lý thuyết xấp xỉ và ứng dụng

8/2/2019 L thuyt xp x v ng dng

1/91

L t h u y t x p x v n g d n g

G S . T S K H i n h D n g

2 - 2 0 0 7


2/91

M c l c

1

C c n h l W e i e r s t r a s s 3

1 . 1 C c k h i n i m c b n . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 . 1 . 1 a t h c B e r n s t e i n . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 . 1 . 2 C h u i F o u r i e r . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1 . 2 X p x b n g t o n t t c h p h n . . . . . . . . . . . . . . . . . . . . . . . 8

1 . 2 . 1 n h l W e i e r s t r a s s t r o n g k h n g g i a n B a n n a c h . . . . . . . . . . 1 1

1 . 2 . 2 C c h x y d n g n h n . . . . . . . . . . . . . . . . . . . . . . . . 1 1

1 . 3 n h l K o r o v k i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2

1 . 4 B i t p c u i c h n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4

2 X p x t t n h t 1 5

2 . 1 X p x t t n h t t r o n g k h n g g i a n n h c h u n . . . . . . . . . . . . . . . . 1 5

2 . 1 . 1 S t n t i c a p h n t x p x t t n h t . . . . . . . . . . . . . . . 1 6

2 . 1 . 2 T n h d u y n h t c a x p x t t n h t . . . . . . . . . . . . . . . . . . 1 6

2 . 1 . 3 T n h l i n t c c a p h n t x p x . . . . . . . . . . . . . . . . . . 1 7

2 . 2 X p x t t n h t t r o n g k h n g g i a n H i l b e r t . . . . . . . . . . . . . . . . . 1 8

2 . 2 . 1 X p x p h i t u y n t r o n g k h n g g i a n H i l b e r t . . . . . . . . . . . . . 1 9

2 . 3 X p x t u y n t n h t r o n g k h n g g i a n n h c h u n . . . . . . . . . . . . . . 2 0

2 . 4 B i t p c u i c h n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3

3 C c k h n g g i a n H m 2 4

3 . 1 M t s k h i n i m c b n , k h n g g i a n Lp(A), C(A) . . . . . . . . . . . 2 4

3 . 1 . 1 K h n g g i a n C(A)

. . . . . . . . . . . . . . . . . . . . . . . . . . 2 4

3 . 1 . 2 K h n g g i a n Lp(A) . . . . . . . . . . . . . . . . . . . . . . . . . 2 5

3 . 2 K h n g g i a n c c h m k h v i : K h n g g i a n S o b o l e v . . . . . . . . . . . . . 2 5

3 . 2 . 1 a t h c T a y l o r v b t n g t h c o h m . . . . . . . . . . . . . 2 6

3 . 2 . 2 P h n h o c h n v v o h m s u y r n g . . . . . . . . . . . . . 2 9

3 . 3 M o d u l l i n t c v m o d u l t r n . . . . . . . . . . . . . . . . . . . . . . . 3 2

3 . 3 . 1 M o d u l l i n t c . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2

3 . 3 . 2 M o d u l t r n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4

3 . 4 K h n g g i a n B V ( A ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7

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3/91

3 . 5 K h n g g i a n L i p s c h i t z v k h n g g i a n H o

l d e r . . . . . . . . . . . . . . . 3 8

3 . 6 K - P h i m h m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2

3 . 7 B t n g t h c B e r n s t e i n . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7

3 . 8 B i t p c u i c h n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1

4

C c n h l t r u n g t m c a l t h u y t x p x 5 5

4 . 1 C c n h l t h u n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5

4 . 2 X p x n g t h i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 9

4 . 3 C c n h l n g c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2

4 . 4 X p x b n g a t h c i s . . . . . . . . . . . . . . . . . . . . . . . . . 6 3

4 . 5 B i t p c u i c h n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5

5

S n g n h 6 8

5 . 1 S n g n h H a a r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9

5 . 2 P h n t c h a p h n g i i v c s s n g n h t r c c h u n . . . . . . . . . . . . 7 4

5 . 3 S n g n h S h a n n o n - k o t e l n i k o v . . . . . . . . . . . . . . . . . . . . . . . 7 9

5 . 4 H m t h a n g b c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3

5 . 4 . 1 G i m n h i u k i n t r c c h u n . . . . . . . . . . . . . . . . . . . 8 3

5 . 4 . 2 X y d n g p h n t c h a p h n g i i . . . . . . . . . . . . . . . . . . 8 4

5 . 5 B i t p c u i c h n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 0

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C h - n g 1

C c n h l W e i e r s t r a s s

T r o n g c h n g n y c h n g t a s x p x c c h m s t r o n g k h n g g i a n

C(A), k h n g g i a n c c h m l i n t c x c n h t r n t p A, t r o n g A l

[a, b],T := [0, 2) , h o c m t t p c o m p a c t t r o n g Rn , h o c t n g q u t h n l

k h n g g i a n t p c o m p a c t H a u s d o r f f , b i c c a t h c l n g g i c k h i A = T

v a t h c i s t r o n g n h n g t r n g h p c n l i .

1 . 1 C c k h i n i m c b n

G i s X l m t k h n g g i a n c a c c h m x c n h t r n A , f X. T a c n t m h m n g i n ( t h u n t i n c h o t n h t o n )

t m t t p c o n

c a

Xs a o

c h o f r t g n v i .

K h n g g i a n X t h n g l k h n g g i a n n h c h u n h o c l k h n g g i a n B a n n a c h

c a c c h m x c n h t r n A

, c h n g h n n h C(A), Lp(A) v i 1 p . K h i

X l k h n g n h c h u n t h k h o n g c c h g i a f v c o b n g f X . i l n g f X c g i l s a i s x p x f b i . T p c o n l m t t p c c h m s c t n h c h t n g i n , t h u n t i n c h o t n h t o n . c g i l

k h n g g i a n x p x . D i y l m t s k h n g g i a n x p x q u a n t r n g .

( a )

= Pnl m t t p c c a t h c i s b c n h h n h o c b n g

n, t c l

t p c c h m c d n g

Pn(x) =nk=0

akxk.

Pn t h n g d n g x p x c c h m x c n h t r n [a, b] .( b )

= Tn l t p c c a t h c l n g g i c b c n h h n h o c b n g n, t c l c c h m x c n h t r n T c d n g

Tn(x) =a0

2+

n

k=1

(ak cos kx + bk sin kx).

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H o c

Tn(x) =|k|n

akeikx.

Tn t h n g d n g x p x c c h m x c n h t r n T.( c ) L p c c h m s p l i n e .

( d ) L p c c s n g n h .

C h n g t a b i t r n g k h i A l t p c o m p a c t t h C(A) l k h n g g i a n

B a n n a c h v i c h u n

f := maxxA

|f(x)|.

H a i n h l d i y s g i q u y t v n t r n c h o t r n g h p X = C(A)

v i

A = [a, b]h o c T.

n h l 1 . 1 . 1

. (W e i e r s t r a s s - 1 ) M i h m f l i n t c t r n o n [a, b] c t h

x p x b n g a t h c i s v i c h n h x c t u , n g h a l v i m i > 0,

t n t i a t h c i s P

s a o c h o

f PC([a,b]) .

n h l 1 . 1 . 2

. (W e i e r s t r a s s - 2 ) M i h m f l i n t c t r n T c t h x p x b n g

a t h c l n g g i c v i c h n h x c t u , n g h a l v i m i > 0 , t n t i

a t h c l n g g i c

Ts a o c h o

f TC(T) .

H a i n h l n y c c h n g m i n h t r o n g c c m c s a u , d a v o c c t n h

c h t c a m t s t o n t t u y n t n h c b i t .

1 . 1 . 1 a t h c B e r n s t e i n

G i s f

C([0, 1]) , c n g t h c

Bn(f, x) :=nk=0

n

k

f(

k

n)xk(1 x)nk, n = 0, 1.2....

x c n h m t n h x t C([0, 1])

v oPn . T a g i Bn(f) l a t h c B e r n s t e i n

b c n c a f. M n h s a u c h o t a b i t c c t n h c h t c a Bn :

M n h 1 . 1 . 3

.

( i ) Bn l t o n t t u y n t n h b c h n v i c h u n 1, x c n h d n g , t c l

Bn = 1, Bn(f) 0 v i f(x) 0 x A.

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( i i ) K h i u ek(x) := xk, k = 0, 1, 2. T a c

Bn(e0) = e0, Bn(e1) = e1, Bn(e2, x) = e2(x) +x(1 x)

2.

C h n g m i n h . ( i ) . H i n n h i n Bn l t o n t t u y n t n h v x c n h d n g . V i

m if C([0, 1])

t a c

|Bn(f, x)| nk=0

n

k

xk(1 x)nkf = f,

d o

Bn(f) f. t b i t , v i

f = 1t h Bn(f) = f . S u y r a Bn = 1.

( i i ) . T a c e0 = 1 n n Bn(e0) = e0.

T a c n g c

Bn(e1, x) =nk=1

n

k

k

nxk(1 x)nk

= xn1s=0

n 1

s

xs(1 x)ns1 = x

( 1 . 1 )

v

nk=0

k(k 1)

n

k

xk(1 x)nk = n(n 1)x2

n2s=0

n 2

s

xs(1 x)ns2

= n(n 1)x2.( 1 . 2 )

T ( 1 . 1 ) s u y r a

nk=1 k

nk

xk(1 x)nk = nx, k t h p v i ( 1 . 2 ) t a c

n

k=0

k2n

kxk(1 x)nk = n2x2 + nx(1 x).

V yBn(e2, x) = e2(x) +

x(1x)n

.

1 . 1 . 2 C h u i F o u r i e r

G i s f L1(T) - k h n g g i a n B a n n a c h c c h m k h t c h c p 1 t r n T . K h i c h u i

S[f]

nZf(n)einx, t r o n g f(n) :=

1

2

f(x)einxdx,

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c g i l c h u i F o u r i e r ( d n g p h c ) c a f, v f(n) l h s F o u r i e r c a

f. C h u i F o u r i e r d n g t h c c a

fl c h u i c d n g

a0

2 +

k=1(a

k cos kx + bk sin kx),

t r o n g

ak :=1

f(x)cos kxdx, bk :=1

f(x)sin kxdx,

l h s F o u r i e r c a f.

T a c

f(k) + f(k) = 12

f(x)(eikx + eikx)dx

= 1

f(x)cos kxdx = ak,

v

f(k) f(k) = 12

f(x)(eikx eikx)dx

=i

f(x)sin kxdx = ibk,

n h v y ak , bk c t h b i u d i n q u a f(k) v f(k) . N g c l i t a c n g c

f(k) = (ak ibk)/2.G i s f L1(T) , i l n g

Sn(f, x) :=|k|n

f(k)eikx

c g i l t n g F o u r i e r b c n c a f. V Sn(f) fL1(T) c t h k h n g h i t n k h n g k h i n , n n t a k h n g d n g Sn(f) x p x f. T a c t h k h c p h c n h c i m n y n h s a u :

V if, g

L1(T), t c h c h p c a h a i h m f v g l h m f

g

c x c n h

b i

(f g)(x) = 12

T

f(x y)g(y)dy,x T.

T a c

Sn(f, x) =|k|n

1

2

f(t)eik(tx)dt

=1

2

f(x t)|k|n

eiktdt

= (Dn f)(x) t r o n g Dn(t) :=|k|n

eikt.

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T a g i Dn(t) l n h n D i r i c h l e t . t

n(f, x) :=1

n + 1

n

k=0Sk(f, x),

v

Fn(x) :=1

n + 1

nk=0

Dk(x).

K h i t a c

n(f, x) =1

n + 1

nk=0

(Dk f)(x)

= (Fn f)(x). ( 1 . 3 )

T a g i Fn(x) l n h n F e j e r . D i y l c c t n h c h t n g i n c a n h n

F e j e r v n h n D i r i c h l e t .

M n h 1 . 1 . 4

.

( i ) Dn v Fn l c c a t h c l n g g i c b c n.

( i i )

Dn(x) =sin (2n+1)x

2

sin

x

2

v Fn(x) =sin2 (n+1)x

2

(n + 1)sin2 x

2

( i i i ) n l t o n t t u y n t n h x c n h d n g , Dn i d u .

( i v ) n = 1 .

C h n g m i n h . ( i ) Dn(x) v Fn(x) l a t h c l n g g i c v

Dn(x) =|k|n

eikx v Fn(x) =|k|n

(1 |k|n + 1

)eikx

( i i ) T a c

Dn(x) = 1 +nk=1

eikx +1

k=n

eikx

= 1 +nk=1

eikx +nk=1

eikx

= 1 + 2Re(nk=1

eikx)

= 1 + 2nk=1

cos kx,

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n h n c h a i v v i sin x

2t a c Dn(x)sin

x2

= sin (2n+1)x2

. V y

Dn(x) =sin (2n+1)x

2

sin x2

.

M t k h c t a l i c

Fn(x) =1

n + 1

nk=0

Dk(x)

=1

(n + 1)sin x2

nk=0

sin(2k + 1)x

2=

1

(n + 1) sin x2

sin2 (n+1)x2

sin x2

.

V y ( i i ) c c h n g m i n h .

( i i i ) H i n n h i n .

( i v ) T h e o n h n g h a c a t c h c h p , t ( 1 . 3 ) t a s u y r a

n(f, x) =1

2

Fn(x t)f(t)dt,

d o

n(f)C(T) fC(T) 12

Fn(x t)dt = fC(T).

L y f = 1 t h n(f) = f, s u y r a n = 1.

1 . 2 X p x b n g t o n t t c h p h n

T r o n g m c n y t a x t A = [a, b] h o c A = T. C h o Kn(x, y), n = 1, 2,..., l m t

d y c c h m l i n t c t r n AA. T a x c n h m t t o n t t c h p h n b i c n g t h c

fn(x) :=

A

Kn(x, y)f(y)dy. ( 2 . 4 )

C h n g t a m u n b i t k h i n o fn(x)

f(x) . G i t h i t r n g A

Kn(x, y)dy 1 u t h e o x k h i n , ( 2 . 5 )

v v i m i > 0,|xt|

|Kn(x, y)|dy 0 u t h e o x k h i n . ( 2 . 6 )

( K h i A = T, m i n l y t c h p h n ( 2 . 6 ) c t h a y b i |x t| . ) T a c n h l s a u :

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n h l 1 . 2 . 1

.G i t h i t ( 2 . 5 ) v ( 2 . 6 ) c t h o m n v

A

|Kn(x, y)|dy M(x), x A, n = 1, 2,.... ( 2 . 7 )

K h i v i m i

f C(A), x A, fn(x) f(x) k h i n . S h i t n y l u n u M(x) k h n g p h t h u c x.

C h n g m i n h . D o f l h m l i n t c t r n t p c o m p a c t n n v i m i > 0, t n

t i > 0 s a o c h o

|f(x) f(y)| , |x y| . t

n(x) :=

A

f(x)Kn(x, y)dy f(x),

t ( 2 . 5 ) s u y r a

n(x)h i t u v

0k h i

n .T a c , v i x c n h ,

|fn(x) f(x)| = |A

(f(y) f(x))Kn(x, y)dy + n(x)|

A

|(f(y) f(x))||Kn(x, y)|dy + |n(x)|

=

|xy|

|(f(y) f(x))||Kn(x, y)|dy+

|xy| |

(f(y)

f(x))

||Kn(x, y)

|dy +

|n(x)

| (M(x) + 1) + 2fC(A), v i n l n .

V y fn(x) f(x), k h i n . N u M(x) k h n g p h t h u c x , t h h i n n h i n s h i t n y l u .

T n g t n h n h l t r n t a c n g c n h l s a u v i c c h c h n g m i n h

c h c n t h a y t c h p h n b n g t n g .

n h l 1 . 2 . 2

.C h o Kn(x, s), x [0, 1], s = 0, 1, 2, .... l d y c c h m l i n t c

t r n

[0, 1], v

f C([0, 1]), t

fn(x) =nk=0

f(k

n)Kn(x, k).

G i t h i t

ns=0

Kn(x, s) 1 u t h e o x k h i n . ( 2 . 8 )

|s/nx|

Kn(x, s) 0 u t h e o x k h i n , > 0. ( 2 . 9 )

nk=0

|Kn(x, k)| M(x). ( 2 . 1 0 )

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K h i fn(x) f(x), k h i n . S h i t n y l u n u M(x) k h n g p h t h u c

x.

T c c n h l t r n t a c c c h q u s a u :

H q u 1 . 2 . 3

.C h o f C(T) . K h i n(f, x) h i t u n f(x) k h i n .

C h n g m i n h . T a c n(f, x) =

12

f(y)Fn(x y)dy , d o c h n

Kn(x, y) =1

2Fn(x y) = 1

2(n + 1)

sin2(n + 1) xy2

sin2 xy2

.

T a c

|Kn(x, y)| 12(n + 1) sin2

v i |x y| .( 2 . 1 1 )

v T

Kn(x, y)dy = 1, n = 1, 2..., ( 2 . 1 2 )

T ( 2 . 1 1 ) t a s u y r a ( 2 . 6 ) , ( 2 . 1 2 ) s u y r a ( 2 . 5 ) v ( 2 . 7 ) . T h e o n h l 1 . 2 . 1

s u y r a k h n g n h t r n .

H q u 1 . 2 . 4

.C h o f C([0, 1]) . K h i Bn(f, x) h i t u n f(x) k h i

n .

C h n g m i n h . T h e o n h n g h a c a Bn(f, x) t a c

Bn(f, x) =nk=0

n

k

f(

k

n)xk(1 x)nk,

d o c h n Kn(x, s) =ns

xk(1 x)nk =: pn,s(x). T a c

| snx|

Kn(x, s) 1

2

ns=0

(s

n x)2pn,s(x).

=1

2 (Bn(e2, x) 2xBn(e1, x) + x2

Bn(e0, x)).

C h r n g Bn(e0, x) = e0(x); Bn(e1, x) = e1(x); Bn(e2, x) = e2(x) +x(1x)n

,

d o t a s u y r a | sn1|

Kn(x, s) x(1 x)

n2

1

4n2.

M t k h c t a c

ns=0 Kn(x, s) = 1 v v i m i s , Kn(x, s) d n g . V y t h e o

n h l 1 . 2 . 2 s u y r a h q u t r n .

N h n x t 1 . 2 . 5

.T c c h q u t r n t a l n l t s u y r a c c n h l c a

W e i e r s t r a s s c h o h m t u n h o n v k h n g t u n h o n .

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1 . 2 . 1 n h l W e i e r s t r a s s t r o n g k h n g g i a n B a n n a c h

C h o A = [a, b] h o c A = T , X l k h n g g i a n B a n n a c h c c h m s x c

n h t r n A

.

n h l 1 . 2 . 6

.G i s k h n g g i a n B a n n a c h X t h o m n c c i u k i n s a u :

( i ) C(A) l t r m t t r o n g X.

( i i ) C(A) c n h n g l i n t c t r o n g X, t c l , v i m i f C(A), fX CfC(A), t r o n g C l h n g s .

K h i v i f X, v i m i > 0 , t n t i a t h c g ( i s k h i A = [a, b],l n g g i c k h i A = T) s a o c h o

f

g

X .

C h n g m i n h . D o ( i ) n n t n t i h C(A) s a o c h o f hX . T h e o n h l

W e i e r s t r a s s t n t i a t h c g ( i s k h i A = [a, b] , l n g g i c k h i A = T )

s a o c h o h gC(A) .T a c

f gX f hX + h gX + Ch gC(A)

(1 + C).

V y k h n g n h c c h n g m i n h .

H q u 1 . 2 . 7

. n h l W e i e r s t r a s s n g t r o n g k h n g g i a n Lp(A) , 1 p < .

1 . 2 . 2 C c h x y d n g n h n

T r o n g p h n n y c h n g t a s t r n h b y c c h x y d n g c c n h n Kn(x, y),

n = 1, 2, .... c a n h l 1 . 2 . 1

G i s r n g L1(R) , v R (u)du = 1. V i m i > 0, t a l y (u) =

1

(u/).

K h i v i m i > 0 , t a c |xy|

|(x y)|dy =|u|/

|(u)|du 0, K h i 0.

B y g i t Kn(x, y) := 1/n(x

y). K h i t t c c c i u k i n c a n h l

1 . 2 . 1 c t h o m n .

1 1


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14/91

C h n s a > 0 s a o c h o 1 = e0(x) aP(x), x A. K h i

Un(e0, x) aUn(P, x) aP(x), n .

V v y t n t i m t s

M0 > 0s a c h o

Un(e0) M0.T a c n n k t q u s a u : C h o fy C(A), y A l m t h c c h m s a o c h o

fy(x) l h m l i n t c t h e o (x, y) A A v fy(y) = 0, y A. K h i Un(fy, y) 0, u t h e o y k h i n . ( 3 . 1 7 )

c h n g m i n h ( 3 . 1 7 ) , x t > 0 v t p n g c h o c a A A, B := {(y, y) :y A}

. M i m t i m (a, a)

c aB

c m t l n c n Va t r o n g A A s a o c h o

|fy(x)| < , v i m i (x, y) Va . G i G = aA

Va , v G l m t t p m , n n p h n

b F c a l t p n g , d o F l t p c o m p a c t ( v A c o m p a c t ) . T a x c n h

c c s m, M b i

m := min(x,y)F

Py(x) > 0; M := max(x,y)F

|fy(x)|.

N u (x, y) G, t h |fy(x)| < . N u (x, y) G, t h |fy(x)| MmPy(x) . V v y

|fy(x)| + Mm

Py(x) ( 3 . 1 8 )

T ( 3 . 1 8 ) t a c

|Un(fy, y)| Un(e0, y) + Mm

Un(Py, y)

M0 +M

mUn(Py, y) (M0 + 1), v i n l n .

T y s u y r a ( 3 . 1 7 ) .

B y g i t a c t h h o n t h n h v i c c h n g m i n h n h l . V i m i f C(A), t

fy(x) := f(x) f(y)

P(y) P

(x).

C h n g t a v a m i c h r a Un(f, y) f(y)P(y)Un(P, y) h i t u v k h n g t h e o yk h i n v v Un(P, y) h i t u n P(y), n n t a t h u c ( 3 . 1 6 ) . N h n x t

1 . 3 . 2

.S d n g n h l K o r o v k i n v i c c h m t h g1 = 1, g2 =

x, g3 = x2

t r n [0, 1] v

Py(x) = (y x)2 = y2g1 2yg2 + g3, Un = Bn,t r o n g Bn l t o n t B e r n s t e i n , t a s u y r a H q u 1 . 2 . 4 .

T n g t , t r n T, t a c t h x t g1 = x, g2 = cos x, g3 = sin x, v Py(x) =

1 cos(y x), Un = n , p d n g n h l K o r o v k i n t a s u y r a H q u 1 . 2 . 3 .

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1 . 4 B i t p c u i c h n g

B i t p 1

.G i t h i t n h l W e i e r s t r a s s c h o h m t u n h o n l c h n g

m i n h , h y c h n g m i n h

n(f, x)h i t u n

f(x)k h i

n .B i t p

2

.C h o {Qn}n=1 l a t h c i s , Qn c b c mn , Qn(x) h i t u

n f(x) k h i n , x [a, b]. C h n g m i n h r n g n u f k h n g p h i a t h c t h

mn , k h i n .

B i t p 3

.

X t t o n t t c h p h n

Wa(f, x) =1

a1/2

R

e(xt)2/a2f(t)dt.

T a t h c t r i n m i h m f C[0, 1] t h n h h m l i n t c t r n R v i g i c o m p a c t . C h n g m i n h r n g , v i m i > 0, k h i a n h t a c

|f(x) Wa(f, x)| < , x [0, 1].

B i t p 4

.

C h o C0 = {f C[0, 1] : f(0) = f(1) = 0}. C h n g m i n h r n g v i f C0 ,

Bn(f) (1 21n)f

B i t p 5

.

C h n g m i n h r n g h m l i n t c f t r n [0, 1] c x p x u b i c c a t h c

v i h s n g u y n k h i v c h k h i f(0) v f(1) l h a i s n g u y n .

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C h - n g 2

X p x t t n h t

T r o n g c h n g n y , c h n g t a n g h i n c u m t s v n l i n q u a n n

x p x t t n h t t r o n g k h n g g i a n n h c h u n , k h n g g i a n B a n a c h , k h n g

g i a n H i l b e r t n h l : S t n t i c a p h n t x p x t t n h t , t n h d u y n h t ,

t n h l i n t c .

2 . 1 X p x t t n h t t r o n g k h n g g i a n n h c h u n .

G i s X l k h n g g i a n n h c h u n , Y l k h n g g i a n c o n n g c a X,

f X. C h n g t a m u n x p x f b i c c p h n t c a Y .S a i s x p x

fb i c c p h n t c a

Y c o b n g

E(f) := E(f , Y , X ) := infY

f

c t r n g c h o x p x t t n h t f

b n g c c p h n t c a Y

. N u i n f i m u m t

c t i 0 Y , t h t a n i r n g 0 l m t x p x t t n h t f t Y . K h i Yl m t k h n g g i a n v e c t c o n c c h i u n, n h n m n h s p h t h u c c a

E(f) v o n, c h n g t a k h i u En(f) t h a y c h o E(f) . T n h l i n t c c a E(f)

d d n g c t r l i b n g n h n x t s a u :

N h n x t 2 . 1 . 1

.E(f)

l h m l i n t c t h e o f

.

T h t v y , g i s f, g X

v Y

, t a c

f f g + g .

L y i n f i m u m h a i v c a b t n g t h c n y t a c

E(f) f g + E(g).

T h a y i v a i t r c a

fv

gt a s c

E(g) f g + E(f).

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H a i b t n g t h c c u i c n g n i r n g

|E(f) E(g)| f g.

V y E(f) l h m l i n t c c a f.

2 . 1 . 1 S t n t i c a p h n t x p x t t n h t

n h l 2 . 1 . 2

. C h oY

l k h n g g i a n c o n h u h n c h i u c a k h n g g i a n

B a n a c h X, f X. K h i t n t i 0 Y s a o c h o

E(f) = E(f , Y , X ) = f 0

C h n g m i n h . T h e o n h n g h a c a

inf, t n t i d y {n}

n=1

s a o c h o

limn fn = E(f) . T a c n f n + f. D o d y {f n}n=1 h i t n n t n t i C > 0 s a o c h o

f n C n.V v y n C + f. N h n g d o Y l k h n g g i a n h u h n c h i u n n t n t i 0 Y v m t d y c o n {nk}k=0 s a o c h o nk 0 0 k h i k . K h i

f nk f 0 , v v y m E(f) = f 0 .

2 . 1 . 2 T n h d u y n h t c a x p x t t n h t .

V n t n t i x p x t t n h t k h n g p h i b a o g i c n g g n l i n v i t n h

d u y n h t c a n . T r c h t t a c n x t k h i n i m s a u :

n h n g h a 2 . 1 . 3

.C h u n c a X c g i l c h t n u v i m i f, g X,

, > 0t h o m n

f = g = 1, f = g, + = 1, t h

f + g < 1.

K h i t a g i X l k h n g g i a n n h c h u n c h t

V d 1

. K h n g g i a n R2 v i c h u n (x1, x2) =

|x1|2 + |x2|2 l n h c h u n c h t .

V d 2

. K h n g g i a n Lp(A) v i 1 < p < , l n h c h u n c h t . K h i p = 1

h o c p = t h Lp(A) k h n g n h c h u n c h t

n h l 2 . 1 . 4

.C h o X l k h n g g i a n n h c h u n c h t , Y l k h n g g i a n c o n

n g c a X, f

X. K h i p h n t x p x t t n h t f t Y n u t n t i l

d u y n h t .

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C h n g m i n h . G i s t n t i h a i p h n t k h c n h a u 1, 2 Y s a o c h o

f 1 = f 2 = E(f).

V i m i (0, 1), x t h m = 1 + (1 )2 Y, t a c

E(f) f f 1 + (1 )f 2 = E(f).

V y c n g l p h n t x p x t t n h t .

N u t g1 =f1E(f)

, g2 =f2E(f)

, g = g1 + (1 )g2 t h

g1 = g2 = 1 v g = 1.

i u n y m u t h u n v i t n h n h c h u n c h t c a X.

2 . 1 . 3 T n h l i n t c c a p h n t x p x

T n h l i n t c c a p h n t x p x c t r l i b n g n h l s a u :

n h l 2 . 1 . 5

.G i s X l k h n g g i a n B a n n a c h , Y l k h n g g i a n c o n h u

h n c h i u c a X. N u v i m i f X t n t i d u y n h t P(f) l x p x t t n h t f t Y , t h P l n h x l i n t c .

C h n g m i n h . G i s fk

f k h i k

. K h i d o X l k h n g g i a n B a n n a c h

n n t n t i M s a o c h o

fk M, k = 1, 2, 3....T a c E(fk) = fk P(fk) fk s u y r a

P(fk) fk P(fk) + fk 2M,

d o P(fk) b c h n v i k = 1, 2....

G i s r n g P(fk) P(f) k h i k . V d y {P(fk)}k=1 l d y b c h n

t r o n g k h n g g i a n h u h n c h i u n n t n t i d y c o n {P(fks)}

s=1s a o c h o

P(fks) = P(f) k h i s .

M t k h c

P(fks) fks P(f) fks,c h u y n q u a g i i h n k h i s t a c

f f P(f).

D o P(f) l x p x t t n h t n n f = f P(f). i u n y m u t h u n v i t n h d u y n h t c a

P(f).

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2 . 2 X p x t t n h t t r o n g k h n g g i a n H i l b e r t

G i s H l k h n g g i a n H i l b e r t v i t c h v h n g , . K h i Hc n g l k h n g g i a n n h c h u n , v i c h u n c m s i n h b i t c h v h n g ,

x = x, x. n h l

2 . 2 . 1

.C h o H l k h n g g i a n H i l b e r t v H0 l k h n g g i a n c o n c a

H,

f H\H0 . K h i H0 x p x t t n h t f t H0 k h i v c h k h i

f , h = 0 h H0.

C h n g m i n h . G i s l x p x t t n h t , v h l m t p h n t t u , c n h

t r o n g H0 . K h n g g i m t n g q u t , t a c t h g i t h i t f , h 0 . V i m i

> 0, t a c

f f hf 2 f 2 2f , h + 2h2h2 2f , h,

c h o 0+ t a c f , h = 0. N g c l i , v i m i g H0 , t a c

f g2 = f g + 2=

f

2 +

g

2

f

2.

S u y r a f f g, g H0. V y l x p x t t n h t f.B y g i x t k h n g g i a n H i l b e r t t c h c H. K h i t n t i c s

t r c c h u n m c {k}k=1 H. M i f H c m t b i u d i n

f =k=1

f, kk. ( 2 . 1 )

C c i l n g fk = f, k c g i l h s F o u r i e r c a f. C h u i k=1f, kk

c g i l c h u i F o u r i e r c a

f. M i

f H t h o m n n g t h c P a r s a v a l

f2 =k=1

|f, k|2 =k=1

|fk|2.

X tn

h m u t i n t r o n g c s t r n , {1,...,n}. C h n g t a m u n x p x

f b i k h n g g i a n c o n

Hn := s p a n {1,...,n}.K h i s a i s x p x s l

En(f) = inf {ck}

nk=1

fnk=1

ckk.

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n h l 2 . 2 . 2

.G i s l x p x t t n h t f t Hn . K h i

=nk=1 fkk ,

v s a i s x p x

En(f) = (

k=n+1

|fk|2)1/2.

C h n g m i n h . G i s =

nk=1 ckk l x p x t t n h t . T h e o n h l 2 . 2 . 1 t a

c

f , j = 0 j = 1, ...n.D o

k=1

fkk nk=1

ckk, j = 0, j = 1, ...n.

nk=1

(fk ck)k, j = 0, j = 1, ...n.

S u y r a fk = ck, k = 1, ...n. T n g t h c P a r s a v a l , s a i s x p x l

En(f) = ||f || = (

k=n+1

|fk|2)1/2.

V y n h l c c h n g m i n h .

N h n x t 2 . 2 . 3

. K h i n h s F o u r i e r u t i n c a f l r t n h h o c t r i t

t i u , k h n g s u y r a f n h . N h n g c n g r t n h . K h i En(f) f,v t h m t r n n k h n g c n g h a . C h n g t a c t h k h c p h c k h i m

k h u y t n y b n g c c h d n g x p x p h i t u y n .

2 . 2 . 1 X p x p h i t u y n t r o n g k h n g g i a n H i l b e r t

C h o k h n g g i a n H i l b e r t H

,M H

t a b i t s a i s x p x t t n h t f H

b n g c c p h n t c a M

l

E(f) = E(f , M , H ) = infgM

f g.

K h i M k h n g c c u t r c t u y n t n h t h t a g i l x p x p h i t u y n , c n k h i

M l m t a t p t u y n t n h t h t a g i l x p x t u y n t n h .

X t a t p p h i t u y n

Mn = { =kQ

ckk, |Q| = n, Q N}.

T a c t h c h n r a n

h s F o u r i e r l n n h t v s p x p c h n g t h e o t h t

t n g d n :

|fk1| ... |fkn|.

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t =nj=1 fkjkj . T a c h n g m i n h

l x p x t t n h t .

T a c , v i Mn ,

f

2 =

kQ |fk

ck|2 +

kQ |fk|2

j=n+1 |

fkj |

2.

L y i n f i m u m h a i v t a t h u c

E(f, Mn, H) = (

j=n+1

|fkj |2)1/2

v l x p x t t n h t .

2 . 3 X p x t u y n t n h t r o n g k h n g g i a n n h c h u n

G i s r n g X l m t k h n g g i a n n h c h u n . K h i k h n g g i a n l i n

h p X c a X c n g l k h n g g i a n n h c h u n , c h u n c a p h i m h m t u y n

t n h X c x c n h b i

= supf=0

|(f)|f = supf=1

|(f)|.

X t h h m = {1,...,n} X, t Y := s p a n v

= { X : () = 0, }.C h n g t a m u n x p x f X b n g c c p h n t c a Y . T r c h t t a c b s a u :

B 2 . 3 . 1

.N u h h m f1,...,fn X l c l p t u y n t n h , t h t n t i h

s o n g t r c c h u n 1,...,n X s a o c h o

i(fj) = i,j, i , j = 1,...,n.

C h n g m i n h . C h n g m i n h b n g q u y n p .

K h i n = 1, t h e o n h l H a h n - B a n n a c h k h n g n h n g .

G i s k h n g n h n g v i k < n. C h n g t a c n c h n g m i n h k h n g n h

n g v i k = n.

T h e o g i t h i t q u y n p , h f2,...,fn, c h s o n g t r c c h u n l 2,...,n. t

Y =s p a n

{f2,...,fn}. i u c n c h n g m i n h t n g n g v i t n t i 1 Ys a o c h o

1(f1) = 0.G i t h i t p h n c h n g v i m i Y , 1(f1) = 0. ().T a c v i m i

X ,

nk=2

(fk)k Y.

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K h i t h e o ( * ) , s u y r a (nk=2 (fk)k)(f1) = 0. D o f1 = nk=1 k(f1)fk,

v l v i t n h c l p t u y n t n h .

n h l 2 . 3 . 2

. ( N i k o l s k y ) S a i s x p x f b n g c c p h n t c a Y c x c

n h b n g c n g t h c

E(f , Y , X ) = inf {ck}

nk=1

fnk=1

ckk = sup

=1

|(f)|.

C h n g m i n h . T a c Y l k h n g g i a n t u y n t n h s i n h b i n n Y = . V i

m i g Y v , = 1, t a c |(f)| = |(f g)| ||f g = f g

|(f)

|

f

g

|(f)| E(f , Y , X ),d o

sup

=1

|(f)| E(f , Y , X ).

c h n g m i n h c h i u n g c l a i c a b t n g t h c n y k h n g m t t n h

t n g q u t t a c t h g i t h i t d y 1,...,n c l p t u y n t n h . K h i l

c s c a Y , v t h e o b t r n , t n 1,...,n l h s o n g t r c c h u n c a .

M i

f X c x t n h m t p h i m h m t u y n t n h l i n t c t r n

X

.K

h i u f l h n c h c a f t r n Y , t c l (f) = (f) v i m i Y .V i m i d y c = {ck}nk=1 , t

gc := fnk=1

ckk.

T a c , v i m i Y ,

(gc) = (f) n

k=1ck(k) = (f) = (f),

d o gc l m t t h c t r i n c a f l n X

. N g c l i g i s g l m t t h c t r i n

c a f l n X . T a c v i m i X t h

nk=1

(k)k Y.

D o g l t h c t r i n c a f v f l h n c h c a f t r n Y n n t a c

( nk=1

(k)k)(g) = ( nk=1

(k)k)(f)

= ( nk=1

(k)k)(f), X.

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S u y r a

(g) = (f) nk=1

[k(g) k(f)](k),

n g t h c n y n g v i m i X. D o t a p h i c

g = fnk=1

k(g f)k.

V y m i t h c t r i n c a f l n X u c d n g

gc = fnk=1

ckk.

M t k h c t h e o n h l H a h n - B a n n a c h , t n t i t h c t r i n g c a f s a o c h o

f = g. T h e o c h n g m i n h t r n t n t i gc = fnk=1 ckk s a o c h o gc = g .

S u y r a

g = ||fnk=1

ckk|| E(f , Y , X ).

N h n g

g = f = sup=1

Y

|(f)| = sup=1

Y

|(f)|.

T y t a s u y r a k h n g n h c c h n g m i n h .

T a b i t Tn l t p c c a t h c l n g g i c b c n, v

E(f, Tn, Lp(T)) = En(f) = infTn

f Lp(T) =: En(f)p

l s a i s x p x t t n h t f b i Tn . H n n a limn En(f)p = 0. T a m u n b i t k h i n o c t c h i t b n g O(n1) . C u t r l i l n h l s a u :

n h l 2 . 3 . 3

.N u f

Lp

(T) v f

Lp

(T), t h En

(f)p C1

n.

T a s c h n g m i n h c h o t r n g h p p = 2, k h i p = 2 s c c h n g m i n h t r o n g C h n g

4. T r o n g n h l 2 . 2 . 2 t a t n h c s a i s x p x

En(f)2 = (|k|>n

|fk|2)1/2.

V fk =

f

ik, n n

En(f)2 = (|k|>n |

f

ik |2)1/2

1

nf

2.

V y k h i p = 2 n h l c c h n g m i n h .

2 2


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N h n x t 2 . 3 . 4

. B n g q u y n p t a c h n g m i n h c r n g n u f(r) L2(T),

t h En(f)2 Cn

r.

T h e o n h n x t n y t a t h y h m c n g t r n t h t c x p x c n g n h a n h .

T r o n g c c c h n g t i p t h e o t a s n g h i n c u v n n g c l i . B i t o n t m

m i q u a n h g i a t r n c a h m s v t c x p x l m t t r o n g n h n g

b i t o n t r n g t m c a l t h u y t x p x .


B i t p 1

.

C h n g m i n h f

g

Tn v i m i f

L1(T) v g

Tn

B i t p 2

.

C h oT

l m t a t h c l n g g i c c b c n h h n n

. C h n g m i n h n g t h c

1

2

20

T(x)dx =1

n

nk=1

T(2k

n).

B i t p 3

.

C h o g(x) = 1 X := L1[1, 1] , U = {h = g : R}, f(x) = sign(x), C h n g m i n h r n g m i

[1, 1],

hl x p x t t n h t

ft

U.

B i t p 4

.

C h o H l k h n g g i a n H i l b e r t v h n c h i u v i c s t r c c h u n {en}nN , l y {n}nN l m t d y t r o n g (1, ) s a o c h o n 1 . K h i u C = {xn = nen :n N}. C h n g m i n h r n g C l m t t p n g , v k h n g t n t i p h n t x p x t t n h t c a 0 t C.

B i t p 5

.

K h i u c0 l k h n g g i a n B a n a c h c c d y v h n f = (1, 2,

) v i

k 0, k , v f = maxk

|k|.

K h i u U0 = {f = (1, 2, ) : (f) :=k=1 2

kk = 0}. C h n g m i n h r n g v i m i

f c0\U, p h n t x p x t t n h t f t U0 k h n g t n t i . T m s a i s x p x t t n h t f b i U0 .

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C h - n g 3

C c k h n g g i a n H m

C h n g n y l m t b c c h u n b d n t i c c n h l t r u n g t m c a l

t h u y t x p x . C h n g t a s n g h i n c u m t s k h n g g i a n h m c l i n q u a n

n t r n c a h m s , n h l : k h n g g i a n S o b o l e v , k h n g g i a n L i p s c h i t z ,

k h n g g i a n H o l d e r .

3 . 1 M t s k h i n i m c b n , k h n g g i a n Lp(A), C(A)

N u k h n g n i g k h c , t a v n x t m i n x c n h c a h m s l R,R+,T,

h o c [a, b].

3 . 1 . 1 K h n g g i a n C(A)

K h n g g i a n C(A) g m t t c c c h m t h c ( h o c p h c ) , x c n h v l i n

t c t r n A. C(A) l k h n g g i a n n h c h u n , v i c h u n

f = supxA

|f(x)|.

K h i u

C(A) l k h n g g i a n c o n c a C(A), g m t t c c c h m f l i n t c

u t r n

A. R r n g , n u

A =T h o c

A = [a, b], t h

C(A) =

C(A).N u

Ac o m p a c t t h

f = maxxA

|f(x)|.K h n g g i a n Cr(A) g m t t c c c h m k h v i l i n t c c p r t r n A . C c h m

| | v x c n h b i |f| := f(r)

v

f := f + |f|,l n l t l m t n a c h u n v c h u n t r n Cr(A). K h i u C(A) l k h n g

g i a n t t c c c h m k h v i v h n l n t r n A .

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3 . 1 . 2 K h n g g i a n Lp(A)

K h n g g i a n Lp(A) g m t t c c c h m f k h t c h c p p t r n A, t c l

i l n g s a u l h u h n

fp := fp(A) =

(A

|f(x)|pdx)1/p n u 0 < p < ; ,ess supxA

|f(x)| n u p =

K h i 1 p , Lp(A) l k h n g g i a n B a n a c h . V i 1 < p < , Lp(A) l k h n g g i a n p h n x . N u 1 p < , t h k h n g g i a n i n g u c a Lp(A) l Lp(A)v i

1p

+ 1p

= 1.

D n g r i r c c a Lp l p

g m c c d y x = {xi}i=1 s a o c h o

xp :=(

i=1 |xi|p)1/p n u 0 < p <

supi

|xi| n u p =

H a i b t n g t h c c t r n g c a k h n g g i a n Lp(A) l

( a ) B t n g t h c H o l d e r . V i 1 p,q , 1p

+ 1q

= 1, t a c

A

|f(x)||g(x)|dx fpgq, f Lp(A), g Lq(A).

( b ) B t n g t h c M i n k o w s k i . N u g(), f(, ) l c c h m d n g , o c t r n B v A B t n g n g , t h

{A

(

B

g(y)f(x, y)dy)pdx}1/p B

g(y){A

f(x, y)pdx}1/pdy.

N u|A| <

, t h t b t n g t h c H o

l d e r t a s u y r a c c p h p n h n g l i n t c

c a k h n g g i a n Lp(A) v p

:

( c ) V i

p q, t a c

Lq(A) Lp(A), fp |A|1/p1/q

fq.( d ) V i p q , t a c p q, xq xp.

3 . 2 K h n g g i a n c c h m k h v i : K h n g g i a n S o b o l e v

T r o n g m c n y c h n g t a s n g h i n c u m t v i t n h c h t c b n c a k h n g

g i a n S o b o l e v .

T a b i t r n g h m f

x c n h t r n A

l l i n t c t u y t i n u v i m i

> 0 , t n t i > 0 s a o c h o v i m i x1,...,xn, A,m1i=1 |xi+1 xi| , t h m1

i=1 |f(xi+1) f(xi)| . H m f l i n t c t u y t i t r n A k h i v c h k h i f

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t n t i h u k h p n i .

G i s X

l k h n g g i a n B a n a c h c c h m x c n h t r n A

, k h i u Wr(X)

l k h n g g i a n t u y n t n h c c h m f X

s a o c h o f(r1)

l i n t c t u y t i v

f(r)

X. N a c h u n v c h u n t r n Wr

(X)l n l t l

|f|Wr(X) := f(r)X , fWr(X) := fX + |f|Wr(X).

N u X = C(A), t h Wr(C(A)) = Cr(A). K h i X = Lp(A), 1 p , t h k h n g g i a n Wrp (A) := W

r(Lp(A)) c g i l k h n g g i a n S o b o l e v .

3 . 2 . 1 a t h c T a y l o r v b t n g t h c o h m

C h o f

Wr

p

(A) . K h i f c c c o h m l i n t c c p k = 0, 1,..,r

1. V

v y v i m i c A , n g t h c

Tr1(x) := Tr1(f , c, x) :=r1k=0

f(k)(c)(x c)k

k!

h o n t o n c x c n h , v c g i l a t h c T a y l o r c a f t i c. B n g

q u y n p v t c h p h n t n g p h n t a c

f(x)

Tr1(x) =

x

c

f(r)(t)(x t)r1

(r 1)!dt.

( 2 . 3 )

C h n g t a s t h n g x u y n s d n g c l n g s a u i v i p h n d f Tr1 .

M n h 3 . 2 . 1

.N u f Wrp (A), A = [a, b], v 1 p, q , t h i v i a

t h c T a y l o r Tr1(f , c, x), c A, t a c

f Tr1q(A) 1(r 1)! |A|

r 1p+ 1

qf(r)p(A).

C h n g m i n h . T h e o b t n g t h c H ol d e r , t a c

|f(x) Tr1(x)| xc

|f(r)(t)(x t)r1

(r 1)! |dt

f(r)p(A) |A|r 1

p

(r 1)! , 1/p + 1/p = 1.

L y c h u n t r o n g k h n g g i a n Lq(A) t a t h u c k t l u n t r n .

T n h n g h a c a Wrp (A) , t a c f(r) Lp(A), f Wrp . T a m u n b i t

f(k), k = 1,...,r

1 c t h u c Lp(A) h a y k h n g ? n h l s a u t r l i c u h i

n y .

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n h l 3 . 2 . 2

.V i r 2, 1 p , t n t i h n g s C p h t h u c v o r s a o

c h o

ukf(k)p C(fp + urf(r)p), f Wrp , k = 0,...,r, ( 2 . 4 ) t r o n g u > 0 t u , n g o i t r k h i A = [a, b] , t r o n g t r n g h p n y 0 u b a. T r o n g t r n g h p A = [a, b] , t a c n g c ( 2 . 2 ) v i 0 u c, t r o n g c > 0 l h n g s t u , n h n g C p h t h u c v o r v c/(b a) .C h n g m i n h .

T r c h t t a c h n g m i n h k h n g n h t r n c h o t r n g h p A =

[a, b].

N uA = [a, b]

, t h v i x [a, a+b

2], 0 u |b a|/2, t a c

f(x + u) = f(x) + uf(x) + ... +ur1

(r 1)! f(r1)(x)+

+u

0

(u t)r1(r 1)! f

(r)(x + t)dt.( 2 . 5 )

P h n d Rr(x, u) t r o n g ( 2 . 3 ) l m t h m c a x, p d n g b t n g t h c

M i n k o w s k i t a c

Rr(, u)p(A) = urf(r)p(A), || 1/r!, A := [a, (a + b)/2].T a c h n c n h r 1 s 1 =: 1 < ... < r1 := 2. K h i v i 0 u |A|/4,t a c

usf(x) + ...+ur1

(r 1)! r1s f(r1)(x)= f(x + su) f(x) Rr(x, su), s = 1, ...r 1. ( 2 . 6 )

N h n x t r n g ( 2 . 4 ) l m t h p h n g t r n h t u y n t n h c n h t h c l

n h t h c V a n d e r m o n d e k h c k h n g , d o t a c t h g i i c n g h i m

uk

k!f(k)(x), k = 1,...,r 1. V i m i k = 1,...,r 1, uk

k!f(k)(x) l t h p t u y n t n h

c a Cs(x) := f(x + su) f(x) Rr(x, su), x A, s = 1, ...r 1.T a c

Cs

p(A

) 2

f

p(A) +

Rr(

, u)

p(A

),

d o

ukfp(A) C(fp + urf(r)p), k = 0,...,r, ( 2 . 7 ) T h a y

Ab i

A = [(a + b)/2, b], t a c b t n g t h c t n g t ( 2 . 5 )

ukfp(A) C(fp + urf(r)p), k = 0,...,r, ( 2 . 8 ) L y ( 2 . 5 ) c n g v i ( 2 . 6 ) , t a t h u c k h n g n h t r n v i 0 u |A|/4. V i c c g i t r k h c c a u , ( 2 . 2 ) c s u y r a t u = (b a)/4 b n g c c h t h a y t h h n g s

C.

N u A = [a, b], q u t r n h t r n n g v i A c t h a y b i A, v u 0. V y k h n g n h c c h n g m i n h .

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H q u 3 . 2 . 3

.N u f Wrp (A) , t h f(k) Lp(A), k = 0, 1,...,r.

H q u 3 . 2 . 4

. ( B t n g t h c K o l m o g o r o v - S t e i n )

C h oA = T,R,R+ . K h i t n t i h n g s Ck,r, k = 0,...,r, v r = 1,... s a o

c h o

f(k)p Ck,rf1k/rp f(r)k/rp , f Wrp . ( 2 . 9 )

C h n g m i n h . N u f(r) = 0 , t h t h e o n h l 3 . 2 . 2 , t a c ukf(k)p Cfp. D o

f(k)p C 1uk

fp.

B t n g t h c n y n g v i m i u 0 ( v A = [a, b] ) . C h o u + t a s u y r a f(k) = 0

.

N u f(r) = 0 , c h n u = ( fpf(r)p )1/r ( 2 . 2 ) t a s c ( 2 . 7 ) . V y b t n g t h c K o l m o g o r o v - S t e i n c c h n g m i n h .

T n h l 3 . 2 . 2 t a c n h n x t s a u

N h n x t 3 . 2 . 5

.N u fn f t r o n g Wrp , t h s u y r a f(k)n f(k)p(A) 0, 0

k r , v f(k)n f(k), 0 k < r, u t r n m i t p c o m p a c t c o n c a A.

N g c l i , n u A

c o m p a c t , f(r)n f(r)p(A) 0, v t n t i a A s a o c h o

f(k)n (a) f(k)(a), k = 0,...,r 1 ( K h i A = T , c h c n k = 0 ) , t h fn f t r o n g

Wrp

.

C h n g m i n h . T r c h t t a c h n g m i n h p h n t h u n .

N u fn f t r o n g Wrp , t h t h e o n h n g h a c a c h u n t r n Wrp , t a s u y r a fn f t r o n g Lp(A) v f(r)n f(r) t r o n g Lp(A). T h e o n h l 3 . 2 . 2 s u y r a f(k)n f(k)p(A) 0, 0 k r.G i s

Bl t p c o m p a c t t u t r o n g

A, v

g W11 . T a c g(y) = g(x) +yx

g(t)dt,x,y B. T h e o b t n g t h c M i n k o w s k i t a c

|B|1/p

|g(y)| gp + y

xg

(t)dtp, y B gp + |B|gp, v |

yx

g(t)dt| |B|1/pgp.

V i 0 k < r, l y g = f(k)n f(k) . T t a s u y r a p h n t h u n c a n h n x t .

B y g i t a c h n g m i n h p h n o .

G i s r n g A l t p c o m p a c t v {gn}n l m t d y h m l i n t c t u y t i , n u ( i ) gn 0 t r o n g Lp(A) , v ( i i ) t n t i a c h o gn(a) 0. K h i d o gn(x) = gn(a) + xa gn(t)dt, n n s u y r a gn(x) 0. N u A = T , d y c c h m {gn}n l i n t c t u y t i v ( i ' ) gn 0 t r o n g Lp(T) , ( i i ' ) gn(a) 0 , t h t n t i hn s a o c h o gn = h

n . K h i t a c

T

gn(t)dt = 0, d o t n t i an T s a o c h o

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gn(an) = 0 . V v y gn(x) =xan

gn(t)dt 0 u t r n A. N i r i n g t a l u n l u n c

f(k)n (x) f(k)(x) u t r n A , 0 k < r .T a c

fn fWrp (A) = fn fp(A) + f(r)n f(r)p(A) |A|1/pmax

xA|fn(x) f(x)| + f(r)n f(r)p(A) 0, n .

V y t a c i u p h i c h n g m i n h .

3 . 2 . 2 P h n h o c h n v v o h m s u y r n g

T r o n g p h n n y t a x t n h n g k t q u b t r c h o v i c n g h i n c u m t s

c c t n h c h t k h c c a k h n g g i a n S o b o l e v , n g t h i s o s n h c c t n h c h t

c a o h m t h n g t h n g v o h m s u y r n g .

n h n g h a 3 . 2 . 6

. M t p h n h o c h n v i v i A l m t d y h u h n

h a y v h n c c h m {j}j x c n h t r n A, s a o c h o

j

j(x) = 1, x A.

B 3 . 2 . 7

. i v i m i o n I = [a, b] v 0 < < (b a)/2 , t n t i

C0 (R) s a o c h o t n g t r n [a, a + ], g i m t r n [b , b], b n g 0 b n n g o i [a, b] , b n g 1 t r n [a + , b ] , v t n t i Ck = C(k, ), (k) Ck|I|k ,k = 0, 1,.... c b i t , v i m i 0 < < 1, v i m i I, t n t i Ck := Ck, s a o c h o

v i 2 |I|,(k) Ck,|I|k, k = 0, 1,...

C h n g m i n h . K h n g m t t n h t n g q u t t a c t h g i t h i t [a, b] = [0, 1] , v

t r n g h p t n g q u t c s u y r a t t r n g h p n y b n g m t p h p t h

t u y n t n h . t

g(x) =

e1/x2 n u x > 0;

0n u

x 0,

r r n g g C(R). V i 0 < < 1/2 , l y G(x) := Cg(x)g( x), t r o n g C

c c h n s a o c h o

10

G(x)dx = 1. T a c

G(x)l h m k h n g m , t r i t t i u

b n n g o i [0, ] . N u t

(x) :=

xx1+

G(t)dt,

t h t h a m n c c i u k i n t r n .

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n h l 3 . 2 . 8

.C h o

0 < < 1 v d y c c o n {Ij}jZ, Ij := [aj, bj] t h a m n A := R =

jZ Ij , v

aj < bj1 aj+1 < bj, bj1 aj 1

2(|Ij1| + |Ij |), j Z

.

K h i t n t i d y c c h m j, j Z s a o c h o

( i )

jZ j(x) 1, x R.

( i i ) 0 j 1, v s u p p (j) = Ij, j Z.

( i i i ) (k)j C(k, )(|Ij1|k + |Ij|k + |Ij+1|k) .

C h n g m i n h . V i m i j

Z , t j =

12

|Ij

|. K h i t h e o B 3 . 2 . 7 , t n

t i j C0 (R) s a o c h o s u p p (j) = Ij v 0 j 1. T g i t h i t t a s u y r a j + j+1 < bj aj+1 , n n R =

jZ[aj + j , bj j] ( * ) .

t :=

jZ j , d o B 3 . 2 . 7 v ( * ) n n t a c C v (x) 1, x R.

N u t j =

j

, t h h i n n h i n ( i ) v ( i i ) c t h a m n . T a c h n g m i n h

d y {j}jZ t h a m n ( i i i ) . T h t v y , v i m i j v x Ij , c h c t i a b a s h n g j1, j, j+1 k h c k h n g . V v y p d n g B 3 . 2 . 7 t a s u y r a

|(k)| C(k, )(|Ij1|k + |I|kj + |Ij+1|k) =: j(k).

K h i u 1 = 1

. o h m h a i v p h n g t r n h 1 = 1, t a c ki=0 C

ki

(ki)(1)(i)

= 0. B n g q u y n p t a c |(1)(k)(x)| Ckj(k), x Ij.C h r n g j =

j

, n n p d n g c n g t h c L e i b n i z c h o o h m c a m t

t c h t a t h u c ( i i i ) .

n h l n y v n c n n g c h o c c t r n g h p c n l i c a A

, b n g c c h

c h c h n n h n g h m j k h n g n g n h t t r i t t i u t r n A .

T n h l n y v n h l 3 . 2 . 2 t a c h q u s a u ( c h n g m i n h x e m n h

m t b i t p . )

H q u 3 . 2 . 9

.V i r = 1,..., k h n g g i a n C(A) t r m t t r o n g Wrp (A), 1 p 0 s a o c h o r(f, t0)p = 0.( i i ) N u

r

h(f, )p = o(hr

), k h i

h 0, t h t n t i P Pr1 s a o c h o f = Ph u k h p n i .

( i i i ) N u f Pr1 , t h r(f, t)p Ctr, C 0 , 0 < t 1 .

C h n g m i n h . T h e o ( e ) c a M n h 3 . 3 . 4 t a c r(f, t)p = r(f,tt0/t0) (t/t0+

1)rr(f, t0)p. D o t a c ( i ) .

T a c

rh(f, x) = (I + h I)r(f, x)

=rk=0

rk

(1)k(I + h)rk(f, x) =

rk=0

(1)k

rk

f(x + (r k)h).

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L y I l m t o n c o n t r o n g Arh v i m i h n h . G i s C0 (I). T a c

|R

f(x)hrrh(, x)dx| = |R

hr(1)rf(x)rh(, x rh)dx|

= |R

hr(1)rf(x + rh)rh(, x)dx|

= |R

hrf(x + rh)rk=0

(1)k

r

k

(x + kh)dx|

= |R

hr(x)rk=0

(1)k

r

k

f(x + (r k)h)dx|

hrrh(f.)pp, 1/p + 1/p = 1. ( 3 . 1 2 )

C h oh

0+

, v s d n g g i t h i t ( i i ) t a t h y v p h i c a ( 3 . 9 ) h i t v

k h n g , v t r i h i t v (1)r R

f(x)(r)(x)dx ( d o H q u 3 . 3 . 6 ) . D o R

f(x)(r)(x)dx = 0,

s u y r a f c o h m s u y r n g c p r b n g 0. T h e o n h l 3 . 2 . 1 1 t h f =

P Pr1, h u k h p n i t r n I. B n g c c h l y c c o n c h o n h a u , t h t t c c c a t h c

P u t r n g n h a u . V y ( i i ) c c h n g m i n h .

M t k h c , v i 0 < t 1, d o ( e ) c a M n h 3 . 3 . 4 v ( i ) , ( i i ) , n n t a c

r(f, t)p (1 + 1/t)rr(f, 1)p = Ctr.

i u n y c h n g m i n h ( i i i ) .

3 . 4 K h n g g i a n B V ( A )

P h n n y c t h x e m n h m t p h l c n h m n h c l i m t s t n h c h t c

b n c a b i n p h n c a m t h m v k h n g g i a n c c h m c b i n p h n b

c h n .

G i s f l m t h m s x c n h t r n A , b i n p h n c a f c n h n g h a

b i

V ar(f) := V ar(f, A) = supni=1

|f(xi) f(xi1)|, {xi}ni=0 A,

t r o n g s u p r e m u m c l y t r n t t c c c d y h u h n {xi}ni=0 t n g t r o n g A

.

K h i u BV(A) l t p t t c c c h m x c n h t r n A s a o c h o V ar(f) < +.M n h s a u c h o t a b i t m t s t n h c h t c a

BV(A).

M n h 3 . 4 . 1

.

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( 1 ) BV l k h n g g i a n t u y n t n h .

( 2 ) V ar() l m t n a c h u n . ( 3 ) M i h m n i u v b c h n u c b i n p h n b c h n .

( 4 ) M i h m c b i n p h n b c h n u c t h b i u d i n d i d n g h i u c a

h a i h m n i u t n g . C t h , t Ax = (, x] n u A = R; Ax = [a, x]n u A = [a, b] ; Ax = [0, x] n u A = T. K h i c c h m V(x) := V ar(f, Ax)

v h m (x) := V(x) f(x) l c c h m k h n g g i m .

T t n h c h t ( 4 ) t a s u y r a n u f BV(A)

, t h f

c h c k h n g q u m

c c c i m g i n o n . H n n a , n u c l m t i m g i n o n c a f,

t h g i i h n t r i f(c) v g i i h n p h i f(c+) t n t i . T r o n g n h i u p d n g ,

c h n g t a c n c c h m h i u c h n h t

fb n g c c h t h a y i g i t r c a

ft i

c c i m k h n g l i n t c c. K h i u f l m t h i u c h n h c a f s a o c h o t i

c c i m k h n g l i n t c c g i t r f(c) n m g i a f(c) v f(c+). G i f0 l

m t h i u c h n h k h c c a f, f0 c h k h c f t r o n g t r n g h p A = [a, b] , t r o n g

t r n g h p n y y u c u t i c c i m a, b

h mf0 p h i l h m l i n t c . t

V ar(f) = V ar(f0), g i t r n y m i p h n n h c t r n g c a b i n p h n .

3 . 5 K h n g g i a n L i p s c h i t z v k h n g g i a n H ol d e r

C h n g t a k h i u v i c x t c c k h n g g i a n n y b n g c c h x t t r n g h p

n g i n n h t , L i p , 0 < 1. K h n g g i a n n y g m t t c c c h m f

C(A)

s a o c h o

|t(f, x)| = |f(x + t) f(x)| Mt, t > 0. i u n y t n g n g v i (f, t) Mt . V m o d u l l i n t c c t h n h

n g h a c h o h m x c n h t r n k h n g g i a n m e t r i c b t k , n n y A c t h

l k h n g g i a n m e t r i c . T u y n h i n c h n g t a v n h n c h b n t r n g h p c a

An h c c p h n t r c . N a c h u n v c h u n c a L i p

l n l t l

|f|Lip := supt>0

t(f, t), fLip := fC(A) + |f|Lip.

S

c g i l t r n c a f

L i p

. D t h y C1(A)

L i p

.

L y c h u n c a s a i p h n t(f, x) t r o n g k h n g g i a n n h c h u n X, t a c

k h n g g i a n L i p ( , X) . V d , k h n g g i a n L i p ( , Lp ) g m t t c c c h m

f Lp(A), 0 < p , s a o c h o

t(f,

)

p =

{At |f(x + t)

f(x)

|pdx

}1/p M t, t > 0.

N a c h u n l c n y s l |f|Lip(,Lp) := supt>0

t(f, t)p . T a c W1p L i p ( , Lp ) .

3 8


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N h n x t 3 . 5 . 1

.N u > 1, p 1 v |f|Lip(,Lp) < , t h f = const.

T h t v y ,

0 limt0+

t(f, )pt

limt0+

M t1 = 0

t(f, )p = o(t), k h i t 0+ . ( 5 . 1 3 ) T h e o M n h 3 . 3 . 7 s u y r a n h n x t t r n .

V v y i v i k h n g g i a n L i p

t a h n c h 0 < 1 . i v i c c g i

t r k h c c a

, c h n g t a c h a i c c h n h n g h a L i p (, Lp).

C c h m t : V i m i > 0 t a v i t = r + , r Z+ , 0 < 1 . K h i v i

p > 0, T a n h n g h a L i p (, Lp) := Wr(L i p (, Lp)) . K h n g g i a n n y g m t t

c c c h m f

L i p (, Lp) s a o c h o f

(r1)l i n t c t u y t i v f(r)

L i p ( , Lp ) .

N a c h u n c a L i p (, Lp) l

|f|Lip(,Lp) := f(r)Lip(,Lp) = supt>0

(t(f(r), t)p)

C c h h a i : ( K h n g g i a n H o l d e r , Hp (A).) V i > 0, 0 < p , t r := []+ 1.

T a n h n g h a k h n g g i a n Hp (A) l t p t t c c c h m f Lp(A) s a o c h o

rt (f, )p =

Art

|rt (f, x)|pdx1/p

Mt, t > 0, 0 < M = const < .

N a c h u n c a Hp (A) l

|f|Hp := supt>0

(tr(f, t)p).

N h n x t 3 . 5 . 2

. K h i p = c h n g t a c n t h a y L(A) b i

C(A) . V

( i ) N u 0 < < 1

, t h Hp = L i p (, Lp) .

( i i ) N u 1, t h L i p (, Lp) Hp .( i i i ) N u = 1, p = , t a c L i p (1,

C) = L i p 1 .

T h t v y , ( i ) v ( i i i ) l h i n n h i n . G i s

= k + , k Z+, 0 < 1, v r = [] + 1 . K h i r = k + 1 . N h n g

tk+1(f, t)p tk(f(k), t)p = t

(f(k), t)p.

L y sup t r n t p {t > 0}, t a c ( i i ) . K h n g g i a n Z(A) := H1(A) c g i l k h n g g i a n Z y g m u n d . T ( i i ) v

( i i i ) c a n h n x t t r n t a s u y r a L i p 1 H1 , n h n g n g c l i k h n g n g , v d , v i A = [0, 1] h m g(x) := xlogx n u x = 0; g(x) = 0 n u x = 0. D t h y

|2t (g, x)

| 2(log2)t, n n g

Z[0, 1]

, n h n g g

L i p 1 .

C c n h l t i p t h e o c a m c n y s n g h i n c u s l i n h g i a c c

k h n g g i a n k h c n h a u . C h n g t a b t u b n g b s a u :

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B 3 . 5 . 3

.BV Lip(1, L1) v |f|Lip(1,L1) V ar(f).

C h n g m i n h . T a c h c h n g m i n h c h o t r n g h p A = [a, b]. G i s f BV ,

x t h m S t e k l o v fh, 0 < h < b

a, c a f c n h n g h a b i

fh(x) :=

h1

x+hx

f(t)dt n u a x b h;fh(b h) n u b h x b.

K h n g m t t n g q u t t a c t h x e m f = f0 v l h m t n g , t r o n g f0

l h m h i u c h n h c a f n h t r o n g m c 3 . 4 . D t h y r n g fh W11 v A

|fh(t) f(t)|dt 0, h 0+ . H n n a , v fh = h1(f(x + h) f(x)) h u k h p n i t r n

[a, b h]v

= 0t r n

[b h, b], n n t a c

V ar(fh) = bh

a

fh(x)dx = h1

{b

bh

f(x)dx

a+h

a

f(x)dx

} f(b) f(a) = V ar(f).

V v y m i f BV u c t h x p x b i m t h m g W11 s a o c h o fg1 v V ar(g) V ar(f). D o t a c h c n c h n g m i n h b t r n i v i t r n g

h p f W11 .V i f W11 , t a c bh

a

|h(f, x)|dx baa

x+hx

|f(t)|dt =ba

ba

|f(t)|[x,x+h](t)dtdx

=ba |f(t)|

ba [x,x+h](t)dxdt h

ba |f(t)|dt = hV ar(f).

S u y r a |f|Lip(,L1) V ar(f) . V y b c c h n g m i n h . n h l

3 . 5 . 4

.C h o r = 1, 2.... K h i h m f L i p (r, Lp) k h i v c h k h i f

c t h h i u c h n h t r n m t t p c o b n g k h n g t r t h n h m t h m

g Wrp , 1 < p , h o c g Wr1(BV), p = 1. H n n a , |f|Lip(r,Lp) = |f|Wrp , 1 < p ( 5 . 1 5 ) |f|Lip(r,L1) = V arf(r1), p = 1. ( 5 . 1 6 )

C h n g m i n h . T r c h t t a x t t r n g h p 1 < p . G i s f L i p ( r , Lp ) .

G i M l s n h n h t s a o c h o r(f, t)p M tr, t > 0. T h e o t n h c h t c a

m o d u l t r n t a c r(f, t)p tr1(f(r1), t)p . D o M l s n h n h t n n s u y r a

M t1(f(r1), t)p t > 0.

V y t a c M |f|Lip(,Lp). M t k h c c n g t h e o t n h c h t c a m o d u l t r n t a

c (f(r1), t) t|f|Wrp , d o |f|Lip(r,Lp) |f|Wrp . c ( 5 . 1 1 ) c h n g t a c n p h i c h n g m i n h M |f|Wrp . X t h m

gh(x) :=

hrrh(f, x), x Arh;0, x Arh.

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T c c h x c n h c a M v n h n g h a m o d u l t r n s u y r a ghp M v im i

h > 0. V

1 < p n n k h n g g i a n Lp(A) l k h n g g i a n l i n h p c a Lq , 1/p + 1/q = 1 , n h n g t p {gh, h > 0} l b c h n t r o n g Lp(A) n n t n t i

hn 0 , s a o c h o d y ghn h i t y u * t r o n g Lp(A) n h m g Lp(A) v gp M. c b i t , v i m i C0 (A) , t a c A

g(x)(x)dx = limn

A

(x)ghn(x)dx = limn

A

hrn rhn

(, x)f(x)dx

( x e m k t h u t b i n i ( 3 . 9 ) )

= (1)rA

(r)(x)f(x)dx.( 5 . 1 8 )

( 5 . 1 3 ) n i n n r n g g l o h m s u y r n g c p r c a f, v v y g = f(r) h u

k h p n i ( x e m n h l 3 . 2 . 1 3 ) . S u y r a f(r)

p = gp , h a y |f|Wrp = gp M.B y g i x t

p = 1. T r o n g h p n y t a c

M |f|Lip(r,L1) = |f(r1)|Lip(1,L1) V ar(f), ( B t r n ) .

T a p h i t c h l m h a i t r n g h p v k h n g g i a n L1(A) k h n g p h i l k h n g

g i a n l i n h p . k h c p h c i u n y , x t h m Gh(x) :=xc

gh(t)dt, t r o n g

c c n h t r o n g A, v t a v n c gh1 M. V h m Gh BV , m BV l k h n g g i a n i n g u c a C(A), n n t n t i Ghn h i t y u * n G BV v

V ar

(G) M. t b i t , v i m i

C

0 (A), t a c

A

(x)dG(x) = limn

A

(x)dGhn(x) = (1)rA

r(x)f(x)dx.

V

A

(x)dG(x) = A

g(x)(x)dx, n n

Gl o h m s u y r n g c p

r 1c a

f,

f(r1) = Gh u k h p n i , v

V ar(f(r1)) M.

n h l d i y c c h n g m i n h n g a y m c s a u .

n h l 3 . 5 . 5

.N u k h n g p h i l m t s n g u y n v 1 p , t h h m

f Lip(, Lp) k h i v c h k h i f b n g m t h m g Hp h u k h p n i , v i s t n g n g c a c c n a c h u n .

N h n x t 3 . 5 . 6

. Q u a n h f g f = g

h u k h p n i , l m t q u a n h t n g

n g t r n k h n g g i a n c c h m s x c n h t r n A. V v y n u t a n g

n h t f v [f] , ( k h i u [f] l l p t u n g n g c i d i n l f) t h t a c

Lip(, Lp) = Hp , 1 p , > 1, Z ( 5 . 1 9 )

v

Lip(r, Lp) =

Wrp , 1 < p , r = 1, 2,...;Wr1(BV), p = 1, r = 1, 2...

( 5 . 2 0 )

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3 . 6 K - P h i m h m

M o d u l t r n v K - p h i m h m l c c h m c a m t t h a m s t > 0, c h a i

c n g t h h i n m t s t n h c h t t h i t y u c a

f. T r o n g c c p h n t r c c h n g

t a b i t c c t h n g t i n t r c t i p v t r n c a f, p h n n y s c u n g c p

t h m m t s t h n g t i n v K - p h i m h m v m i q u a n h c a n v i m o d u l

t r n c a h m s .

n h n g h a 3 . 6 . 1

. K - p h i m h m c a f Lp(A) , 1 p , c n h

n g h a b i

Kr(f, t) := K(f , t , Lp, Wrp ) := inf

gWrp{f gp + tg(r)p}, r N.

T n h n g h a t a t h y r n g , n u Kr(f, t) , t h f c t h x p x b n g

m t h m g c t r n r s a o c h o s a i s x p x f gp , v o h m c a g k h n g q u l n g(r)p t1 .V i c c h n g m i n h m n h d i y c x e m n h m t b i t p

M n h 3 . 6 . 2

. K - p h i m h m c c c t n h c h t s a u :

( i ) N h m t h m c a t

, K - p h i m h m l m t h m t n g , l m , l i n t c v

d i c n g t n h : Kr(f, t1 + t2) Kr(f, t1) + Kr(f, t2) .

( i i ) V i c n h t, Kr(f, t) l m t n a c h u n t r n Lp(A) + Wrp (A).

G i a K - p h i m h m v m o d u l t r n r(f, t)p c s l i n h v i n h a u . C h n g

t a c t h c h r a r n g K - p h i m h m v m o d u l t r n l t n g n g . T r c

h t t a c b s a u :

B 3 . 6 . 3

. C h oI1 = [a1, b1], I2 = [a2, b2], t r o n g a1 < a2 < b1 < b2 ,

I = I1

I2 , J = I1

I2 . K h i t n t i h n g s C p h t h u c v o r v |I|/|J|s a o c h o

Kr(f, t)(I) C{Kr(f, t)(I1) + Kr(f, t)(I2)}, t |I|. ( 6 . 2 1 )

C h n g m i n h . T a p d n g B 3 . 2 . 7 i v i o n J =: [c, d]

v = |J|/4

. K h i

t n t i h m := C0 (R) v h n g s C = C(k, ) s a o c h o

|(k)| C|J|k, k = 0, 1, ...r.

V i m t h n g s t h c h C/|J|, h m (x) := x

(x)dx t r i t t i u v i

m i x < c, v l h m t n g t r n J, b n g 1 v i m i x > d. H n n a t a c n g c

|(k)(x)| C|J|k, k = 0, 1,...,r.

4 2


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G i s gi l h m t u t r o n g Wrp (Ii), i = 1, 2, v f Wrp (I), t |I| . T a c , v i

g := (1 )g1 + g2

f

gp

(I) ={I |f(x) g(x)|pdx}1/p

= {(ca1

+

dc

+

b2d

)|(1 (x))f(x) + (x)f(x) g|pdx}1/p

( C h n c c t n h c h t c a (x) )

{I1

|f(x) g1(x)|pdx +I2

|f(x) g2(x)|pdx}1/p

f g1p(I1) + f g2p(I2). ( 6 . 2 2 )

C n g t t n h c h t c a , t a c g(r)(x) = (1 )g(r)1 (x) + g(r)2 (x) t r n I\J. V v y

trg(r)p(I\J) tr{g(r)1 p(I1) + g(r)2 p(I2)}.T r n J = I1

I2 , t a v i t g = g1 + (g2 g1), v p d n g q u y t c L e i b n i z

t n h o h m c a m t t c h , t a c

trg(r)p(J) Ctr{g(r)1 p(J) + max0jr

[|J|(rk)g(r)2 g(r)1 p(J)]}

C{trg(r)1 p(J) + max0jr

(tkg(k)2 g(k)1 p(J))}, t |I|. ( 6 . 2 3 )

T h e o b t n g t h c o h m t a c

tk(g1 g2)(k)p(J) C{g1 g2p(J) + t(r)(g1 g2)(r)p(J)} C{f g1p(I1) + trg(r)1 p(I1) + f g2p(I2) + trg(r)2 p(I2)} ( 6 . 2 4 )

T ( 6 . 1 7 ) , ( 6 . 1 8 ) v ( 6 . 1 9 ) c n g v i

trg(r)P(I) tr{g(r)p(J) + g(r)p(I\J)}

v n h n g h a c a K - p h i m h m , t a s u y r a i u p h i c h n g m i n h .

n h l 3 . 6 . 4

. ( J o h n e n [ 1 9 7 2 ] ) T n t i c c h n g s C1, C2 > 0 , c h p h t h u c

v o r s a o c h o v i m i f Lp(A) t a c

C1r(f, t)p K(f, tr; Lp, W

rp ) C2r(f, t)p, t > 0. ( 6 . 2 5 )

C h n g m i n h . T h e o c c t n h c h t c a m o d u l t r n t a c , v i g Wrp ,

r(f, t)p = r(f g + g, t)p r(f g, t)p + r(g)p 2rf gp + trg(r)p.

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T s u y r a 2rr(f, t)p K(f, t

r; Lp, Wrp ) .

c h n g m i n h b t n g t h c n g c l i , t a s o s n h f

v i h m

g(x) := f(x) + (

1)r+1 R

rtu

(f, x)Mr(u)du.

( 6 . 2 6 )

T r c h t x t A = [a, b], k h i g l h m x c n h t r n A , v p d n g b t n g t h c M i n k o v s k y t a c

f gp(A) R

rtu(f, )p(A)Mr(u)du r(f,rt)p rrr(f, t)p, ( 6 . 2 7 )

( d o s u p p Mr = [0, r] v R

Mr(u)du = 1. )

G i F l t c h p h n b c r c a f t r n A , t c l F(r) = f. T n h n g h a c a

t o n t s a i p h n

rtu

, t a s u y r a v p h i c a ( 6 . 2 1 ) l t h p c a c c t c h p h n R

f(x + jtu)Mr(u)du =

R

f(x + u)Mr((jt)1u)(jt)1du

= (jt)rrjt(F, x), j = 1, 2,...,r,

M ((jtr)rjt(F(r), x)) = (jt)rrjt(f, x). V v y

g(r)(x) = trr

j=1(1)j+1Crjjrrjt(f, x).

V r n g rjt(f, x)p r(f,jt)p jrr(f, t)p , n n

trg(r)p(A) 2rr(f, t)pf gp(A) + trg(r)p(A) Cr(f, t)p. ( 6 . 2 8 )

n y t a d d n g s u y r a ( 6 . 2 0 ) v i A = [a, b].

B y g i t a x t A = [a, b] . T r o n g t r n g h p n y , c h n g i c a Mr t a s

t h y g(x) c x c n h n u t (b a)/4r2 =: t0, x [a, b ba4 ] =: I1 . T u y n h i n t a v n c

f gp(I1) + trg(r)p(I1) r(f, t)p, t t0.

D o t n h i x n g , v i I2 := [a +ba4

, b] , t a c n g c

f gp(I2) + trg(r)p(I2) r(f, t)p, t t0.

T h e o B 3 . 6 . 3 t a s u y r a 6 . 2 0 v i i u k i n t t0. K h i u g0 l h m

t r o n g Wrp s a o c h o

f g0p(A) + tr0g(r)0 p(A) r(f, t0)p,

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s u y r a f g0p Cr(f, t0)P . G i P l a t h c T a y l o r b c r 1 c a g0 t i i m g i a c a

A. K h i t h e o c l n g p h n d

g0 P t r o n g M n h 3 . 2 . 1 t a c

f g0p |A|rg(r)0 p Ctr0g(r)0 p Cr(f, t0)p.

T a c

f Pp f g0p + P g0p Cr(f, t)p, t > t0.V Kr(f, t) f Pp + tP(r)p = f Pp n n k h n g n h h o n t o n c c h n g m i n h .

n h l d i y l m t p d n g c a n h l t r n , c h n g t a s s o s n h

c c m o d u l t r n k h c n h a u .

n h l 3 . 6 . 5

. ( J o h n e n v S c h e r e r [ 1 9 7 6 ] ) T n t i h n g s C c h p h t h u c

v o r = 2, ...., s a o c h o v i m i f Lp(A), 1 p < h o c f C(A), p = ,v

1 k < r,

rk(f(k), t)p Cr

to

r(f, s)psk+1

ds, t > 0,( 6 . 2 9 )

t h e o n g h a : b t c k h i n o v p h i h u h n t h f Wkp (A) ( h o c f Ck(A)k h i

p = ) v ( 6 . 2 4 ) n g .

C h n g m i n h . C n h t > 0 , N u A = [a, b] , t h rk(f(k), t)p Crk(f(k), |A|)p .

V v y k h i A = [a, b] t a c t h g i s t |A|. H n n a , n u r(f, t0)p = 0 t h c h a i v c a ( 6 . 2 4 ) u b n g k h n g , n n t a c t h g i t h i t

r(f, t)p > 0, t > 0,

v t h e o n h l J o h n e n 3 . 6 . 4 s u y r a Kr(f, t) > 0, t > 0. K h i u j := 2

jt.

C h n d y {gj}j=0 Wrp s a o c h o

f gjp + rjg(r)j p 2K(f, rj ; Lp, Wrp ) Cr(f, j)p, ( 6 . 3 0 )

t r o n g C c h p h t h u c v o r . T a v i t f = g0 +j=0(gj+1 gj) v i s h i t t r o n g Lp . S d n g b t n g t h c o h m , t a c

kjg(k)j+1 g(k)j p C{gj+1 gjp + rjg(r)j+1 g(r)j p} C{f gjp + f gj+1p + rjg(r)j p + rjg(r)j+1p} C{r(f, j)p + r(f, j+1)p} Cr(f, j)p. ( 6 . 3 1 )

C h i a c h a i v c h o kj r i l y t n g h a i v t a c

j=0

g(k)

j+1 g(k)

j p Cj=0

r(f, j)pkj C

t0

r(f, s)psk+1 ds.

( 6 . 3 2 )

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V v y f = g0 +j=0(gj+1 gj) v i s h i t t r o n g Wrp . T a c f(k) Lp(A)

(f(k) C(A), p = )v

f(k)

g(k)

0 p Ct

0

r(f, s)p

sk+1 ds.

M t k h c , t ( 6 . 2 5 ) t a c trg(r)0 p Cr(f, t)p , v t n h l J o h n e n t a c

rk(f(k), t)p CK(f

(k), trk; Lp, Wrkp )

C{f(k) g(k)0 p + trkg(k)0 p}

C{t0

r(f, s)psk+1

ds + tkr(f, t)p}. ( 6 . 3 3 )

M

tkr(f, t)p 2rtkr(f,t/2)p = 2

rk(t/2)kr(f,t/2)p

C

tt/2

r(f, s)psk+1

ds C

t0

r(f, s)psk+1

ds( 6 . 3 4 )

K t h p v i ( 6 . 2 8 ) t a t h u c k h n g n h t r n .

n h l c a J o h n e n v S c h e r e r c h o p h p t a s o s n h c c k h n g g i a n

L i p s c h i t z v k h n g g i a n H ol d e r . H q u s a u c h n g m i n h n h l 3 . 5 . 5

H q u 3 . 6 . 6

.C h o 1 p , v > 0 .

( i ) N u

k h n g p h i l s n g u y n , t h Hp = Lip(, Lp).

( i i ) N u = k + 1 l m t s n g u y n , t h f Hp k h i v c h k h i f(k) H1p .

C h n g m i n h . N u

k h n g p h i l m t s n g u y n , t h t a c t h v i t = k +

,

k Z, v

0 < < 1. t

r := k + 1, t h e o t n h c h t c a m o d u l t r n t a c

r(f, t)p tk(f(k), t) , d o

|f|Hp |f|Lip(,Lp).M t k h c , t M := |f|Hp , t a c r(f, t)p tM. T h e o n h l t r n t a c

(f(k), t)p C

t0

r(f, s)psr

ds MC

t0

s1ds = M Ct,

d o

Lip(, Lp) CM.

V y

H

p = Lip(, Lp).T n g t v i

= k + 1v

r := k + 2, t a s c ( i i ) .

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3 . 7 B t n g t h c B e r n s t e i n

B t n g t h c B e r n s t e i n l m t c n g c c s d n g p h b i n c h n g

s a u . l l d o m c h n g t a n g h i n c u b t n g t h c n y . M t a t h c

l n g g i c b c n b i n p h c l m t b i u t h c c d n g

Tn(z) :=|k|n

c(k)eikz, z C.

D o c n g t h c E u l e r n n Tn(z) c n c t h v i t d i d n g

Tn(x) =a02

+nk=1

(a(k)cos kz + b(k)sin kz).

K k i u Tn l t p c c a t h c l n g g i c d n g t r n . T n h c h t c a n g h i m

( k h n g i m ) c a Tn(z) c t h h i n t r o n g m n h s a u :

M n h 3 . 7 . 1

.

( i ) N u a(n) = 0

v b(n) = 0

, t h Tn c n g 2n k h n g i m ( k c b i )

t r o n g m i n

Da = {z = x + iy,a z a + 2}, a R.

( i i ) N u z1,...,z2n l c c k h n g i m c a Tn , t h Tn c b i u d i n

Tn(z) = A2nk=1

sin(z zk)

2( 7 . 3 5 )

( i i i ) C n g t h c ( 7 . 3 0 ) x c n h m t a t h c l n g g i c t h u c Tn .

C h n g m i n h . X e m n h b i t p

T M n h t r n t a s u y r a : N u t n t i a

R s a o c h o Tn c n h i u h n

2n k h n g i m t r o n g Da , t h Tn 0. V v y t t c c c h s a(k), b(k) ub n g 0.

D o n u Tn(z) = 0, v i m i z , t h a(k) = 0, b(k) = 0, |k| n. N i c c h k h c , c c h {1, cos z, sin z, ..., cos nz, sin nz} v {eikz}|k|n , c l p t u y n t n h . C h n g t a m u n b i u d i n o h m T

n q u a Tn . n h l s a u t r l i c u h i

n y

n h l 3 . 7 . 2

. ( C n g t h c n i s u y R i c z ) C h o Tn Tn , k h i

T

n(z) =1

4n

2nk=1

(1)k+1 1sin2( zk

2)

Tn(z + zk), zk := (2k 1)2n

( 7 . 3 6 )

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C h n g m i n h . G i s

Tn(x) =a02

+n

k=1(a(k)cos kz + b(k)sin kz).

T a c zk l k h n g i m c a cos nz Tn , n n t h e o M n h t r n

cos nz = A2nk=1

sin(z zk)

2.

t

Qm(z) :=cos nz

2n(1)mcotan(z zm)

2

= cos nz2n

(1)m sin(z (zm ))/2sin(z zm)/2 ,

d o Qm(z) c n g c 2n k h n g i m .

tTn(z) :=

2nk=1 Tn(zk)Qk(z) Tn. T a c Tn(zk) = Tn(zk) , n n s u y r a

Tn(z) = c cos nz + Tn(z).

V i m i k = 1, ..., 2n t a c

T

Qk(z)cos nzdz = 0,

m Tn(z) l t h p t u y n t n h c a Qk(z) n n h s F o u r i e r c a Tn n g v i

cos nz c n g b n g k h n g . V v y c = a(n). N h t h t h Tn c v i t d i

d n g

Tn(z) = a(n)cos nz +cos nz

2n

2nk=1

(1)kcotanz zk2

Tn(zk).

T n g t h c n y d d n g t n h c

T

n(0) =1

4n

2nk=1

(1)k+1 1sin2 zk

2

Tn(zk) ( 7 . 3 7 )

M t k h c v i m i z C, c n h , t h

Tn(u) := Tn(z + u) c n g l a t h c

l n g g i c c a b i n u . S d n g ( 7 . 3 2 ) v i

Tn(u) t a c

T

n(0) =1

4n

2nk=1

(1)k+1 1sin2 zk

2

Tn(zk).

N h n g

T

n(0) = T

n(z), d o k h n g n h c c h n g m i n h .

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T c n g t h c n i s u y R i c z t a c h q u s a u

H q u 3 . 7 . 3

. ( B t n g t h c B e r n s t e i n . ) N u1 p v r N, t h t a c

T(r)n p n

r

Tnp, Tn Tn.C h n g m i n h .

T a c h c n c h n g m i n h v i r = 1 , n u r > 1, l p l i r l n t a s u y

r a k t q u . S d n g c n g t h c n i s u y R i c z v i sin nz t a c

n =1

4n

2nk=1

1

sin2 zk2

,

v

T

np 1

4n

2nk=1

1

sin2 zk2

Tn( + zk)p= nTnp

T r o n g k h n g g i a n C(T) , t a c n g c b t n g t h c B e r n s t e i n

T(r)n C(T) nrTnC(T) r = 1,...,Tn Tn. ( 7 . 3 8 )

c h n g m i n h b t n g t h c n y c h n g t a c n k t q u s a u

n h l 3 . 7 . 4

. ( S z e g o

[ 1 9 2 8 ] ) V i m i a t h c l n g g i c Tn , t r o n g c h u n

c a C(T) t a c

Tn(t)2 + n2Tn(t)

2 n2Tn2

C h n g m i n h . T r c h t t a x t Tn < 1. T a c t h g i s r n g t = 0 v Tn(0)

0. T a x c n h s t h c , || /(2n) v a t h c l n g g i c Sn x c n h b i c c n g t h c

sin n = Tn(0), v

Sn(x) := sin n(x + ) Tn(x). t tk := + (2k1)n . V |Tn(x)| < 1 v sin (2k1)2 = (1)k+1 , n n s u y r a s i g n Sn(tk) = (1)k+1, k = 0, 1,.... V v y t r n m i k h o n g (tk, tk+1), t n t i d u y n h t m t s

ck (tk, tk+1) s a o c h o Sn(ck) = 0 . T h e o c c h x c n h t r n t a c Sn(0) = 0 , n h n g 0 (t0, t1) n n c0 = 0.M t k h c Sn(t1) > 0, n n n u S

n(0) 0, t h t n t i m t n g h i m k h c c a Sn

t r n (t0, t1). V t h t a c Sn(0) > 0. S u y r a

0 Tn(0) = n cos n S

n(0) < n cos n = n

1 Tn(0)2.

= Tn(t)2 + n2Tn(t)2 n2, Tn < 1. ( 7 . 3 9 )

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G i s Tn l m t a t h c l n g g i c t u . T a c h n > Tn. S d n g ( 7 . 3 4 ) v i

Tn/ t a c

Tn(t)2 + n2Tn(t)

2 n22.

C h o Tn, t a s u y r a k t q u t r n .

T n h l S z e g o d d n g s u y r a ( 7 . 3 3 ) . N g o i r a

H q u 3 . 7 . 5

. V i m i Pn Pn , t a c

|Pn(x)| nPn

1 x2 , 1 < x < 1.

C h n g m i n h . tx := cos t

, t a c v i Tn(t) := Pn(cos t) Tn,

|Tn(t)| nPn.

N h n g

Tn(t) = Pn(x)

1 x2

n n t a c h q u t r n .

C h n g t a k t t h c c h n g n y b n g n h l N i k o l s k i i , c h o p h p c h n g

t a s o s n h c h u n c a Tn t r o n g c c m e t r i c k h c n h a u .

n h l 3 . 7 . 6

.C h o 0 < q p v Tn Tn t r n T . K h i

Tnp (2nr + 12

)1/q1/pTnq, ( 7 . 4 0 )

t r o n g r := r(q)

l s n g u y n n h n h t , l n h n h o c b n g q/2.

C h n g m i n h . D o Tn Tn n n Trn Trn . T a c

Trn(t) =1

2 TTrn(x)Dm(x t)dx, m := rn,

d o

2Trn = 2Tnr Tnrq/2T

|Tn(x)|q/2|Dm(x t)|dx. 2Tnrq/2 Dm2Tnq/2q= 2Tnrq/2 Tnq/2q

2(2m + 1),

v Dm2 =

2(2m + 1),

s u y r a

Tn

(

2nr + 1

2

)1/q

Tn

q.

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V y b t n g t h c N i k o l s k i i c c h n g m i n h k h i p = . K h i p = q h i n n h i n b t n g t h c n g . K h i

p > q, t a s d n g b t n g t h c s a u t h u

c k t l u n

Tnp = (T

|Tn(x)|p)1/p

= (

T

|Tn(x)|pq|Tn(x)|q)1/p

(TnpqT

|Tn(x)|qds)1/p = Tn1q/p Tnq/pq .


B i t p 1

.

( a ) C h o 1 p q , |A| < . C h n g m i n h r n g

fp |A|1/p1/qfq.

( b ) C h o p1, p2,...,pm (1, ) v mi=1 1/pi = 1, A = T, fj Lpj(T). C h n g

m i n h

T|mi=1

fi(t)|dt mi=1

fipi.B i t p

2

.

C h n g m i n h c c b t n g t h c

( i ) V i A = [a, b] , fp 2f1/2p f1/2p , f W2p , 1 p .( i i ) V i A = R, f2 2ff, f W2.( i i i ) K h i A = T, p = r = 2,

f

2

f1/2

2 f

1/2

2 , f W2

2 .

B i t p 3

.

C h o f, fn, n = 1, 2,... l c c h m t r o n g k h n g g i a n S o b o l e v Wrp (T) . B i t r n g

( a ) f(r)n f(r)p 0 k h i n .( b ) T n t i a T s a o c h o fn(a) f(a) k h i n .

C h n g m i n h r n g

( a )f(k)n (x) h i t u n f(k)(x) t r n T k h i n , k = 0, 1,...,r 1.

( b ) fn fWrp 0 k h i n .

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V n d n g k t q u n y h y c h n g m i n h r n g C(T) t r m t t r o n g Wrp (T) v i

m ir = 1, 2..., p 1

.

B i t p 4

.B n g q u y n p c h n g m i n h b i u d i n s a u c a p h n d f

Tr1(f , c, x) v i f Wrp (A) ,

f Tr1(f , c, x) =xc

f(r)(t)(x t)r1(r 1)! dt.

B i t p 5

.C h o l m t h m l m l i n t c t r n R+ v t n g v i (0) = 0.

C h n g m i n h r n g

l m t m o d u l l i n t c . T m m t v d c h n g t r n g c

h m l m o d u l l i n t c m k h n g p h i h m l m .

B i t p 6

. C h n g m i n h r n g n u

l m o d u l l i n t c , t h t n t i m o d u l l i n

t c s a o c h o

(t) (t) 2(t).

B i t p 7

.C h o f(x) = sin x v r = 1, 2, , p [1, ]. C h n g m i n h

( i ) T n t i t1 > 0 v c c h n g s C1(r), C2(r) > 0 s a o c h o v i t (0, t1),

C1(r)tr r(f, t)p C2(r)t

r.

( i i )

r(f)

1 = 2

r+2.

B i t p 8

.

( a ) ( X e m l i H q u 1 . 2 . 4 ) C h n g m i n h r n g

nk=0

(k

n x)2

n

k

xk(1 x)nk 1

4n

( b ) C h o f l m t h m b c h n t r n [0, 1] , c h n g m i n h

( b 1 )

|f(x)

f(y)

| (f,

|x

y

|)

v i m i x

= y

[0, 1]

.

( b 2 ) f Bn(f) 32(f, n1/2)

B i t p 9

.C h o f L2(T), g = f( + t) f v En(f)2 = infTTn( 12

20

|f(x) T(x)|2dx)1/2 . C h n g m i n h r n g

( a )|g(k)|2 = 2|f(k)|2(1 cos kt)

( b ) En(g)22 = 2En(f)

22 2

|k|>n |f(k)|2 cos kt .

( c ) T n t i h n g s c k h n g p h t h u c v o n,

En(f)22 c.(f, t)22 +

|k|>n

|fk|2 cos kt,

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B i t p 1 0

.C h o A = [0, 1], Ax = [0, x], V(x) = V ar(f, Ax) . C h n g m i n h

V(x) f(x)l h m k h n g g i m .

B i t p 1 1

. C h oA = [0, 1]

v

f0(x) =

1 n u x [0, 1/2);1 n u x [1/2, 1], v g0(x) =

0 n u x = 0;

x cos x

n u x = 0,( a ) H y t n h V ar(f0, A) . T s u y r a , W

11 (A) l t p c o n t h c s c a BV(A).

( b ) C h n g m i n h V ar(g0, A) = . T s u y r a W11 (A) l t p c o n c a

BV(A)C(A)

.

B i t p 1 2

.C h o A = [a, b] v c (a, b) . t Ac = [a, c] v Ac = [c, b] . C h n g

m i n h

V ar(f, A) = V ar(f, Ac) + V ar(f, Ac).

B i t p 1 3

.K i m t r a l i r n g

( a ) V i A = [0, 1]

h m

f(x) =

x log x n u x (0, 1];0

n ux = 0,

t h u c k h n g g i a n Z[0, 1].

( b )

Lip(, Lp) Hp

v i m i

1.

B i t p 1 4

.C h o A = R, 1 p v c c h m f v g x c n h t r n A n h

s a u

f(x) =

1

n u x (0, );

0 n u x (0, ), v g(x) =

1 n u x (1, );x n u x (0, 1];0

n ux (, 0].

C h n g m i n h :

( a ) V i = 1/p, (f, t)p C.t

.

( b ) V i 1 1/p, (g, t)p C.t .B i t p

1 5

.C h o B = [0, 1) v

M1(x) =

1 n u x B;0 n u x B,

l h m c t r n g c a t p B

. T a x c n h d y c c h m M2, M3, , Mr n h

s a u :

Mr(x) = (Mr1 M1)(x) = R Mr1(x y)M1(y)dy.H y t m M2, M3 d i d n g b i u t h c t n g m i n h .

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B i t p 1 6

.C h n g m i n h

( a ) Mr(x) > 0, v i m i x (0, r) .( b ) Mr(x) l i x n g q u a x = r/2 :

Mr(x) = Mr(r x);

( c )

kZ Mr(x k) = 1 v i m i x R.

( d )

Mr(x)dx = 1.

B i t p 1 7

. C h op 1, r = 1, 2, ,{fn}n, f l m t d y h m t r o n g Wrp . B i t

r n g fn fWrp (A) 0 k h i n . C h n g m i n h r n g v i m i 0 k < r, f(k)n

h i t u n f(k) t r n m i t p [c, d]

A .

B i t p 1 8

.C h o p 1, r = 1, 2, . C h n g m i n h r n g Wrp (A) l k h n g g i a n

B a n a c h .

B i t p 1 9

.C h o ak, bk l c c h s F o u r i e r ( d n g t h c ) c a f L2(T) . B i t

r n g

En(f)22

1

2(f, t)22 +

k>n

(a2k + b2k)cos kt.

C h n g m i n h

( a ) En(f)22

n4

/n0

(f, t)22 sin nt dt.

( b ) En(f)2 C.(f,/n)2, t r o n g C l h n g s k h n g p h t h u c v o f v

n.

B i t p 2 0

.C h o f Wr2 (T). C h n g m i n h r n g

En(f)22 n

2rEn(f(r))22.

B i t p 2 1

.C h n g m i n h :

( a ) C c t n h c h t c a K P h i m h m . ( b ) V i p [1, ], r = 1, 2, .

r(f, t) CK(f, tr; Lp, W

rp ),

t r o n g C l h n g s c h p h t h u c v o r .

B i t p 2 2

.A = [a, b],T . C h o f Lp(T) v i p [1, ) . C h n g m i n h t n t i

d y a t h c ( l n g g i c k h i A = T) s a o c h o limn

f

Pn

p = 0.

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C h - n g 4

C c n h l t r u n g t m c a l t h u y t x p x

C h n g t a i q u a m t b c c h u n b t n g i d i i n c c

n h l t r u n g t m c a l t h u y t x p x . C c n h l n y s g i i q u y t v n

t r n g t m c a l t h u y t x p x . V n n y c t r a n h s a u :

( i ) X c n h t c x p x k h i b i t t r n c a h m s f.

( i i ) X c n h t r n t h e o t c h i t c a En(f)p := infTn

f p .K h i

p = , t a c h a i n h l s a u

n h l 4 . 0 . 1

. ( J a c k s o n ( 1 9 1 2 ) ) N uf Cr(T), t h

En(f) Crnr

(f(r)

, n1

), n = 1, 2...

n h l 4 . 0 . 2

. ( B e r n s t e i n ) N u t n t i 0 < < 1 s a o c h o

En(f) Crnr, n = 1, 2, ...,

t h

f(r) Lip

C c n h l n y s c c h n g m i n h t r o n g c c m c s a u . T h a i k h n g

n h t r n t a s u y r a

f Lipk h i v c h k h i

En(f) Cn, 0 < < 1.

4 . 1 C c n h l t h u n

T r o n g p h n n y c h n g t a s n g h i n c u c c d n g k h

lý thuyết xấp xỉ và ứng dụng

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