матан Бесов - весь 2012

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Московский государственный физико-технический университет (МФТИ)

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[НЕСОРТИРОВАННОЕ]

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t, 0 t π, a x b,

(t) = f a +

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[a, b]

{ai}mi=0 [a, b] a = a0 < a1 < a2 < . . . < am = = b f

[ai−1, ai]

2π !

R R

[−π, π]

f 2π

f f

sup \|Sn(x; f ) − f (x)\| C	ln n	n 2,
	n
x R

C n

0 < δ = δn < π

Sn(x; f ) − f (x) =

δ ! π

+ gx(t) sin n +

dt = In + Jn,

π 0

gx(t)

f (x + t) + f (x − t) − 2f (x)

2 sin

M1 = max |f |

! " #$%

& ! '& 0 < t π

|f (x + t) + f (x − t) − 2f (x)| 2M1t.

(

t < sin t < t 0 < t <

0 < t π

|gx(t)|

2M1t

πM1

2 sin

" ) % $

* t

gx(t)

(x + t) − f

− t)|

2 sin

+|f (x + t) + f (x −t) −2f (x)|

cos

πM1

2πM1

2 t

4 sin

2 tg 2

+ |In| δM1

, ! " $' '

!π

cos

n +

cos n +

Jn = −

gx(t)

dt.

n +

+ "

2M1 ln

1 + 2 ln

|Jn|

n +

$' δ = δn =

n 2

sup |Sn(x; f ) − f (x)| |In| + |Jn|

C ln n

x R

$ C n -

$ .

' α = 1 /

0 .

" & ' 1 0

' " 2$ ' 3 ' & ' 1 *

% ) '& ) ) ! & . ' *

'"! & " 40

E = R E = [a, b] 0 < < α 1 4 ' ' f 5 E → R '

α %

α = 1 Mα > 05

|f (x) − f (y)| Mα|x − y|α x, y E.

α α ! f

[a, b] " # [a, b]

$ % % &

2π f

R α 0 < α 1

f R

sup |Sn(x; f ) − f (x)| Cα

ln n

n 2,

nα

x R

Cα

' (

) ( *+,

! π

f (x + t) − f (x)

sin n +

t dt.

Sn(x; f ) − f (x) =

−π

2 sin

f (x + t) − f (x)

hx(t)

, λ = λn

= n +

2π

δ < π.

2 sin

. ( / %

! π−

! π

hx(t) sin λt dt = −

hx t +

sin λt dt =

−π

−π−

! π

! π ! −π

= − hx t +

sin λt dt + −

t +

sin λt dt =

−π

π−

−π−

hx(t) − hx

t +

sin λt dt + Kn,

−π

πMα

π +

|Kn|

λ2 sin

2λ

|Sn(x; f ) − f (x)|

−π

t +

− hx(t) dt + |Kn| =

2π

! δ

−δ ! π

. . . dt + |Kn| +

. . . dt + |Kn| =

2π

−δ

−π

- |t| 2δ

= Iδ,n(x) + Jδ,n(x) + |Kn|.

*0,

πMα|t|α

|hx(t)|

Mα|t|α−1,

2|t|

! 2δ

Iδ,n(x) Mα

tα−1 dt =

Mα2αδα.

*1,

δ < |t| < π (

' Jδ,n(x)

2π

f x + t +

− f (x)

f (x + t) − f (x)

hx t +

−hx(t) =

−

2 sin

t +

2 sin

f x + t +

− f (x + t)

t +

2 sin

−

(f (x + t) − f (x)),

t + λ

sin

t +

− hx(t)

C1 f x + t +

− f (x + t)

C2|f (x + t) − f (x)|

|t|

t2λ

C2Mα|t|α

C1Mα

CMα

|t|

|t|λα

*2,

t2λ

Jδ,n(x)

! π CMα dt

CMα

λα

πλα

δ =

! "

# ! !

2π f

[−π, π]

[a , b ] f

f f

[a, b] (a , b )

$ % λ = λn = n + 12 2λπ δ [a −2δ, b + 2δ] [a , b ] x [a, b] & % " '

& ( α = 1

Iδ,n(x) 2δ max |f |.

)

[a ,b ]

$ * Jδ,n % % +

, -

Jδ,n(x)

−π f

u +

− f (u)

du+

πδ

|f (u)| du + 2π max |f | .

−π

2πδ2λ

[a,b]

% ε > 0 & δ = δ(ε) > 0

* sup Iδ,n <

2π

[a,b]

nδ N : sup Jδ,n <

n nδ .

[a,b]

- % ' ) ./ *

sup |Sn(x; f ) − f (x)| → 0 n → ∞,

x [a,b]

§ ! " #

0 * 1 2 3 "

% % * +

" # 3 f

% [a, b] * % 4 " 3 2 ε+ (a − ε, b + ε) 4 %

ε > 0

	∞
5% 1 *	sin kx
5% 1 *	k=1	k	%
	k=1

. % [ε, 2π − ε] ε > 0

3 f (x) = π − x

- 1 ! % * +

" 33 3 67 +

α > 0 [a , b ]

A0			n
A0
		+	Ak cos kx + Bk sin kx (An2 + Bn2 > 0)
	2	+	Ak cos kx + Bk sin kx (An2 + Bn2 > 0)
			k=1

f 2π

! ε > 0 " #

# T $

max |f (x) − T (x)| < ε.

x R
$ % 8 ε > 0	τ =
= {xj }jJ=0 9 % % [−π, π] xj	= −π + j	2π
		J

3 3 f +

* (xj , f (xj )) 3 3 f % 0 % * * % ΛJ 2π+ * +

3 3 " [−π, π]

ΛJ

[−π, π]

\|f (x ) − f (x )\| <	ε	\|x − x \|					2π	,

4							J
J = J(ε) N
max \|f (x) − ΛJ (x)\| <			ε		.

x [−π,π]		2
! ΛJ "# "$
% ! & ' ΛJ		R (
& ) n = n(ε)
max \|ΛJ (x) − Sn(x; ΛJ )\| <				ε		.

x R		2

* ' '

max |f (x) − Sn(x; ΛJ )| < ε,

x R

T (x) = Sn(x; ΛJ ).

$ +

& ,) -

1 f

[−π, π] f (−π) = f (π)

ε > 0

max |f (x) − T (x)| < ε.

−π x π

% -& -& f (−π) = f (π)

. $

T & /-) 0 & Sn(x; f ) / , ! & f 0 &

§ ! " # $

! & - '

& / - + ' & 0

f T + & σn(x; f ) /

f 0 -&1 n

σn(x; f ) = S0(x; f ) + S1(x; f ) + . . . + Sn(x; f ) n + 1

! &

	f	2π

	n → ∞.
σn(x; f ) f (x)	n → ∞.

- &

! ' & !

! + , )% ,) -

! " 2π f

f (x) # #

% + & ' ' ) '

& ' &

' ' ' ' 3 ' )

4 ' ) 1 − 1 + 1 − 1 + . . .

' ' 12 ( )&, $ /5 1 0

+ & -+ ,- &,

' ) -

[a, b]

$% ε > 0 %

max |f (x) − P (x)| < ε.

a x b

[0, π]

[a, b]

f [−π, 0]

2π f ! "

# f R → R	2π"
R $	% ε > 0	"
% % T
max \|f (t) − T (t)\| max \|f (t) − T (t)\| <			ε	.

0 t π	t R	2

& # cos kt sin kt ' T (t)(

R = +∞ ) "

*
* + n = n(ε)
max \|T (t) − Pn(t)\| <		ε	,

0 t π	2

% Pn , % - ! # T

max \|f (t) − Pn(t)\| <	ε	+	ε	= ε,
	2		2
0 t π

' + x(


			x − a
max	f (x) − Pn	π	x − a	< ε.
max	f (x) − Pn	π	b − a	< ε.
a x b			b − a

- / ! + "

[a, b]

f 2π

ak bk ! " #

$ %

		∞			!	π
	a02	+ (ak2 + bk2 )		1
						f 2(x) dx.	'$(
2			π
		k=1				−π
f							2π"

" !"

! # ! #			01 0 0 "
+ R &
a0		∞

f (x) =		+	ak cos kx + bk sin kx.	'0(
	2
		k=1

2 '0( f (x) % "

' + ( "

3 ! '01$0( + *!! # "

		∞			!	π
a02		2 2	1			π
a02		2 2	1			2
	+	(ak + bk) =				−π f	(x) dx,	'/(
2	+	(ak + bk) =		π		−π f	(x) dx,	'/(
		k=1

% '$(

! # f "

ΛJ R → R , 2π"

! # [−π, π] "

3 4 01 /$ '% ! ! " # ΛJ % ! ! # f ( ak(f ) bk(f ) *!! # & ! # f

	a02(ΛJ )		n			!		π
	a02(ΛJ )	+ (ak2 (ΛJ ) + bk2 (ΛJ ))				1				ΛJ2 (x) dx n N.
2		+ (ak2 (ΛJ ) + bk2 (ΛJ ))				π		−π		ΛJ2 (x) dx n N.
2			k=1			π		−π
	n N J → ∞
ak(ΛJ ) → ak(f ),					bk(ΛJ ) → bk(f ),
				! π	!	π
				ΛJ2 (x) dx →			f 2(x) dx.
				−π		−π
	! J → ∞
										! π
			a02(f )	n					1	! π
			a02(f )	+ (ak2 (f ) + bk2 (f ))					1	f 2(x) dx.
		2		+ (ak2 (f ) + bk2 (f ))					π	f 2(x) dx.
		2		k=1					π	−π

! n →

→∞ "

#$ %

& ' "

§ () ' * " (−π, π) +

! , "! ! %

2π f

	a0		∞
	a0
f (x) =		+	ak cos kx + bk sin kx
f (x) =	2	+	ak cos kx + bk sin kx
			k=1

∞

f (x) −kak sin kx + kbk cos kx,

k=1

∞

f (x)

α0

αk cos kx + βk sin kx.

k=1

f (x) dx =

[f (π) − f (−π)] = 0.

α0 =

−π

. , *

! π

(x) cos kx dx =

αk =

−π f

f (x) cos kx

f (x) sin kx dx = kbk,

−π

βk =

f (x) sin kx dx =

−π

f (x) sin kx

−

f (x) cos kx dx = −kak.

−π

2π f

m − 1 "

" " m N

# f


	1
\|ak\| + \|bk\| = o		k → ∞.	)

	km

- m N

∞

f (m)(x) αk cos kx + βk sin kx.

k=1

|αk| + |βk| = km(|ak| + |bk|), k N.

/ + " 0 αk βk → 0 k → ∞

)

! " "

# $%& ' f (m)(

∞		!	π
k2m(ak2 + bk2 )	1
			(f (m)(x))2dx < ∞.
	π
k=1			−π

! " )

+ ' , - + +

2π

' ' '

∞

˜	$.&
S(x; f ) ak sin kx − bk cos kx,	$.&
k=1

- ak bk / f

						1
	n	cos	x	− cos n +		1	x
˜		cos	2	− cos n +		2	x
Dn(x) = k=1 sin kx =				2 sin	x			.
Dn(x) = k=1 sin kx =				2 sin	2

0 " $1 % 2&

3- $1 % 4&
		n
		n
	˜			ak sin kx − bk cos kx
	˜	(x; f ) =
	Sn	(x; f ) =
		k=1
$.& "
	!	π
˜	1	˜
Sn(x; f ) = −	π	Dn(t)[f (x + t) − f (x − t)] dt =
		0		! π
		1		! π
		1				1		˜
		=	π	0	hx(t) cos	n +	2	t dt + f (x),

hx(t)			f (x + t) − f (x − t)				,
					t
				2 sin
					2
˜	1 ! π f (x + t) − f (x − t)
f (x) −		0						dt.
	π			2 tg		t

				2

2π f

ak bk

$.& C >

> 0 n 2
	∞			ln n

sup	ak sin kx − bk cos kx		C		.	$5&

x R	n+1			n
,		0 " M1 max \|f \|				6
				R

7 + 7 ' -"

|f (x + t) − f (x − t)| 2M1t, 0 < t π,

"- x 7

f (x) $ - ' -' (0, π]

& $.& +

0 2 n < p

Sp(x, f ) − Sn(x; f ) =

hx(t) cos

p +

t dt−

−

! π

hx(t) cos n +

t dt,

|hx(t)| πM1,

hx(t)

(x + t) + f

(x − t)|

2 sin

+|f (x + h) − f (x − h)|

cos

πM1

2πM1

2 t

4 sin 2

2 tg 2

	˜	ln n		ln p
˜
sup \|Sp(x; f ) − Sn(x; f )\| C			+ C		2	n < p,

x R		n		p
p → ∞				!"
#	$					% &

& ' " &
∞
	akbk '
k=1		∞

				n
						("
			akbk	\|a1\| sup	bk	("
		k=1		n N
		k=1		k=1

)

◦ {ak}

◦ ("

n ∞

k=1 n=1

m N 2π

m − 1

f(m)

f f

R ε > 0
max \|f (x) − Sn(x; f )\| = O		ln n		=
max \|f (x) − Sn(x; f )\| = O				=
x R			nm				n → ∞.
				1
			= o
			= o		nm−ε
					nm−ε
m = 1
ϕ f (m−1) α β
							k k !""
#$ % " # ϕ
	∞
	∞					ln n
						ln n
sup	αk cos kx + βk sin kx C						n 2.	&'
sup	αk cos kx + βk sin kx C					n	n 2.	&'
x R	k=n+1					n

ak bk !"" #$ % " # f

m − 1 ( ) m − 1 (

$ & x R

∞

|rn(x; f )|

ak cos kx + bk sin kx =

k=n+1

∞

(αk cos kx + βk sin kx) .

km−1

k=n+1

* + &'

|rn(x; f )| C

ln n

1 m−1

ln n

n (n + 1)

m − 1 ( )

∞

|rn(x; f )|

ak cos kx + bk sin kx =

k=n+1

∞

(αk sin kx −

βk cos kx) .

km−1

k=n+1

∞

αk sin kx − βk cos kx .- / 0

k=n+1

1 +

|rn(x; f )| C

ln n

1 m−1

ln n

n (n + 1)

) $ 2 3 $.

" #- f 2 3 4 .- ( - %

5 6 7

" - " # f 3

[−π, π] 2 - # .

48 $ - ( 2π 7

- $ $ *- " # f 9 [−π, π] → R $ $

48 $ -

f+(j)(−π) = f−(j)(π) j = 0, 1, . . . , m − 1.

f [−π, π] → R

f (−π) = f (π)

! 2 " # $

" ! ! $

2π$

# % & ' 2

# !% (

& " ) * + , ! (

-# #

2 m N 2π

m − 1

f (m)

max \|f (x) − Sn(x; f )\| = o		1		n → ∞. * +
x R	n	m−
			2
'			. #

f % &

/ & f

	∞


\|rn(x; f )\| =		ak cos kx + bk sin kx

k=n+1	∞	∞
	(\|ak\| + \|bk\|)	(\|αk\| + \|βk\|)	1	,
	(\|ak\| + \|bk\|)	(\|αk\| + \|βk\|)	km	,
	k=n+1	k=n+1	km

( αk" βk 0 - & f (m)" $

m$ $

1 2 3 4) ( * 5 +

N		N		N
	1				1
(\|αk\| + \|βk\|)	1		(\|αk\| + \|βk\|)2		1	.
(\|αk\| + \|βk\|)	km		(\|αk\| + \|βk\|)2			.
k=n+1	km	k=n+1		k=n+1	k2m

# N → → ∞ " % "

% N ∞ 6 3 $

|rn(x; f )|

∞

2 (α

+ β

)

k2m

= εn

k2m

, * +

k=n+1

∞

% εn → 0 n → ∞ # (αk2 +

k=1

+ βk2)" )

f (m) 7 "

∞

k dx

∞ dx

k−1

(2m − 1)n2m−1

k2m

k=n+1

x2m

k=n+1

/ * + * +

[−π, π]

∞

f (x)

+ ak cos kx + bk sin kx

k=1

! "

x a

∞

f (t) dt =

(ak cos kt + bk sin kt) dt =

k=1

∞

a0x

sin kx +

(1 − cos kx), * 8+

k=1

! x

F (x) =

f (t) −

dt.

F [−π, π]

F (π) − F (−π) =

! π

f (t) dt − πa0 = 0.

−π

F (t) = f (t) −

[−π, π]

" F # " F

[−π, π]$

∞

F (x) =

Ak cos kx + Bk sin kx.

%&'(

k=1

) ! *"" Ak Bk

" F *"" " f

Ak =

!π F (x) cos kx dx =

!π

−π

sin kx

F (x)

−

f (x) −

sin kx dx = −

, k N.

−π

kπ

−π

, Bk =

k N

# A0 x = 0 %&'(

∞

Ak = 0,

k=1

∞

F (x) =

sin kx +

(1 − cos kx),

k=1

%&.(

§ 2l

l > 0 f / 2l0 " 10

2 [−l, l] fl(x) =

= f	lx	3 fl / 2π0 " 1 0
	π

2 [−π, π] fl

1 2 x

πx

" f

∞

f (x)

+ ak cos

kπx

+ bk sin

kπx

k=1

a0 =

−l f (x) dx,

kπx

ak =

−l f (x) cos

dx,

bk =

−l f (x) sin

dx,

f 2l

1 1 ! 2l0 #

" ! # 0

! 2l0 # " ! 1 1 0

2 [−l, l]

# " !

cos

sin

cos

2π

sin

2π

x, . . .

2 #

24 2 l = π

51 # # 0

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