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Московский государственный физико-технический университет (МФТИ)

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[НЕСОРТИРОВАННОЕ]

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f (x0) =	f (x0 − 0) + f (x0 + 0)	x0
	2
f

f x0

f x0 x0

(−∞, +∞) a(y) b(y)

"◦ x0 f

! ∞

S(x0) = S(x0, f ) = [a(y) cos x0y + b(y) sin x0y] dy =

0	f (x0	+ 0) + f (x0 − 0)
=			;
		2

◦ x0 f

S(x0, f ) = f (x0).

Sη (x0) −

f (x0 + 0) + f (x0 − 0)

∞ 2

sin ηt

! 0∞ [f (x0 + t) − f (x0 + 0)]

dt+

[f (x0 − t) − f (x0 − 0)]

sin ηt

dt =

J+(η) +

J−(η).

η > 0 J+(η)

(η) =

! 1 f (x0

+ t) − f (x0 + 0)

sin ηt dt+

! ∞0

! ∞

sin u

f (x0 + t)

sin ηt dt − f (x0 + 0)

du =

= J1+(η) + J2+(η) − f (x0 + 0)J3+(η).

J1+(η) J2+(η) → 0 η → +∞
! " " # $ % % J3+(η) → 0
"	∞
η → +∞ &		sin u	du '
	0	u

J+(η) → 0 J−(η) → 0 η → +∞

( $ "

) * +

,- * +

* . / 0

" ! " +

+ % %

& * 12 * &

0 % &

+ +

3 * 0 % *

w(t) = u(t) + iv(t), u(t), v(t) R t a, b ,

# $* #

, / * * 0 %

! b	! b	! b
	w(t) dt =	u(t) dt + i v(t) dt,
a	a	a

& -,

!	b		! b

			\|w(t)\| dt,
	a	w(t) dt	\|w(t)\| dt,
	a		a

* - #

$, & $*

0 % f 5 (−∞, +∞) → R

# ,$ [−η, η] η > 0

"
−∞+∞ f (x) dx
	! +∞	! η
v.p.	−∞	f (x) dx lim	f (x) dx.
	−∞	η→+∞ −η

! " #$ ϕ% [a, b] \ {x0} → R& x0 (a, b)&

' ( ) ' *+ ' ' , [a, b] \ Uε(x0)& ε > 0

"ab ϕ(x) dx

! b	ϕ(x) dx lim	!
v.p.	ϕ(x) dx lim	ϕ(x) dx.
a	ε→0	[a,b]\Uε(x0)

- . # # + /& ' & 0 & 0 & ( *1 +

' ' 2+"		" 3 ( ' &
	∞	1	dx
0	−∞ sin x dx&	−1	x	1 *

*& ' . # # +

! " #$ f + * ' (−∞, +∞)

# , / 0# ' + (

0 & ( (−∞, +∞) ( ' 4

*+ x R

	!		∞ ! +∞
f (x) =		1	f (t) cos y(x − t) dt dy =
f (x) =	π 0		f (t) cos y(x − t) dt dy =
	π 0		−∞	!		+∞	! +∞
				!		+∞	! +∞
			=		1		f (t) cos y(x − t) dt dy.
			=	2π		−∞	f (t) cos y(x − t) dt dy.
				2π		−∞	−∞

5 , ' 06 " #$ sin x

! +∞ ! +∞
0 = v.p.	f (t) sin y(x − t) dt dy.
−∞	−∞

7' , + 0 (						i	,
						2π
( 0 ( 1 '& ( 0 '
!			+∞ !	+∞
1				iy(x−t)
f (x) =		v.p.	−∞	−∞ f (t)e	dt dy.			8
	2π

9 ' 8

! " #$ f + * ' (−∞, +∞)

# , / 0# x ' ( f+(x)& f−(x) 0 & ( (−∞, +∞)

f ' , + ! , 9 ! (

9 ! : , & ' *1
# '( # / " ' ;8 4 8 & ' , ( ( ! #%
!			+∞	!	+∞
f (x) = v.p.	√1		−∞	√1	f (t)e−iyt dt	eixy dy. 4

		2π		2π	−∞

0 ! 4 ( + / ! . (

! ( ' 6 . ! . ( + /

! # '( # 0 " #$ f % (−∞, +∞) → C + * ' *+ ' #

[−η, η] (−∞, +∞)

" #$ f ( " '

" #$ f (

" ' /

+∞

−1

[f ](y) v.p.

√1

f (x)e

iyx

dx. <

f (y) = F

2π

−∞

5 0 & f = # '( # 0 + *

' (−∞, +∞) " #$&

√1

+∞

F [f ](y) =

f (x)e−iyx dx,

2π

!−∞+∞

F −1[f ](y) =

√1

f (x)eiyx dx.

2π

−∞

M =

f (−∞, +∞) → R

(−∞, +∞)



F −1[F [f ]] = f, F [F −1[f ]] = f.

f (x) f (−x)

" #

$ f %

$ f & (−∞, +∞) → C '

( ( ) * ! ! !

* f (x) =	g(x) + ih(x)) ( g, h&
(−∞, +∞) → R) '
g h
+ * , *	'

($ , F F −1 - $ '

* ( '

.					$
/			,
, ! (			$
f1) f2) f				, ! (
(−∞, +∞)
# (
◦
		F [λ1f1 + λ2f2] = λ1F [f1] + λ2F [f2] λ1, λ2 C;
0◦		ˆ	(−∞, +∞))
		f = F [f ]
		ˆ			y → ±∞;
1◦		f (y) → 0
		ˆ
		f ( (−∞, +∞)
1◦ 2 ,
, ( (
2	◦		03 0)		√		ˆ

						2πf (y) = a(y) − ib(y)

§
3◦ 2◦
	!		+∞

ˆ	√1		\|f (x)\| dx < ∞.

sup \|f (y)\|		2π
−∞<y<+∞			−∞

4 , $ '

(−∞, +∞) f

! f

F [f ](y) = (iy)F [f ](y), y (−∞, +∞).

! f

f (x) = f (0) +

f (t) dt.

4 $ (

" +0∞

(t) dt 5 '

−∞ f

lim f (x))

lim f (x)

% (

x→+∞

x→−∞

$ (

−+∞∞ |f (x)| dx

6 5 ! ( '

! +∞

√1

F [f ](y) =

f (x)e−ixy dx =

2π −∞

+∞

−ixy

√

f (x)e

√

f (x)e

dy = iyF [f ](y).

2π

x=−∞

2π

−∞

(−∞, +∞) f

n ! f (n)

(−∞, +∞)

F [f (n)](y) = (iy)nF [f ](y)

M |F [f ](y)| |y|n ,

y (−∞, +∞),	7
sup \|F [f (n)]\|.	3
(−∞,+∞)

(−∞, +∞) f

f1 f1(x) = xf (x)

(−∞, +∞)

dy F [f ](y) = F [−if1](y) = F [−ixf (x)](y).

! " y# $

%						!	∞
			√1			!	∞
	d	F [f ](y) =	√1		d		f (x)e−iyx dx =
		F [f ](y) =					f (x)e−iyx dx =
dy				2π dy			−∞		! ∞
							=	√1	(−ix)f (x)e−iyx dx.
							=		(−ix)f (x)e−iyx dx.
								2π	−∞

& # % ! '

(−∞, +∞) ( ) *

ϕ(x) = |xf (x)|

(−∞, +∞) f

n N fn fn(x) = xnf (x)

(−∞, +∞)

dn F [f ](y) = F [(−i)nfn](y) = F [(−ix)nf (x)](y). dyn

§ D D

+ , ,-. ) %

. *

% % #

% % ! # ! ! % % !

% ( % # * )

- % , ,-.

* / 0

% * % ( . # # * # - [−1, 1]

+ , ' ! # % 1#

* % x = 0 2 % , % # %

/	−	ε	,		ε	# !
				2
	2

ε > 0 3 ! * δε(x) .

			1		\|x\|		ε
			1				ε	,
δε(x) =			ε				2	,
δε(x) =		0		\|x\| >			ε	.
		0		\|x\| >			2	.
							2
4 # *
!	1			! ε/2
m =	δε(x) dx =			−ε/2		δε(x) dx = 1.
	−1			−ε/2
+ . ε → 0 3 ! % 5
6
δ(x) lim δε(x) =				+∞ x = 0,
δ(x) lim δε(x) =				0		x = 0.
	ε→0			0		x = 0.

(* , , *

δ(x) dx = 1.

−1

δ(x)

! " #

$% # & ' &

' ( )* +* , ,

- #

, , . # ,

/ ϕ 0 # (−∞, +∞)

$ ! +∞

lim δε(x)ϕ(x) dx = ϕ(0).

ε→0 −∞

1 2 # 3 4
!	+∞
(δ, ϕ) = ϕ(0)	δ(x)ϕ(x) dx = ϕ(0).

−∞

5 . . ,

6 # δ 0

% .

(−∞, +∞) $ .

% # * 1 δ # 3 δ 1 3 δε . .

! +∞

ϕ → δε(x)ϕ(x) dx,

−∞

, . , 7

# − 2ε , 2ε δε → δ ! #

6 % 8

9 f 4 R → R #

f = 0 , #

§ D D

f 4 R → R # #

. x R f (x) = 0 5 #

supp f

-f 4 R → R

, , , % !

# , . "

& C0∞ # .

: . C0∞

. .

-% C0∞

6 {ϕk}∞k=1

ϕk C0∞ # ϕ C0∞

;◦ [a, b]4 supp ϕk [a, b] k N

<◦ sup |ϕ(ks) − ϕ(s)| → 0 k → ∞ s N0

' C∞

% #

6 f 0 D

= f ϕ D # # (f, ϕ)

1 f D #

(f, αϕ + βψ) = α(f, ϕ) + β(f, ψ) ϕ, ψ D, α, β R.

1 f D #


k → ∞ # ϕk → ϕ	D	(f, ϕk) → (f, ϕ).
	-

D #

! "

#◦ (αf + βg, ϕ) = α(f, ϕ) + β(g, ϕ) f, g D α, β R

ϕ D$

%◦ {f				}∞		f D	k N
			k		k=1	k
D					f D k → ∞
(fk, ϕ) → (f, ϕ)						k → ∞	ϕ D.
&	D
f	→ f	D				k → ∞.
	k
' (
' ! a > 0

		e	a2		\|x\| < a,
		e	x2−a2		\|x\| < a,
ϕ(x) = 0					\|x\| a
C∞
ϕ § #)(* (		0			(
ϕ § #)(* (
+ C∞
							0
, (
	'	f "				R → R

( ! ,

[a, b] (−∞, +∞)( -,

! +∞

f (x)ϕ(x) dx ϕ D,

−∞

( ( . D (

! ϕ D

# ! ,

f (

§ D D

(

1 ,

# f

! , f ( -

! , (

(δ, ϕ) = ϕ(0) ϕ D,

, ( ' . ( 2

! +∞

(δ, ϕ) =

f (x)ϕ(x) dx ϕ D

−∞

! ,

f ( -, ϕ #

! a

f (x)e x2−a2 dx = ϕ(0) = e−1 a (0, 1).

−a

3 .

a ( ( , , ,

! a	a → 0 + 0.
\|f (x)\| dx → 0	a → 0 + 0.
−a
& δ ,
(	{fk}∞
'	{fk}∞
	k=1

! , (−∞, +∞)
δ
#◦	" +∞
	−∞ "fk(x) dx = 1 k N$
%◦	lim −+εε fk(x) dx = 1 ε > 0(
	k→∞

k fk(x) = 2 0

|x| k1 ,

|x| > k1 .

{fk}∞k=1 δ

f	→ δ D	k → ∞,
k
D
! +∞		k → ∞ ϕ D.
fk(x)ϕ(x) dx → ϕ(0)		k → ∞ ϕ D.
−∞

f	!
(−∞, +∞) " ϕ D
! +∞	! +∞
f (x)ϕ(x) dx = −		f (x)ϕ (x) dx.
−∞		−∞
# $ ! "%
	f D & %'
f
(f , ϕ) −(f, ϕ )		ϕ D,	(

! %' f

) " *

% ( !

! ! D %'

%' '

(◦

(αf + βg) = αf + βg f, g D , α, β R;

+◦

k → ∞ fk → f D fk → f D .

,1◦ , " ϕ D

((αf + βg) , ϕ) = −(αf + βg, ϕ ) = −α(f, ϕ ) − β(g, ϕ ) =

=α(f , ϕ) + β(g , ϕ) = (αf + βg , ϕ).

2◦ fk → f D k → ∞		-* "
ϕ D
(fk, ϕ) = −(fk, ϕ ) → −(f, ϕ ) = (f , ϕ)			k → ∞.
θ

0	x < 0,
θ(x) =	x 0.
1	x 0.
. θ %' " " ' '
" ϕ D -*
! +∞
(θ , ϕ) = −(θ, ϕ ) = −	ϕ (x) dx = ϕ(0) = (δ, ϕ).
0
/ θ = δ
f D n N			& %'
f (n)
(f (n), ϕ) (−1)n(f, ϕ(n)) ϕ D,			+

! n %'

- 0 n = 1

(f, ϕ(n)) + !

! ! D %'

f (x) = |x|

f D

" f (n) ! " n " " " #

$ $ % D

			∞
& f	D	k N '"
& f	D	k N '"	f
k			k
			k=1

" ( " #

" D f D

Sn fk → f D n → ∞.

k=1

& % )

∞

fk = f.

k=1

+ " $#

2◦ , "$ -"$" D "

- $ * #

∞ fk = f , k=1

$ " -" D

& f D " λ

(−∞, +∞) λf #

" " " " $

(λf, ϕ) (f, λϕ) ϕ C0∞.

& λf / $ $ #

$ D " " D

§ S S

0 " $ D D - - #

- $ - - !#

S S -

- - $ ( #

" 1 .

ϕ S F [ϕ] S, f S F [f ] S .

S #

" $ -

- (−∞, +∞) $ ϕ " -

ϕ n,m sup |xnϕ(m)(x)| < ∞ n, m N0

−∞<x<+∞

x → ±∞ ϕ S

! |x| " !

! !

#$ C0∞ S$ S C0∞ "$

ϕ(x) = e−x2 S$ C0∞ % S

& & {ϕk}∞

k=1

ϕk S ϕ S$

ϕk − ϕ n,m → 0 k → ∞ n, m N0. '

%( & ' $ !

n, m N0

xnϕ(km)(x) xnϕ(m)(x) (−∞, +∞) k → ∞.

S !

" f

#$ A > 0$ n N

|f (x)| A(1 + |x|n), x (−∞, +∞).

% #	! +∞
	! +∞
	(f, ϕ)	f (x)ϕ(x) dx ϕ S
		−∞
! #
		S

! #

&$ &

'$ & !

(

)◦ (αf + βg, ϕ) = α(f, ϕ) + β(g, ϕ) f, g S $ α, β C$

ϕ S*

◦ {f			}∞		f	k	S	k N$
		k	k=1$			k
S f S						k → ∞$
(fk, ϕ) → (f, ϕ)					k → ∞			ϕ S.
+ S

(f , ϕ) −(f, ϕ ) ϕ S.
, S $
' k → ∞ f	k	→ f S f		→ f S
	k		k

§ S S

- $ ' fk$ f S

		∞	S			∞	S
			S				S
		fk = f					fk = f .
		k=1				k=1
+ S &
# ' p = p(x)
	(pf, ϕ) (f, pϕ)					ϕ S, f S .
" . F [ϕ]
. F −1[ϕ]		ϕ S
				√1		! +∞
		F [ϕ](x) =		√1			ϕ(y)e−ixy dy,
		F [ϕ](x) =					ϕ(y)e−ixy dy,
					2π	!−∞+∞
	F −1[ϕ](x) =			√1		−∞	ϕ(y)eixy dy.

					2π
					2π
/ !
. (
)◦ ϕ S F [ϕ], F −1[ϕ] S*
◦	. '
0◦	& S S*
0◦	.							F [ϕ] #
	. F −1[ϕ] S
	$ ' k → ∞
	S			S				S
	ϕk → ϕ F [ϕk] → F [ϕ]						F −1[ϕk] → F −1[ϕ].
1
! +∞	! +∞
−∞	f (y)e−ixy dy			ϕ(x) dx =
−∞	−∞				! +∞ ! +∞
					! +∞ ! +∞
				=			ϕ(x)e−iyxdx f (y) dy
		ϕ			−∞		−∞
		ϕ	S					#

(−∞, +∞) f

f S

F [f ] F −1[f ]

(F [f ], ϕ) (f, F [ϕ]) ((F −1[f ], ϕ) (f, F −1[ϕ])) ϕ S.

! "

#◦ f S F [f ] S F −1[f ] S $

%◦ S ↔F S $

&◦ k → ∞

fk → f S F [fk] → F [f ] S , F −1[fk] → F −1[f ] S ;

'◦	F [f (n)] = (ix)nF [f ] f S $
(◦	(F [f ])(n) = F [(−ix)nf ] f S )

(xn) = nxn−1 n = 0, ±1, ±2, . . .$

(ax) = ln a · ax

a > 0$

(xa) = axa−1

x > 0$

(loga x)

loga e

x > 0 a > 0

a = 1$

(ln x)

x > 0$

(ln |x|) =

x = 0$

(sin x) = cos x$

(cos x) = − sin x$

= sec

(tg x)

cos2 x

(ctg x) = −

= − cosec2 x$

sin

(arcsin x) =				√ 1

					1 − x2				$

(arccos x) = −						√ 1

							1 − x2			$
		1

(arctg x)	=							$
			1 + x2

(arcctg x) = −1 +1 x2 $

(sh x) = ch x$


(ch x) = sh x$
(th x) =	1		$
	2	x
	ch
(cth x) = −		1			)
		2		x
	sh

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