. , , ,

,,,

(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 .

(x) , g(x) , x&Icirc;R1 [-p, p] , 2p- , . f*g(x) f*g(x) =dt ,

 

 

 

! , , , .
. , :