,,,
(x) , g(x) , xÎR1 [-p, p] , 2p- , . f*g(x)
f*g(x) =dt
, [-p,p]
cn ( f*g ) = cn ( f )× cn ( g ) , n = 0, 1 , 2 , ... ( 1 )
{ cn ( f )} -- f ( x ) :
cn = -i n tdt , n = 0, 1, 2,¼
Î L1 (-p, p ) . 0 £ r < 1
r ( x ) = n ( f ) r| n | ei n x , x Î [ -p, p ] , ( 2 )
(2) r , 0 £ r < 1 . r ()
cn ( fr ) = cn × r| n | , n = 0 , 1, 2, ¼ , (1) , r ( x ) :
r ( x ) = , ( 3 )
, t Î [ -p, p ] . ( 4 )
r (t) , 0 £ r <1 , t Î [ -p, p ] , , (3) -- .
,
Pr ( t ) = , 0 £ r < 1 , t Î [ -p, p] . ( 5 )
Î L1 ( -p, p ) - , , ,
c-n ( f ) = `cn( f ) , n = 0, 1, 2,¼, (2) :
fr ( x ) =
= , ( 6 )
F ( z ) = c0 ( f ) + 2 ( z = reix ) ( 7 )
- . (6) , Î L1( -p, p ) (3)
u ( z ) = r (eix ) , z = reix , 0 £ r <1 , x Î [ -p, p ] .
u (z) v (z) c v (0) = 0
v (z) = Im F (z) = . ( 8 )
1.
u (z) - ( ) | z | < 1+e ( e>0 ) (x) = u (eix) , xÎ[ -p, p ] .
u (z) = ( z = reix , | z | < 1 ) ( 10 ).
Pr (t) - , (10) , u (z) - :
=, | z | < 1+ e .
(10) (2) (3).
r (x) r1 , :
) ;
) ;
) d>0
) ) (5), ) (2) (3) () º 1.
1.
() ( -p, p ) , 1 £ p < ¥ ,
;
(x) [ -p, p ] (-p) = (p) ,
.
.
(3) )
( 12 )
, ,
.
,
.
e > 0 d = d (e) , . r , ,
.
.
1 .
" " " ", .
1.
(-, ), > 0 .
I , .
2.
(,) , y > 0
.
2 ().
- .
.. .
.
,
, ( 13 )
- , M ( f, x ) - f (x) [*]. (5)
( - ).
- ,
.
.
(13) . (1,1) , ,
,
( 14 )
.. .
(13) xÎ (-2p,2p)
, 1 xÎ [-p, p] (14)
n¥.
2 .
.
(13) (59), , , .. xÎ [-p, p] , reit eix .
[*]
, f (x)
[-2p,2p] (..
f (x) = f (y) , x,y Î [-2p,2p] x-y=2p) 蠠 f (x) = 0 , 蠠 |x| > 2p .
Copyright (c) 2025 Stud-Baza.ru , , , .