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Московский государственный физико-технический университет (МФТИ)

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[НЕСОРТИРОВАННОЕ]

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матан Бесов - весь 2012

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x + y 2 + x − y 2 = 2( x 2 + y 2),

! "

# $%

& $

'$ !

% C([a, b]) CL1([a, b]) ( )* )* )

( ( ! & $ $ &

(

, -

( ( . / (x, y)

◦
	(x, x) 0 (x, x) = 0 x = 0
)◦ (x, y) =
)◦ (x, y) =		(y, x)
◦ (x1 + x2, y) = (x1, y) + (x2, y)
0◦	(λx, y) = λ(x, y)		λ C

, % %

Cn 1

! $ ( % % $

x = (x1, . . . , xn) ( n % /

(

(x, y) xiyi, x = (x1, . . . , xn), y = (y1, . . . , yn).

i=1

CL2([a, b]) 1

( % % . /

( [a, b] (

(f, g)	! b
	f (t)	g(t)	dt.
	a

(

, $

&$ (

&$ $

& ! &$

(

- ( - !

(x, y)

x y		(x0 +
+ x, y0 + y) → (x0, y0) x +	y → 0
3 ( ,	x < δ < 1	y <

< δ < 1 4 & ! 5 6 72 & )

|(x0 + x, y0 + y) − (x0, y0)| =

= |(Δx, y0) + (x0, y) + (Δx, y)|x y0 + y x0 + x y

δ( y0 + ( x0 + δ)) δ( x0 + y0 + 1).

xk x a R

◦ xk → x (xk, a) → (x, a) k → ∞

∞	∞
◦ xj = x	(xj , a) = (x, a)
j=1	j=1

l2		∞
x = (x1, x2, . . .), xi R		∞
x = (x1, x2, . . .), xi R	i N,	xi2 < ∞,
		i=1
! "
∞

(x, y)	xiyi
i=1

! ! !

# ! $ % !

& " '
∞		∞	∞
\|xiyi\|	1	xi2	+ yi2 .
	2
i=1		i=1	i=1

( !% !

l2 ! ! !

x ' xi

CL2([a, b]) " ! ! !

) "

' ** ' ! ! !

CL2([a, b]) +

% ,-

- * '

L2((a, b)) (a, b) (−∞, +∞) " . / ! ! !

L2((a, b)) * ' " * ' % ! -

$ " . 0&

* * '- '

! ! !

L2((a, b))

,* R "

+ x, y R " %

$ & (x, y) = 0

{ej }∞j=1

R " %

(ej , ek) = 0 j, k N, j = k.

1 ej = 1 j N !

$& " !

- -

% 1 {ej }∞j=1 2 ! ej > 0 j N $

%& % k N e1 ek

- "

3 )

k N λj R

λj ej = 0.

j=1

1 λs = 0 !

! es " ! %

2
λs es	= 0 4 % es = 0

es -

{12 cos x sin x cos 2x sin 2x . . .}

! π

(f, g) =	f (x)g(x) dx.
	−π

{eikx}+∞

k=−∞

	! 2π
(f, g) =			f (x)¯g(x) dx.
	0
			1 dn(x2 − 1)n
P0(x) = 1, Pn(x) =			1 dn(x2 − 1)n		, n N,

		2nn!		dxn
		2nn!		dxn
					!
{P	}∞
n	n=0 " #

	! 1
(f, g) =			f (x)g(x) dx.

−1

# " # Pn

$ Qm m < n!

% ((x2 −1)n)(k) 0 k n−1 $&

x = ±1 &

! 1 Qm(x)

dn(x2 − 1)n

dx = − ! 1 Qm(x)

dn−1(x2 − 1)n

dx =

−1

dxn

−1

dxn−1

! 1

dn−2(x2 − 1)n

Qm(x)

dxn−2

dx = . . . =

−1

n−m−1

n 1

(m)

(x − 1)

= (−1)

(x)

n−m−1

= 0. '

−1

( "−11 Pm(x)Pn(x) dx = 0 0 m < n!

§
( " #
Pn(x) =	(2n − 1)!!	xn + Qn−1(x),

	n!

Qn−1 ) * n−1! + '

! 1

(2n

−

Pn2(x) dx =

1)!!

Pn(x)xn dx =

−1

(2n − 1)!!

1 dn(x2 − 1)n

−1

dx = . . . =

n!(2n)!!

dxn

= (−1)n−1

(2n − 1)!!

! 1

((x2 − 1)n) x dx =

(2n)!!

! −1

= (−1)

n−1 (2n − 1)!!

n−1

(2n − 2)!!

−1

(x −

x dx =

(2n − 1)!!

= (−1)n−2

(x2 − 1)n−2x4 dx = . . . =

(2n − 4)!!3 −1

! 1

−1 x

dx =

2n + 1

Pn =

2n + 1 !

- R $

) . x

(x, x)

{ej }∞

R! /

j=1 )

> 0 j N!

∞

x R x =

αkek

k=1

(x, ek)

αk =

- ! ( 01!2!'

& R # # ! +

k=1

∞

(x, es) = αk(ek, es) = αs(es, es),

k=1

x R {e

}∞

k=1

αk =

(x, ek )

ek 2

∞

x {e

}∞

k=1

x {ek}k∞=1 Sn

= Sn(x) =

αkek

k=1

}∞

x {e

k=1

! "

x R

∞

αkek.

k=1

& '

x (

∞

αkek % ) x R

x =

k=1

x − Sn(x)

→ 0

n → ∞.

*' + ' ' )

Sn(ek) = ek

Sn(Tn) = Tn,

Tn =

ckek.

(x − Sn(x), ek) = 0

k=1

k n.

x = Sn(x) + (x − Sn(x)), (Sn(x), x − Sn(x)) = 0.

§

x 2 = x − Sn(x) 2 + Sn(x) 2.	.

/0 -

x 2 = (x − Sn) + Sn 2 = ((x − Sn) + Sn, (x − Sn) + Sn) =

=x − Sn(x) 2 + Sn(x) 2.

! "

#$	+		+
	+		+
	+	n	+
min	+		+	= x − Sn(x) .
min	+x −		ckek+	= x − Sn(x) .
c1, ..., cn +		k=1	+

/ Tn = ckek 1

+ , ' '

x − Tn 2 = (x − Tn) − Sn(x − Tn) 2 + Sn(x − Tn) 2 =

= x − Sn(x) 2 + Sn(x) − Tn 2 x − Sn(x) 2.

%
x − Sn(x) x − Sm(x)	n m.
& ' (	x R

αk x

}∞

k=1

∞

α2

x 2.

k=1

{ek} .

+ n

αk

= +

αkek+ = Sn

k=1

αk ek → 0 k → ∞,

{ek}∞k=1

∞

αk2 < ∞, αk → 0 k → ∞.

k=1

{ek}∞k=1

! " x R # "

! $αk !

x%&


◦									n
◦	ε > 0 #								ckek
	{ek}∞								k=1
	{ek}∞
		k=1 !
		+					+
		+	n				+
		+					+	< ε,
		+	c		e		+
		+x −	c		e		+
		+	k=1	k		k	+
		∞	k=1
		∞
◦
◦	x =	αkek
◦		k=1
◦	&
			∞
		x 2 = αk2						ek 2.
			k=1
1◦									2◦

! " 1◦ #

ε > 0 nε N : x − Snε (x) < ε,

x − Sn(x) < ε n nε.

$ 2◦

§ !

& 2◦ 3◦ % $

' " ' ( $

x 2 = x − Sn(x) 2 + αk2 ek 2.

k=1

) * " % % %#

% + " , " , "

%- " , . $

2◦ 3◦ ' / 0 ' %" %

"%$ 2◦ x

1 {xk}∞k=1 '"$#

+ " " " '" " R

, % R ,- #

,- + 2 ' R

{ek}∞k=1

# " ! &

◦ {ek}∞			R
	k=1
	∞
◦		x R
◦	x = αkek	x R
	k=1
	∞
◦	x 2 = αk2 ek 2 x R

k=1

$αk ! x%

' % # " . $% $ x R

! " #$% {ek}∞k=1

' α1 α2 α3	! '
∞
αk2 ek	2	3
k=1
(

				∞

H				αkek
x H				k=1
x H				∞
				∞

				αkek = x.
				k=1

ε > 0 nε N
+	n+p	+		n+p
+	n+p	+2		n+p
+		+	=		2 < ε n nε, p N,
+		αkek+	=	αk2 ek	2 < ε n nε, p N,
+	k=n+1	+		k=n+1

!			" #$ H % #
		∞
	n	αkek
	k=1	αkek	& # # #$ #$
	k=1	n=1

' % # ( H # ) # x H

∞

* ! H αkek = x % % # + (

k=1
{e	}∞
k		k=1
H
x H
H		∞
∞		∞
		(x, ek)
αkek =				ek = x0,
αkek =		k=1	ek 2	ek = x0,
k=1		k=1
(x − x0, ej ) = 0 j N			∞
			∞
,			αk2 ek 2
			k=1		∞
					∞

# # - .		/ #$			αkek

k=1

% 0 , 12 3 % j N

∞

(x − x0, ej ) = (x, ej ) − αk(ek, ej ) =

k=1

= (x, ej ) − αj ej 2 = (x, ej ) − (x, ej ) = 0.

§ ! "
	4 ! # # % #
{ek}∞	R # (

k=1 % ! 5 % #

$ +5 ! x R

(x, ej ) = 0 j N x = 0,

# ' # # ! ) # x R$ !

# # ! ) # ( {ek}∞k=1

H !

{ek}∞k=1 "

6 {ek}∞k=1 % #H % x H * ! # 6

		∞			∞
x	2		2	2	(x, ek)2
		= αk		ek	=			.
						ek 2
		k=1			k=1
6 )$ (x, ek) = 0 k N$ x
						= 0$ x = 0
/ #$ {e }∞
6 {e			}∞	k	k=1 #
							H$ x H α		k 7
		k	k=1 #

)&& 8 # ( 2 ) # x * ! % 9

∞	∞
	αkek =	(x, ek)		ek = x0 H
	αkek =			ek = x0 H
k=1		ek	2
k=1	k=1
# ) # x0 H$ % :
	(x − x0, ek) = 0 k N.
# ( {e					}∞
		∞		k	k=1 + $
		∞
			αkek
x − x0 = 0$ x = x0 =			αkek			}∞
		k=1
3 ( $ {e
					k	k=1 % #
45 # #( %
	6 # # # {xk}∞
						k=0

% # # ## % # & # 8 C([a, b])

C([a, b]) = {f : f 7 # % ( # # [a, b] & # 8,

f = max |f |}

[a,b]



1		, cos t, sin t, cos 2t, sin 2t, cos 3t, sin 3t, . . .	!"#
	2

Cper = {f : f $ 2π% ,

f = max |f |}

(−∞,+∞)

& % &

!"#

C ([−π, π]) = {f : f $ [−π, π] ,

f (−π) = f (π)}

!"#

C([−π, π]) ' ( )

) [−π, π] ) f f (−π) = f (π)

*) *+ + *

( %

Tn Tn(−π) = Tn(π)

, % * * % {xk}∞k=0

& CL1([a, b])( CL2([a, b])( RL1([a, b])( RL2([a, b])( L1([a, b])( L2([a, b]) (

+ & &

&	[a, b]		-!
-.( -/#

!"#		& CL1([−π, π])( CL2([−π, π])(
RL1((−π, π))(		RL2((−π, π))( L1([−π, π])( L2([−π, π])

.( + & &

C0([−π, π]) = {f : f $ [−π, π] , f (−π) = f (π) = 0}.

, + *( !"#

RL1((−π, π + δ)) δ > 0

, * f L2([−π, π]) % f %

% 0 * 1

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

& % 2 f L2([−π, π])(

% % #( %

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

3% * ak( bk $ 4 0 *

( ! #( %

) 5

6 % + !"# L2([−π, π]) 7# + -

( %

+ *

[−π, π] f

, * f L2([−1, 1]) % f %

% 0 * 5 % 1

∞			!	1
		2n + 1		1
f =	αnPn, αn =	2n + 1			f (x)Pn(x) dx
f =	αnPn, αn =	2		−1	f (x)Pn(x) dx
n=0		2		−1
n=0					L2([−1, 1])(
% & %					L2([−1, 1])(

% % #( %

[−1, 1]

! R "

! {e		}∞	e	j	R	j N
	j	j=1		j
R
◦	#$ x R
	∞

	x = λj ej , λj R;
	j=1
%◦
! {e						}∞
					j	j=1 &

' R

( {x	}∞
k	k=0
C([−1, 1]) ) *+ &
	∞
, f (x) =
, f (x) =	λkxk
	k=0
- & . C([−1, 1])

[−1, 1] $ f

(−1, 1)

C([−1, 1])

/ $ ) +

C ([−π, π]) & )

{ek}∞k=1

1 ! {ek}∞

k=1 "

$ R x R

2$ 3

				∞
					(x, ek)
			x = αkek, αk =		(x, ek)	,
			x = αkek, αk =		ek 2	,
				k=1
			x $			4 2

{e	k	}∞		R
	k	k=1 "

{xk}∞k=1

R {ek}∞k=1

◦ {ek}∞k=1

%◦ ! n N

en = an1x1 + . . . + annxn, ann = 0.

" ! # # {ek}∞k=1

1◦ 2◦ $ $ ! ±1

1 5 e1 6 e1 = = a11x17 & a11

(e1, e1) = a112 (x1, x1) = 1, a11 =	±1	.

	x1

! & ek )k = 1 n −1+ #6

1◦ 2◦

/6 & en

en = ann(xn − bn1e1 − . . . − bn n−1en−1).

8 $

xn − bn1e1 − . . . − bnn−1en−1,

6 & xn $

& e1

en−1

(en, ek) = 0 k < n

bnk = (xn, ek) (k = 1, . . . , n − 1).

(en, en) = a2nn xn − bn1e1 − . . . − bnn−1en−1 2 = 1,

ann en

±1

{xk}∞k=1 {ek}∞k=1 !

" 1◦ 2◦

# R {x	}∞	{e }∞
k	k=1	k k=1

	!$b	! ψ(y)
I(y) =	f (x, y) dx, J(y) =	f (x, y) dx
	a	ϕ(y)

% !

&& ' !

f [a, b] ×

× [c, d]
I(y) = "ab f (x, y) dx [c, d]
( y, y +		y [c, d] )
!	b	! b

	f (x, y + y) dx −
\|I(y + y) − I(y)\| =	f (x, y + y) dx −		f (x, y) dx

!	a	a
!	b

|f (x, y + y) − f (x, y)| dx (b − a)ω(| y|; f ),

ω(δ; f ) * & ' f )

ω(δ; f ) → 0 δ → 0

& ' f [a, b] × [c, d]

ϕ ψ [c, d]

ϕ ψ [c, d] G = {(x, y) R2 ϕ(y) x ψ(y) c y d}


f	G
	"	ψ(y)	f (x, y) dx
J(y) =		ϕ(y)	f (x, y) dx

[c, d]

J(y) = f (ϕ(y) + t(ψ(y) − ϕ(y)), y)(ψ(y) − ϕ(y)) dt

! 1

g(t, y) dt.

g [0, 1] × × [c, d]

! " J(y)

[c, d]


"◦	f [a, b] × [c, d]
#◦	I(y) = "ab f (x, y) dx
$◦	y [c, d]		"	d

	[a, b]			c f (x, y) dy x


	! d !	b		! b	! d
	c	a	f (x, y) dx dy =		f (x, y) dy dx.
				a	c

% "&$" "&$"

f [a, b] × [c, d]

f								∂f
f								∂y
[a, b] × [c, d]								∂y
[a, b] × [c, d]
			! b
			I(y) =	f (x, y) dx
			a		! b ∂
[c, d]
	dI(y)	d	! b
			f (x, y) dx =				f (x, y) dx.
	dy	dy	f (x, y) dx =				f (x, y) dx.
	dy	dy	a		a	∂y

y + y

[c, d]

(

)

∂

I(y + y) − I(y)

−

f (x, y) dx

a ∂y

= ! b

f (x, y + y) − f (x, y)

−

∂f

(x, y) dx

∂y

∂f

(x, y + θ y) −

(x, y) dx

− a)ω | y|;

∂y

∂f

δ;

∂f

∂y

[a, b]×[c, d] + !

∂f

[a, b] × [c, d]

∂y

→ 0

y| → 0.

| y|;

∂f

∂y

, -

dI(y)

lim

I(y +

y) − I(y)

! b

∂f

(x, y) dx,

y→0

∂y

∂f

∂y

[a, b] × [c, d] ϕ ψ

[c, d] a ϕ ψ b [c, d]

[c, d]

dJ(y)

! ψ(y)

f (x, y) dx =

ϕ(y)

= ! ψ(y)

∂f

(x, y) dx+f (ψ(y), y)

dψ

(y)−f (ϕ(y), y)

dϕ

(y). /"0

∂y

ϕ(y)

1 [c, d] ×[a, b] ×[a, b]

F (y, u, v)

! v

f (x, y) dx.

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