[a, b], j = i, i + 1 , . . . , i + k .

[x i , . . . , x i+ k ]f = 1k!

f (k)(x i ) , xi = xi+1 = . . . = xi+ k ,

[x i , . . . , x i+ k ]

[x i , . . . , x i+ k ]f =

= [x i , . . . , x s−1, x s+1 , . . . , x i+ k ]f −[x i , . . . , x r −1, x r +1 , . . . , x i+ k ]f

xr −xs, xr

= xs ,

(k + 1)

x i = x i+ k

[x i , . . . , x i+ k ](

−t)k

−2

+ ==

[x i , . . . , x i+ k−1]( −t)k−2+ −[x i+1 , . . . , x i+ k ]( −t)k−2+

x i −x i+ k

(x i+ k −x i)[x i , . . . , x i+ k ]( −t)k−1+ == ( t −x i )[x i , . . . , x i+ k−1]( −t)k−2+ +

+ ( x i+ k −t)[x i+1 , . . . , x i+ k ](

−t)k

−2

+ .


88/412

xxx := ( x j )i+ k j = i , xi ≤x i+1 ≤ · · · ≤x i+ k , f f

|xxx

= ( f j )i+ k j = i [x i , . . . , x i+ k ]f f |xxx

d j , j = i, i + 1 , . . . , i + k xxx f

[x i , . . . , x i+ k ]f =i+ k

j = id j f j .

xxx k = 0 k xi = xi+1 = · · · = xi+ k

k − 1

xi < x i+ k

d j d j f

[x i , . . . , x i+ k−1]f =i+ k−1

j = id j f j =

i+ k

j = id j f j , di+ k := 0 ,

[x i+1 , . . . , x i+ k ]f =i+ k

j = i+1

d j f j =i+ k

j = i

d j f j , di := 0 .

[x i , . . . , x i+ k ]f =i+ k

j = i

d j −d jx i −x i+ k

f j ,

f ∈C k ([a, b]) x j ∈[a, b], j = i, i + 1 , . . . , i + k .

[x i , . . . , x i+ k ]f

[x i , . . . , x i+ k ]f =

=1

0

dt1

t1

0

dt2 · · ·tk − 1

0

f (k)(tk∆ x i+ k−1 + tk−1∆ x i+ k−2 + · · ·+ t1∆ x i + x i )dtk .

x j = xi , j = i + 1 , i + 2 , . . . , i + k

∆ x i = ∆ x i+1 = · · ·= ∆ x i+ k−1 = 0


89/412

f (k)(x i )1

0

dt1

t1

0

dt2 · · ·t k − 1

0

dtk = f (k) (x i)

k! .

k = 1 xi = xi+1 x i = xi+1

[x i , x i+1 ]f =1

0

f (t1∆ x i + x i )dt1 = 1∆ x i

x i +1

x i

f (u)du = f (x i+1 ) −f (x i )

x i+1 −x i.

u u := t1∆ x i + x i k = 1

k k + 1 k + 1

x i+ k−1 xi+ k tk ξ

ξ = tk∆ x i+ k−1 + tk−1∆ x i+ k−2 + · · ·+ t1∆ x i + x i , dξ = ∆ x i+ k−1dtk .

ξ 0 = tk−1∆ x i+ k−2 + · · ·+ t1∆ x i + x i ,

ξ 1 = tk−1(∆ x i+ k−1 + ∆ x i+ k−2) + tk−2∆ x i+ k−3 + · · ·+ t1∆ x i + x i .

1

0

dt1

t1

0

dt2 · · ·ξ1

ξ0

f (k) (ξ ) dξ

∆ x i+ k−1=

= 1

∆ x i+ k−

1

1

0

dt1 · · ·t k − 2

0

f (k−1) (ξ 1) −f (k−1) (ξ 0) dtk−1 =

= 1

∆ x i+ k−1

1

0

dt1 · · ·t k − 2

0

f (k−1) (tk−1(x i+ k −x i+ k−2) + · · ·+ x i )dtk−1−

− 1

∆ x i+ k−1

1

0

dt1 · · ·t k − 2

0

f (k−1) (tk−1∆ x i+ k−2 + · · ·+ t1∆ x i + x i)dtk−1 =

=

[x i , . . . , x i+ k−2, x i+ k ]f

−[x i , . . . , x i+ k−2, x i+ k−1]f

x i+ k −x i+ k−1 ,

[x i , . . . , x i+ k ]f


90/412

f ∈ C k ([a, b]) x j ∈ [a, b], j = i, i + 1 , . . . , i + k

• [x i , . . . , x i+ k ]f x j

• 1k!

minx∈[a,b ]

f (k) (x) ≤[x i , . . . , x i+ k ]f ≤ 1k!

maxx∈[a,b ]

f (k)(x) .

• ξ ∈ mini x i , maxi x i

[x i , . . . , x i+ k ]f = 1k!

f (k)(ξ ) .

f ∈ C n +1 ([a, b]) xxx = ( x i)ni=0 , xi ∈ [a, b],

f (x) = pn (x) + ( x −x0)(x −x1) · · ·(x −xn )[x0, . . . , x n , x]f , x ∈[a, b],

pn (x) =

n

i=0(x −x0)(x −x1) · · ·(x −x i−1)[x0, . . . , x i ]f

f xxx

n = 0 x = x0

f (x) = f (x0) + ( x −x0)f (x) −f (x0)

x −x0,

x = x0 f ∈C 1 ([a, b]) f (x) = f (x0) + ( x −x0) [x0, x]f x→x0−→ f (x0) + ( x0 −x0) f (x0)1! = f (x0).

k < n

f (x) = pk (x) + ( x −x0)(x −x1) · · ·(x −xk )[x0, . . . , x k , x]f . g g(x) := [ x0, . . . , x k , x]f n − k ≥ 1

g g n = 0

x0 →xk+1 g(x) = g(xk+1 ) + ( x −xk+1 )[xk+1 , x]g =

= [x0, . . . , x k+1 ]f + ( x −xk+1 )[xk+1 , x] [x0, . . . , x k , ]f .


91/412

[xk+1 , x] g f

f (x) = pk (x) + ( x

−x0)(x

−x1)

· · ·(x

−xk ) [x0, . . . , x k+1 ]f +

+ ( x −xk+1 )[xk+1 , x] [x0, . . . , x k , ]f == pk+1 (x) + ( x −x0)(x −x1) · · ·(x −xk+1 )[xk+1 , x] [x0, . . . , x k , ]f .

q y

q (y) := pk+1 (y) + ( y −x0)(y −x1) · · ·(y −xk+1 )[xk+1 , x] [x0, . . . , x k , ]f ≤ k + 2 y q k + 2 x j f

q (x j ) = pk+1 (x j ) + 0 = f (x j ), j = 0 , 1, . . . , k + 1 , f y = x

q [x0, . . . , x k+1 , x]f

q (y) = pk+1 (y) + ( y −x0)(y −x1) · · ·(y −xk+1 )[x0, . . . , x k+1 , x]f. f (x) = q (x) k + 1

f

[xk+1 , x] [x0, . . . , x k , ]f = [x0, . . . , x k+1 , x]f, x = xk+1

f ∈C n +1 ([a, b]) pn n + 1 xi ∈[a, b] f x∈[a, b] ξ x ∈[a, b] f (x) − pn (x)

f (x) − pn (x) = ( x −x0)(x −x1) · · ·(x −xn ) 1

(n + 1)!f (n +1) (ξ x ).

f

[x i , . . . , x i+ k ] [y j , . . . , y j + r , ]f = [y j , . . . , y j + r , x i , . . . , x i+ k ]f.

k = 0 k

O n2 O(n)

O(n)


92/412

[x0, x1]f f (x0) f (x1) x0 x1 x0

[ ]f [ , ]f [ , , ]f [ , , , ]f

x0 f (x0) [x0, x1]f x1 f (x1) [x0, x1, x2]f

[x1, x2]f [x0, x1, x2, x3]f x2 f (x2) [x1, x2, x3]f

[x2, x3]f x3 f (x3)

xn−1 f (xn−1) [xn−1, xn ]f xn f (xn )

f (x) = ln( x) [1, 2] f

x = ( x0, x1, x2, x3) = (1 , 1, 2, 2)

[ ]f [ , ]f [ , , ]f [ , , , ]f

−0.306853

−0.193147

ln(x)

1, x −1, (x −1)2, (x −1)2(x −2).

p(x) = 1 ·0 + ( x −1) ·(1 + ( x −1) ·(−0.306853 + ( x −2) ·0.113706)) . (2, 2, 1, 1)

p(x) = 1

·0.693147+( x

−2)

·(0.5 + ( x

−2)

·(

−0.193147+( x

−1)

·0.113706)) .

x = 32 p( 32 ) = 0 .409074 ≈ ln 32 = 0 .405465


93/412

x0, x1, . . . , x n

a i = [x0, . . . , x i ]f, i = 0 , 1, . . . , n .

x p(x) =n

i=0(x −x0)(x −x1) · · ·(x −x i−1) a i

p(x) = b0(x)

bn (x) := an i := n −1, n −2, . . . , 0 bi(x) := a i + ( x −x i )bi+1 (x)

p(x) := b0(x)

bi (x) b0(x) = [x] p = p(x) ai = [x0, . . . , x i ]f = [x0, . . . , x i ] p

bi+1 (x) = bi (x) −a i

x −x i=

[x, x 0, . . . , x i−1] p −[x0, . . . , x i] px −x i

= [x, x 0, . . . , x i ] p

bi(x) = [x, x 0, . . . , x i−1] p i

bi (x)

p

p

p

32 = 1 .5

[ ]f p (1.5) [1.5, 1.5] p = 11! p (1.5) 1.5

p

[ ]f [ , ]f [ , , ]f [ , , , ]f

−0.25

−0.306853

−0.363706

−0.306853 −0.193147

p


94/412

p(x) = 0 .409074 + ( x −1.5) ·(0.664721++ ( x −1.5) ·(−0.25 + ( x −1.5) ·0.113706)) .

n + 1

x ≤ k k + 1

x0 ≤ x1 ≤ ·· · ≤ xn x O(log n) k + 1 x

i

x j , j = i, i +1 , . . . , i + k

f ∈ C ([a, b]) xi j ∈[a, b], j = 0 , 1, . . . , pi0 ,i 1 ,...,i k − 1 ,i k f

x i0 , x i1 , . . . , x ik − 1 , x ik

pi0 ,i 1 ,...,i k − 1 ,i k (x) = 1x ik − 1 −x ik pi0 ,i 1 ,...,i k − 2 ,i k − 1 (x) x −x ik − 1 pi0 ,i 1 ,...,i k − 2 ,i k (x) x −x ik ,

pi0 ,i 1 ,...,i k − 2 ,i k − 1 (x) pi0 ,i 1 ,...,i k − 2 ,i k (x) , f

k

pi0 (x), pi0 ,i 1 (x), . . . , p i0 ,i 1 ,...,i k − 1 ,i k (x),

x ik +1 , f x ik +1

x i0 , x i1 , . . . , x i r − 1 , x i r +1 , . . . , x ik , x ik +1 , 0 ≤ r ≤k + 1 . O(k)

O k2 k + 1

k + 1

k (k + 1)


95/412

x0 f (x0) p01(x)

x1 f (x1) p012 (x) p12(x) p0123 (x)x2 f (x2) p123 (x) p23(x) p1234 (x)

x3 f (x3) p234 (x) p34(x)

x4 f (x4)

k = 3

x0 f (x0) p

01(x)

x1 f (x1) p012 (x) p02(x) p0123 (x)x2 f (x2) p013 (x) p03(x) p0124 (x)

x3 f (x3) p014 (x) p04(x)

x4 f (x4)

k = 3

x i = x0 + i h, i = 0 , 1, . . . , n x t x = x0 + h t t =

x −x0h

x −x j = ( x0 + h t ) −(x0 + j h ) = h(t − j ),

(x −x0)(x −x1) · · ·(x −x i−1) = h i (t −0)(t −1) · · ·(t −(i −1)) == h i i!

ti

.

x j [x0, . . . , x i ]f j

i

r =0r = j

(x j −xr ) =i

r =0r = j

( j h −r h ) = hi (−1)i− j j ! (i − j )!.


96/412


97/412

x i f i ∆ 1f i ∆ 2f i ∆ 3f ix0 f 0 f 1 −f 0x1 f 1 ∆ f 1 −∆ f 0f 2 −f 1 ∆ 2f 1 −∆ 2f 0x2 f 2 ∆ f 2 −∆ f 1f 3

−f 2

x3 f 3

(x −x0)(x −x−1) · · ·(x −x−(i−1) ) == hi (t + 0)( t + 1) · · ·(t + ( i −1)) == ( −1)ih i (−t −0)(−t −1) · · ·(−t −(i −1)) == ( −1)ih i i! −ti .

i

r =0r = j

(x− j −x−r ) =i

r =0r = j

((− j )h −(−r )h) = hi (−1) j j ! (i − j )!

[x0, . . . , x −i]f = 1h ii

j =0(−1) j 1 j !(i − j )!

f (x− j ) = 1h i i!i

j =0(−1) j i j f − j .

p(x) = p(x0 + h t ) =n

i=0(−1)i −

ti

i

j =0(−1) j

i j

f − j =

=n

i=0

(−1)i −t

i ∇if 0 .

∇ r

∇if r :=

i

j =0(−1) j

i j

f r − j .

∇

0f r = f r ,

∇if r = ∇i−1f r −∇i−1f r−1.


98/412

x i f i ∇1f i ∇2f i ∇3f ix0 f 0 f 0 −f −1x−1 f −1 ∇f 0 −∇f −1f −1 −f −2 ∇2f 0 −∇2f −1x−2 f −2 ∇f −1 −∇f −2f

−2

−f −

3x−3 f −3

f ∈ C ([a, b]) n + 1

f − pn ∞ → 0 n → ∞

f

f (x) = ex [−1, 1] x i = −1 + 2n i, i = 0 , 1, . . . , n

α O(eα n )

n α ×10

×10 ×10 ×10 ×10

×10 ×10

n α

ex

f (x) = (1 + 25 x2)−1 [−1, 1]

f − pn ∞ n → ∞ f [−1, 1] ⊂ C

X

S ⊂X P : X →S


99/412

f ∈X f −P f ≤(1 + P ) inf s∈S f −s = (1 + P ) dist( f, S ) .

P P s = s s∈S

f −P f = (f −s) −P (f −s) = (I −P )( f −s) ≤(1 + P ) f −s . s

s∈S

I n : C ([a, b]) →P n : f →I n f := I n f (x i ) = f (x i)n

i=0,

f −I n f ∞ ≤(1 + I n ∞) dist ∞(f, P n ) .

dist ∞(f, P n ) 0 n I n ∞ n

I n ∞ = supf =0I n f ∞f ∞

= supf ∞ =1

I n f ∞ ,

n

I n : C ([a, b]) → P n a ≤ x0 < x 1 < · · · < x n ≤ b I n ∞ = λn ∞ λn (x) :=

n

i=0| i (x)|

f ∈ C ([a, b])

I n f =n

i=0f (x i ) i , i (x) =

n

j =0 j = i

x −x jx i −x j

.

x∈[a, b]

|I n f (x)| =n

i=0

f (x i) i(x) ≤n

i=0|f (x i )|| i (x)| ≤

≤ f ∞n

i=0| i(x)| ≤ f ∞ λn ∞ .


100/412

x x∈[a, b]

I n f ∞ ≤ f ∞ λn ∞ f = 0 I n f ∞f

∞≤ λn ∞ =⇒supf =0

I n f ∞f

∞

= I n ∞ ≤ λn ∞ . λn ∈C ([a, b])

x̃∈[a, b] λn

λn ∞ = |λn (x̃)| = λn (x̃) =n

i=0| i(x̃)|.

C ([a, b]) g, g ∞ = 1 g(x i ) = sgn( i(x̃)) , i = 0 , 1, . . . , n | i (x̃)| = g(x i ) i (x̃)

λn ∞ = λn (x̃) =n

i=0 | i (x̃)| =n

i=0g(x i ) i (x̃) =

= I n g(x̃) ≤ I n g ∞ ≤ I n ∞ g ∞ = I n ∞ .

λn ∞

• λn ∞ λn ∞ ≈

2n +1

e n log n, n → ∞.

• T n [a, b] λn ∞

2π

log(n + 1) + 0 .9625 · · ·< λn ∞ < 2π

log(n + 1) + 1 .

• ( λn ∞)n≥0

• f ∈ C ([a, b]) f

x i i = 0 , 1, . . . , n xi = x j , i = j

n

i=0i(x) ≡1,


101/412

i (x) := n,i (x) :=n

j =0 j = i

x −x jx i −x j

A = {x0, x1, . . . , x n} B = {w0, w1, w2} ⊆ A. f pi ∈ P n−2

f (A \ B )∪{wi}. pi p∈P n f A

Φ(a ,x ,y ) = a0 + a1x + a2y + a3x2 + a4xy + a5y2, a = ( a i)5i=0 .

(x i , yi ) ∈ R 2 Φ(a , x i , yi) = zi , i = 1 , 2, . . . , 6 zi ∈ R

f ∈ C 1([x0, xn ]) x0 < x 1 < · · · < x n Ai , B i ∈P 2n +1

p2n +1 =n

i=0f (x i )Ai +

n

i=0f (x i )B i

p2n +1 (x i) = f (x i ), p2n +1 (x i ) = f (x i ), i = 0 , 1, . . . , n .

f (x) = 41 + x

p∈P 5

p(0) = f (0) , p (0) = f (0) , p (0) = f (0) , p(1) = f (1) , p (1) = f (1) , p (1) = f (1) .

x = 12

maxx∈[0,1]

|f (x) − p(x)|.

p(0) = 1 , p (0) = 2 , p (0) = 3 , p(1) = −1, p (1) = 3 , p(2) = 4 .

[x0, x1, . . . , x n ]f

f (x) = x1 + x

xi = x j i = j


102/412

x i i = 0 , 1, . . . , n

[x0, x1, . . . , x n ]f =n

i=0

f (x i)ω (x i)

,

ω(x) :=n

i=0(x −x i) ω = ddx ω

x i i = 0 , 1, . . . , n

[x0, x1, . . . , x n ]f = 12π i γ f (z)ω(z) dz ,

γ xi f

f ∈C 1([a, b]) x1 ∈[a, b]

ddx

([x1, x]f )∈C ([a, b]) .

f (x) = sin( x) 0, π12

, π6

f p x0, x1, . . . , x n

h := maxi |x i+1 −x i|

I

f − p ∞,I ≤ f (n +1) ∞4(n + 1)

hn +1 .

f

x : 0 0.2 0.4 0.6 0.8 1f (x) : −1 3 2 4 6 1

f (0.5)

x i = x0 + i h i = 0 , 1, . . . , n

∇n f (x0) = ∆ n f (x−n )

∇n f (x ) =

n

j =0

n j

(−1) j f (x − j )

x i = x0 + i h i = 0 , 1, . . . , n i


103/412

p∈P 2m 2m +1

x i = x0 + i h i = −m, −m + 1 , . . . , −1, 0, 1, . . . , m

p(x0 + t h ) = f 0 +m

i=1

t + i −12i −1

δ 2i−1f 12

+t + i −1

2iδ 2if 0 .

δ δf k := f k+ 12 −f k−12 f k := f (xk )

T = {(x i , y j ) | 0 ≤ i ≤n, 0 ≤ j ≤m}, x0 < x 1 < · · · < x n y0 < y1 < · · · < ym

p(x, y ) f T


104/412

3, 4, . . . [a, b]⊂R

xxx := ( x i )ni=0

a =: x0 < x 1 < . . . < x n := b.

x i f

|(x i ,x i +1 ) ∈

P k

∆ x i = xi+1 −x i

∆ x := max0≤i≤n−1

∆ x i .

x i ν i −1 ν i ∈N 0 0 ≤ν i ≤k +1

ν i = 0 xi ν i = 1 xi ν i = k + 1

x i

xxx ∆ x i → 0

r k −r

(k + 1) 0

ν ν ν = ( ν i )n

−1

i=1

≤k P k,xxx,ν ν ν

P k,xxx,ν ν ν = f | f |(x i ,x i +1 ) ∈P k , i = 0 , 1, . . . , n −1;

f ( )(x i −0) = f ( )(x i + 0) , = 0 , 1, . . . , ν i −1, 1 ≤ i ≤n −1 .

dim P k,xxx,ν ν ν = n dim P k −n−1

i=1

ν i = n(k + 1) −n−1

i=1

ν i .


105/412

f (x) :=

9 −x3, 0 ≤x < 32 ,x3 + 12 ,

32 ≤x < 2,

−12 x3 + x2 + 172 , 2 ≤x < 3,2x3 −18x2 + 932 x − 552 , 3 ≤x ≤5.

f k = 3

xxx = 0, 32

, 2, 3, 5 .

f (32 −0) = f (

32 + 0) ,

f (2 −0) = f (2 + 0) , f (2 −0) = f (2 + 0) ,f (3 −0) = f (3 + 0) , f (3 −0) = f (3 + 0) , f (3 −0) = f (3 + 0) ,

f ∈P 3,xxx,ν ν ν ν ν ν = (0 , 1, 2) .

(0, 32

, 32

, 32

, 32

, 2, 2, 2, 3, 3, 5).

1 2 3 4 5 x

2

4

6

8

f


106/412

ν i = k, i = 1 , 2, . . . , n −1,

k + 1 − ν i = 1

S k := S k,xxx

dim S k,xxx = n(k + 1) −(n −1)k = n + k.

(x i , x i+1 ) pi ∈P k

S 1,xxx n+1 (H i )ni=0

S 1,xxx

x i

I 1 : C ([a, b]) →S 1,xxx , f →I 1f := I 1f (x i) = f (x i )n

i=0.

I 1f x i ≤x ≤x i+1

I 1f (x) = f (x i ) xi+1 −xx i+1 −x i + f (xi+1 ) x −x ix i+1 −x i , x

i ≤x ≤x i+1 .

∆ x i = xi+1 −x i ∆ x = max0≤i≤n−1 ∆ x i

f ∈C 2([a, b])

f −I 1f ∞ ≤ 18

∆ x2 f (2) ∞ .

f

f (x) = I 1f (x) + ( x −x i)(x −x i+1 )[x i , x i+1 , x]f .


107/412

x∈[x i , x i+1 ]

|f (x) −I 1f (x)| ≤ |(x −x i )(x −x i+1 )| |f (2) (ξ i )|

2! ≤ (x i < ξ i < x i+1 )

≤ (x −x i)(x i+1 −x)f (2)

∞2 ≤≤

x i + x i+12 −x i x i+1 −

xi + x i+12

f (2) ∞2

=

= 18

∆ x2i f (2)

∞ ≤ 18

∆ x2 f (2) ∞ .

(x −x i)(x i+1 −x) [x i , x i+1 ]

12 (x i + x i+1 )

I 1

(x i , x i+1 ) I 1 ∞ x i ≤x ≤x i+1

|I 1f (x)| = f (x i) xi+1 −xx i+1 −x i

+ f (x i+1 ) x −x ix i+1 −x i ≤

≤ |f (x i )| xi+1 −xx i+1 −x i

+ |f (x i+1 )| x −x ix i+1 −x i ≤

f ∞ .

f = 0 I 1f ∞ ≤ f ∞ ⇒ I 1f ∞f ∞ ≤

1⇒ I 1 ∞ = supf =0I 1f ∞f ∞ ≤

1.

I 1 ∞ ≥ I 1 f ∞f ∞ f = 0

f = 1 I 1f ∞ = f ∞ = 1 I 1 ∞ = 1

f ∈C ([a, b]) f −I 1f ∞ ≤2 dist∞(f, S 1,xxx ) .

I 1f f I 1

S 1,xxx

C ([a, b])

f, g :=b

a

f (x)g(x)dx f 2 = f, f . L1

L1 : C ([a, b]) →S 1,xxx , f →L1f := f −L1f ⊥S 1,xxx .


108/412

S 1,xxx (H i )ni=0

L1f =n

i=0α iH i ,

ααα = ( α i )ni=0 Gααα = bbb G = ( H i , H j )ni,j =0

H i (x i−1, x i+1 ) |i − j | > 1 H i , H j = 0 G

H i , H i =

x i +1

x i − 1

H 2i (x)dx =x i

x i − 1

H 2i (x)dx +

x i +1

x i

H 2i (x)dx =

=x i

x i − 1 x −x i−1x i −x i−12

dx +

x i +1

x i x i+1 −xx i+1 −x i2

dx =

= ∆ x i−11

0

t2dt + ∆ x i

1

0

t2dt = 13

(∆ x i−1 + ∆ x i ) ,

∆ x−1 := 0 ∆ xn := 0 x

→t =

xi+1 −xx i+1 −x i

H i , H i+1 =

x i +1

x i

H i (x)H i+1 (x) dx =

x i +1

x i

x i+1 −xx i+1 −x i

x −x ix i+1 −x i

dx =

= ∆ x i

1

0

t(1 −t)dt = ∆ x i12 −

13

= 16

∆ x i .

G =

13 ∆ x0 16 ∆ x016 ∆ x0

13 (∆ x0 + ∆ x1)

16 ∆ x1

16 ∆ x i−1

13 (∆ x i−1 + ∆ x i)

16 ∆ x i

16 ∆ xn−1

13 ∆ xn−1

.

G


109/412

bbb = ( H i , f )ni=0 L1

L1 : C ([a, b]) → S 1,xxx L1

∞ ≤3 .

L1f

L1f ∞ =n

i=0α i H i

∞≤

n

i=0|α i |H i

∞≤ max0≤i≤n |α i|

n

i=0H i

= 1 ∞= max

0≤i≤n |α i|.

G |α | = max0

≤i

≤n |α i | α −1 α α +1

∆ x −16

α −1 + ∆ x −1 + ∆ x

3 α +

∆ x6

α +1 =

x +1

x − 1

f (x)H (x) dx .

|α | = 3

∆ x

−1 + ∆ x −

∆ x −16

α −1 − ∆ x

6 α +1 +

x +1

x − 1

f (x)H (x) dx ≤

≤ 3

∆ x −1 + ∆ x∆ x −1

6 |α −1|+ ∆ x

6 |α +1 |+ f ∞x +1

x − 1

H (x) dx .

x +1

x − 1

H (x) dx = ∆ x −1 + ∆ x

2

|α ±1| ≤ |α | |α |

|α | ≤ 12 |α |+

32

f ∞ . |α | ≤3 f ∞ L1f ∞ ≤ max0≤i≤n |α i | ≤3 f ∞

L1 ∞ ≤3

f ∈C ([a, b]) f −L1f ∞ ≤4 dist∞(f, S 1,xxx ) .


110/412

[a, b] (x i)ni=0 a = x0 < x 1 < . . . < x n = b

f ∈ S 3,xxx (x i , x i+1 )

≤3

x i f [x i , x i+1 ] pi ∈ P 3

pi f

pi (x i ) = f (x i ) , pi (x i+1 ) = f (x i+1 ) .

pi

pi−1(x i ) = pi (x i )

pi (x i) = si , pi (x i+1 ) = si+1 ,

s i , i = 0 , 1, . . . , n pi pi

pi (x) = f (x i ) + ( x −x i)s i + ( x −x i )2 [x i , x i+1 ]f

−s i

∆ x i +

+ ( x −x i)2(x −x i+1 )s i+1 + s i −2[x i , x i+1 ]f

∆ x2i.

s i f xi

[ ]f [ , ]f [ , , ]f [ , , , ]f

x i f (x i )s ix i f (x i ) [x

i , x i+1 ]f −s i∆ x i[x i , x i+1 ]f

si+1 + s i −2[x i , x i+1 ]f ∆ x2i

x i+1 f (x i+1 ) si+1 −[x i , x i+1 ]f

∆ x is i+1x i+1 f (x i+1 )

pi

s i =

numericna analiza

Documents