26.11.96 , " " .1 P1 P1 P1 q P1 ? P1 ZZZ

26.11.96 ,

" "
1
fen gn=1 .R V
.1
1
n = 1; 2; ::: hf; en i = 2n f 2 V .V
['
25]
kf k ()
.kf + e10 k ()
.
-------------------------------------------:
kf k =
2
P1
n=1 jhf; en ij =
2
14
P1 1
1 2
=
=
= 31 :
(
)
n
n
n=1 4
n=1 2
1? 1
4
kf k = p13
, ()1
P1
.
(. )
()1
kf + e10 k = hf + e10 ; f + e10i = hf; f i + hf; e10 i + he10 ; f i + he10 ; e10 i:
2
kf k + 2hf; e10 i + 1 = + 2 2

kf + e10k =
2
''
1
2
1
3
10
R
+ 1 = + 2?9:
4
3

q
kf k =
.
P1
n=1 n

10 = (1 +
( .
.
10 )
1
2
2

n 6= 10

4
3
V

+ 2?9

n =

)
1 2
2n
:

kf k 6=
P1
n=1 jhf; en ij
. .1

)
:

.2
kf k 6= jhf; en ij
. .3
. , ,
-------------------------------------------. .
1
X
[' 25]
.2
1 x9 ex3 e?inxdx ein=2 ()
n=?1 2 ?
1 1 Z
X
9 x3 ?inx
in ()
x
e
e
dx
e
n=?1 2 ?
Z
--------------------------------------------

f
f (x) = x
.
f (x) 1
X
n=?1
( )
f
cneinx =
9
ex3
''
f : [?; ] ! C
:
()2
1 X
1 Z t9 et3 e?inxdt einx
n=?1 2 ?

x = 2
.

(P 1)
)3

(P 1)
()
.
f( ) =
.
2
? 9
2
e( 2
f0

(P 1)

f ((?)+0)+f (?0) = f (?)+f () = 9 (e3 ?e?3 )
2
2
2

()2

:
:

x ! ?x

.
x

fi( 2 )
h
R
P
9 t3 ?inx dt einx
1
x9 ex3 = 1
n=?1 2 ? t e e

f (? )
2

'
:
x 2 (?; )
. .

--------------------------------------------
.3
a0 + P1 (an cos nx + bn sin nx) ()
n=1
2
[' 35]

x 2 [0; ] u(x) = x x 2 [?; 0) u(x) = 0 " u : [?; ] ! C
[?; ] F f ()
P1
P
A
0
F (x) 2 + n=1(An cos nx + Bn sin nx) f (x) a20 + 1
n=1 (an cos nx + bn sin nx)
.x 2 [?; ] F (x) = f (?x) .
.n bn an n Bn An
.
, ,() () ()
x 2 [?; ] w(x) = x v(x) = jxj " w v
.
--------------------------------------------
R
R
xdx = x22 j0 = 2
a0 = ? u(x)dx = 0
n2N
?
R
R
R sin nx 1
1 x sin nx 1
=
x
cos
nxdx
=
j
an =? ? u(x) cos nxdx
0 ? 0
0
n
n dx
1
cos nx 1
n
= 0 ?R 0 + n2 j0 = n2 ((R?1) ? 1) :
?
R
bn =1 ? u(x) sin nxdx = 1 0 x sin nxdx = 1 ? x cosn nx j0 + 0 cosnnx dx
n+1
n
n
= 1 ? (?n1) + 0 + sinn2nx j0 = ? (?n1) = (?1)n :
1
.
1
,

() :
1
.
)
.

()
n=0
n2N
An = ? F (x) cos nxdx = ? f (?x) cos nxdx:
tR= ?x
R ?
R ?
1
1
1
An = f (t) cos(?nt) (?1)dt = ? f (t) cos ntdt = ? f (t) cos ntdt = an:
1

R
1
R
,

R
R
R ?
,

= ? f (?x) sin nxdx = f (t) sin(?nt) (?1)dt =
BnR = ? F (x) sin nxdx
R
?
1
1
f (t) sin ntdt = ? ? f (t) sin ntdt = ?bn .
1
1

x
1

?x
-----------------------------
' )
1
X
a
0
f (x) 2 + (an cos nx + bn sin nx)
n=1
(P 2)

1
X
a
0
f (?x) 2 + (an cos n(?x) + bn sin n(?x))
(P 3)
n=1
"

(
.

[?; ] x
.
f
F (x) = f (?x) F (x) A20 + P1
n=1 (An cos nx + Bn sin nx)
Bn = ?bn An = an
"

)

bn an

''
.

(P 2)

()
Z Z 1
1
an = f (x) cos nxdx bn = f (x) sin nxdx:

?
(P 2) x
.

?x
(P 4)
?
(P 3)

???f (?x) a20 +

(P 4)

1
X
(an cos nx ? bn sin nx) ???
(P 5)
n=1

( ) .
[?; ]
. .
.
F

(P 5)
.

Bn = ?bn: An = an

''
,
F f
.

-----------------------------
()
w(x) = u(x) ? u(?x) v(x) = u(x) + u(?x)
Bn = ?bn An = an u
b n an
x 2 [?; ]
u(?x)
P1
(a0 +A0 )
v(x) 2 + n=1((an + An) cos nx + (bn + Bn) sin nx)

,
.
( )

a0 + P1 2an cos nx = + P1 2((?1)2n?1) cos nx
n=1
n=1 n
2
2

a0?A0 ) + P1 ((a ? A ) cos nx + (b ? B ) sin nx)
n
n
n
n=1 n
2

v(x) 2
w(x) (
.

P1
w(x) n=1 2bn sin nx =
.
P1

?
n=1 2 n sin nx
( 1)n
---------------------

[?; ]

x
) .
x 2 [?; 0)
(!!!!!!!!

2 [0; ] x

---------------------
-------------------------------------------[' 15]
?
.4
[?; ] (3 ) v(x) = jxj ()
.
?
[?; ] (3 ) w(x) = x ()
.
--------------------------------------------

Dirichlet

.
:

'4
. .
:
f : [?; ] ! C
:
[?; ] f (1)
f (?) = f () (2)
0
x 2 [?; ] f (x) (3)
f 0 (x?) := limh!0;h>0 f 0 (x ? h) f 0 (x+) := limh!0;h>0 f 0 (x + h) ,f 0 2 E (4)
x 2 [?; ] f 0 (x+) = f 0 (x?) (f - ) x 2 [?; ]
.
[?; ] f
.
(4)

.(3)

(2)
,(1)

f 0 (x+) = f 0 (x?) 0 6= x 2 [?; ]
,

v
.

.
''
f
f (x) = v(x) = jxj
f 0 (0+) = 1 f 0 (0?) = ?1
x
?1
x
1

f
.

)
''

()

(2)
'
v
''
w

)

()4

(

:""
. (2) , " ,
: ." '
w

W (x) = w(x) = W (x) x 2 R , ,
, w W W (? ) = W ( ) = 0 x 2 (?; ) x
.x 2 R W (x + 2 ) = w(x)
, [?; ] " w
. W " ,W
" .
W
(." ) .
."

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26.11.96 , " " .1 P1 P1 P1 q P1 ? P1 ZZZ

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