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‚« ¤¨ª ¢ª §áª¨© ¬ â¥¬ â¨ç¥áª¨© ¦ãà­ «
Ÿ­¢ àì{¬ àâ, 2001, ’®¬ 3, ‚ë¯ã᪠1

“„Š 517.98

‹ŽŠ€‹œŽ Žƒ€ˆ—…›… Ž‘’€‘’‚€ ‚…Š’Ž-”“Š–ˆ‰
ˆ …‹ˆ…‰›… Ž…€’Ž› ‚ ˆ•
‚. ƒ. ”¥â¨á®¢,

 

¥¤¨­®©

¬¥â®¤®«®£¨ç¥áª®©

®á­®¢¥

. . ¥§ã£«®¢ 

¨áá«¥¤ãîâáï

­¥«¨­¥©­ë¥

®¯¥à â®àë

⨯ 

áã-

¯¥à¯®§¨æ¨¨, ¨­â¥£à «ì­®£® ®¯¥à â®à  “àëá®­  ¢ ¯à®áâà ­áâ¢ å ¨§¬¥à¨¬ëå ¢¥ªâ®àä㭪樥©.

1. ¥ª®â®àë¥ ®¡®§­ ç¥­¨ï, ®¯à¥¤¥«¥­¨ï ¨ ¢á¯®¬®£ â¥«ì­ë¥
¯à¥¤«®¦¥­¨ï

ãáâì (T; ; ) | ¯à®áâà ­á⢮ á ¬¥à®©, â. ¥. T | ¬­®¦¥á⢮,  |  «£¥¡à  ¥£® ¯®¤¬­®¦¥áâ¢,  | áç¥â­®- ¤¤¨â¨¢­ ï ­¥®âà¨æ â¥«ì­ ï ¬¥à  ­ 
. ¥§ ®£à ­¨ç¥­¨ï ®¡é­®á⨠¬®¦­® ¯à¥¤¯®« £ âì, çâ® ¢á¥  â®¬ë ¤¨áªà¥â­®©
ç á⨠T ïîâáï â®çª ¬¨. ãáâì () (ᮮ⢥âá⢥­­®  ()) ¥áâì ª®«ìæ®
(ᮮ⢥âá⢥­­® -ª®«ìæ®) ¬­®¦¥á⢠¨§ , ¨¬¥îé¨å ª®­¥ç­ãî (ᮮ⢥âá⢥­­®
-ª®­¥ç­ãî) ¬¥àã. ‚áî¤ã ¢ ¤ «ì­¥©è¥¬ ¡ã¤¥¬ áç¨â âì, çâ®:
(a) ¥á«¨ A B  ¨ (B ) = 0, â® A  (¯®«­®â  ¬¥àë );
(b) ¥á«¨ ¤«ï «î¡®£® B () ¨¬¥¥¬ B A , â® A ;

(c) ¤«ï «î¡®£® A  ¨¬¥¥¬ (A) = sup (B ) : B A; B ()S;
(d) áãé¥áâ¢ãîâ ¤¨§êî­ªâ­ë¥ ¬­®¦¥á⢠ Ti â ª¨¥, çâ® (T Ti ) = 0 ¨
0 < (Ti ) < + ¯à¨ «î¡®¬ i;
(e) ¤«ï «î¡®£® A () áãé¥áâ¢ãîâ ¬­®¦¥á⢮ N ¬¥àë
­ã«ì ¨ ­¥ ¡®«¥¥,
S
祬 áç¥â­®¥, ¬­®¦¥á⢮ J ¨­¤¥ªá®¢ i â ª¨¥, çâ® A N = (A Ti ).
i 2J
Š ª ¨§¢¥áâ­® [1], ãá«®¢¨ï ( ){(¥) ¢ë¯®«­¥­ë ¤«ï «î¡®© ¯®«­®© -ª®­¥ç­®©
¬¥àë ¨ ¤«ï ¬¥àë, ¯®à®¦¤¥­­®© áãé¥á⢥­­® ¢¥àå­¨¬ ¨­â¥£à «®¬ ¬¥àë  ¤®­ 
­  «î¡®¬ «®ª «ì­® ª®¬¯ ªâ­®¬ ¯à®áâà ­á⢥. ¥§ ãé¥à¡  ¤«ï ­¥âਢ¨ «ì­®á⨠¢á¥£® ¤ «ì­¥©è¥£® ¨§«®¦¥­¨ï ¬®¦­® áç¨â âì, çâ® (T; ; ) ¥áâì ®â१®ª [0,1]
á ¬¥à®© ‹¥¡¥£   ¨«¨ ¦¥ ®£à ­¨ç¥­­ë© ª®¬¯ ªâ ¢ R n á ¬¥à®© ‹¥¡¥£  .
ãáâì (T; ; ) | ¯à®áâà ­á⢮ á ¬¥à®©, E | ª¢ §¨¡ ­ å®¢® ¨¤¥ «ì­®¥
¯à®áâà ­á⢮ á ¬¥à®© ‹¥¡¥£  , (â. ¥. E | F  -¯à®áâà ­á⢮ á ¨­¢ à¨ ­â­®©
p
-¬¥âਪ®© ¨ F -­®à¬®©
), X | ¡ ­ å®¢®
E ; ­ ¯à¨¬¥à, L ; 0 < p <


2

2

2

2

\

2

2

f



f

2

g

n

1

2

n

k
k

c 2000 ”¥â¨á®¢ ‚. ƒ., ¥§ã£«®¢  . .



g

\

1

1{43

‹®ª «ì­® ®£à ­¨ç¥­­ë¥ ¯à®áâà ­á⢠ ¢¥ªâ®à-ä㭪権

¨¤¥ «ì­®¥ ¯à®áâà ­á⢮. ‘¨¬¢®«®¬ L0 (X ) ®¡®§­ ç ¥¬ ¯à®áâà ­á⢮ (ª« áᮢ
íª¢¨¢ «¥­â­®áâ¨) ¢á¥å X -§­ ç­ëå ¨§¬¥à¨¬ëå ä㭪権 ­  T .
—¥à¥§ E (X ) ®¡®§­ ç¨¬ à¥è¥â®ç­®¥ ª¢ §¨¡ ­ å®¢® ¯à®áâà ­á⢮ ¢á¥å ¨§¬¥à¨¬ëå ¢¥ªâ®à-ä㭪権 f~ : T X â ª¨å, çâ® f~ E(X ) = f~ X E < + . Œë
®£à ­¨ç¨¢ ¥¬áï ¢ ᢮¥¬ ¨§«®¦¥­¨¨ ¢ ®á­®¢­®¬ ¤¢ã¬ï ¬®¤¥«ì­ë¬¨ ¯à¨¬¥à ¬¨
E (X ),   ¨¬¥­­®:
(1) ç¥à¥§ Lp (X ) (¨«¨ Lp (T; X ) [2]) ®¡®§­ ç ¥âáï ¯à®áâà ­á⢮ ¢á¥å ¨§¬¥à¨¬ëå ¢¥ªâ®à-ä㭪権 f~(t) â ª¨å, çâ® F -­®à¬  í«¥¬¥­â  ¢¢®¤¨âáï ä®à¬ã«®©:
!

k

k

1p
0Z
f~ p = @ f~(t) pX d(t)A < +

kk

k

k

1

1

k

k

k

1

k

T

(0 < p < );

(1)

1

(2) ç¥à¥§ L' (X ) (¨«¨ L' (T; X ), [3]) ®¡®§­ ç¨¬ ¯à®áâà ­á⢮ ¢á¥å ¨§¬¥à¨¬ëå ¢¥ªâ®à-ä㭪権 f~(t) â ª¨å, çâ® (á¬. [4])


8
9
Z
<
=

f~ ' = inf :" > 0 : '( f~(t) X =")d(t) "; ; £¤¥ ' (L):
T

k

k

k

k



2

®á«¥¤®¢ â¥«ì­®áâì f~n (t) 1
n=1 í«¥¬¥­â®¢ ¯à®áâà ­á⢠ E (X ) ­ §ë¢ ¥âáï C -¯®á«¥¤®¢ â¥«ì­®áâìî, ¥á«¨ ¤«ï ª ¦¤®© ç¨á«®¢®© ¯®á«¥1
P
¤®¢ â¥«ì­®á⨠n 1
0

(

R
),
àï¤
n f~n (t) á室¨âáï.
n
n=1
n=1
Ž¯à¥¤¥«¥­¨¥ 1.2.
à®áâà ­á⢮ ¢¥ªâ®à-ä㭪権 E (X ) ­ §ë¢ ¥âáï C -¯à®áâà ­á⢮¬, ¥á«¨
¤«ï «î¡®© C -¯®á«¥¤®¢ â¥«ì­®á⨠¥£® í«¥¬¥­â®¢
1
P
f~n (t) á室¨âáï.
f~n (t) 1
n=1 E (X ) àï¤
n=1
‹¥¬¬  1.1.
f~n(t) 1

n=1
E (X )
C
nP
o
1
A0 =
an f~n (t) : an 1
n=1
C
„®áâ â®ç­®áâì.
ãáâì ¨§¢¥áâ­®, çâ® ¬­®¦¥á⢮ í«¥¬¥­â®¢ A0 =
nP
o
1
an f~n (t) : an 1 ï¥âáï ®£à ­¨ç¥­­ë¬. ãáâì cn 1
n=1 0 | ¯à®n=1
¨§¢®«ì­ ï ç¨á«®¢ ï ¯®á«¥¤®¢ â¥«ì­®áâì ¨ " > 0. ‚ ᨫ㠮£à ­¨ç¥­­®á⨠¬­®¦¥á⢠ A0 ­ ©¤¥âáï ­®¬¥à n0 â ª®©, çâ® ¤«ï n > n0 , cn f~n (t) < " ¤«ï ¢á¥å
í«¥¬¥­â®¢ f~n (t) A0 , £¤¥
®§­ ç ¥â F -­®à¬ã ¢ ¯à®áâà ­á⢥ E (X ). Žâá

¤«ï ­®¬¥à®¢ n; m; n0 < n < m ¨¬¥¥¬:



X

X

m c
m

cif~i(t)

=

ck
i~
~

i=n+1 ck fi(t)

= ck f (t) < ";
i=n+1
Ž¯à¥¤¥«¥­¨¥ 1.1.

f

f

g

g

f

#

g

2



„«ï ⮣®, çâ®¡ë ¯®á«¥¤®¢ â¥«ì­®áâì í«¥¬¥­â®¢ f

﫠áì

g



-¯®á«¥¤®¢ â¥«ì­®áâìî, ­¥®¡å®¤¨¬® ¨ ¤®áâ â®ç­®, çâ®¡ë ¬­®-

¦¥á⢮

j

j

j 

¡ë«® ®£à ­¨ç¥­­ë¬.

j 

f

k

2

#

k

k
k

k

g

k

1{44

‚. ƒ. ”¥â¨á®¢,

£¤¥ ck = maxnim ci ¨ f~(t) =

. . ¥§ã£«®¢ 

m c
P
i ~
fi (t) 2 A0. ‡­ ç¨â, ¯®á«¥¤®¢ â¥«ì­®áâì
c
k
i=n+1

ff~n (t)g1
n=1 ï¥âáï C -¯®á«¥¤®¢ â¥«ì­®áâìî.

à¥¤¯®«®¦¨¬, çâ® ¬­®¦¥á⢮ A0 ­¥ ï¥âáï ®£à ­¨ç¥­­ë¬. ’®£¤  áãé¥áâ¢ãîâ á室ïé ïáï ç¨á«®¢ ï ¯®á«¥¤®¢ â¥«ì­®áâì fn g1
n=1 # 0,
1
®£à ­¨ç¥­­ ï ç¨á«®¢ ï ¯®á«¥¤®¢ â¥«ì­®áâì fan gn=1 ; janj  1, ¨ ¯®¤¯®á«¥¤®¢ n P
n
⥫쭮á⨠­®¬¥à®¢ fnk g, fn0k g â ª¨¥, çâ® ¯®á«¥¤®¢ â¥«ì­®áâì k
an f~n (t) :
¥®¡å®¤¨¬®áâì.

0

k

n=nk

o

k 2 N ­¥ á室¨âáï ª ­ã«î, £¤¥ nk < n0k < nk+1 .
®« £ ¥¬:




k ak ; ¥á«¨ nk < n  n0k ;
cn =
0; ¤«ï ¢á¥å ®áâ «ì­ëå ­®¬¥à®¢.

1
P

Œ®¦­® ¢¨¤¥âì, çâ® ¯®á«¥¤®¢ â¥«ì­®áâì fcn g1
cn f~n (t)
n=1 # 0, ­® àï¤
n=1
à á室¨âáï. Žâá ¢¨¤­®, çâ® ff~n (t)1
g
­¥
¡ã¤¥â
C
-¯®á«¥¤®¢ â¥«ì­®áâìî.
n=1
à®â¨¢®à¥ç¨¥. B
‹¥¬¬  1.2 (€. . Š®«¬®£®à®¢ | €. Ÿ. •¨­ç¨­ | ‚. Žà«¨ç).
ãáâì ¤ ­ 

C

-¯®á«¥¤®¢ â¥«ì­®áâì

ff~n (t)g1
n1
=1

¯à®áâà ­á⢠

P ~
ª ¦¤®¬ ¬­®¦¥á⢥ T0 ª®­¥ç­®© ¬¥àë àï¤
jfn(t)j2
n=1
’¥®à¥¬  1.1 (‹. ˜¢ àæ  [5]).

¯à¨

C

0p1

ïîâáï

‘«¥¤á⢨¥ 1.1

C

u

á室¨âáï

(á¬. [3]).

à®áâà ­á⢠ ¢¥ªâ®à-ä㭪権

'

.

’®£¤  ­ 

-¯®ç⨠¢áî¤ã.

à®áâà ­á⢠ ¢¥ªâ®à-ä㭪権

-¯à®áâà ­á⢠¬¨.

-¯à®áâà ­á⢠¬¨ ¯à¨ ãá«®¢¨¨, çâ®

ãá«®¢¨î ¯à¨ ¢á¥å

L0 (X )


-äã­ªæ¨ï ª« áá 

(L)

L' (X )

Lp (X )

ïîâáï

¯®¤ç¨­ï¥âáï


2

-

.

C „®ª § â¥«ìá⢮
á«¥¤á⢨ï 1.1 ¢ë⥪ ¥â ¨§ ⥮६ë 1.1, ¥á«¨ ¯®«®¦¨âì
+
p
'(u) = u ; u 2 R . B
Žâ¬¥â¨¬, çâ® ®¯à¥¤¥«¥­¨ï C -¯®á«¥¤®¢ â¥«ì­®á⨠¨ C -¯à®áâà ­á⢠ ¯à®-

é¥, 祬 (0)-ãá«®¢¨¥, ¢¢¥¤¥­­®¥ ‚. Œ âã襢᪮© ¨ ‚. Žà«¨ç¥¬ ¢ [6], ª®â®àë¥ ­ 
è¨à®ª®¬ ª« áᥠ¬®¤ã«ïà­ëå ¯à®áâà ­á⢠¯®ª § «¨ ­¥®¡å®¤¨¬®áâì (0)-ãá«®¢¨ï
(á¬. [6]).
Ž¯à¥¤¥«¥­¨¥ 1.3 (á¬. [7]). ãáâì ¤ ­ë ¤¢  C -¯à®áâà ­á⢠ E1 (X ) ¨
E2 (X ) ¨ ¯à®¨§¢®«ì­ë© ®¯¥à â®à W : E1(X ) ! L0(X ). Ž¯¥à â®à W ­ §ë¢ ¥âáï -¨­¢ à¨ ­â­ë¬, ¥á«¨ ¤«ï ª ¦¤®£® ¨§¬¥à¨¬®£® ¯®¤¬­®¦¥á⢠ T0  T
¢ë¯®«­ï¥âáï ãá«®¢¨¥:


W (~v) , W (~v + 1T0 ~u) = 1T0
fW (~v) , W (~v + ~u)g ~u; ~v 2 E1 (X ) ; (2)

1{45

‹®ª «ì­® ®£à ­¨ç¥­­ë¥ ¯à®áâà ­á⢠ ¢¥ªâ®à-ä㭪権

¯®ç⨠¢áî¤ã ­  T .
‡¤¥áì T0 ®¡®§­ ç ¥â å à ªâ¥à¨áâ¨ç¥áªãî äã­ªæ¨î ¨§¬¥à¨¬®£® ¯®¤¬­®¦¥á⢠ T0  T .
Žç¥¢¨¤­®, -¨­¢ à¨ ­â­ë© ®¯¥à â®à ï¥âáï H -®¯¥à â®à®¬. „¥©á⢨⥫쭮, W (~v) , W (~v + ~u) = T1
fW (~v) , W (~v + ~u)g + T2
fW (~v) , W (~v + ~u)g
¤«ï «î¡ëå ¤¨§êî­ªâ­ëå ¨§¬¥à¨¬ëå ¯®¤¬­®¦¥á⢠T1 \ T2 = ?, T1 [ T2 = T ,
¯®ç⨠¢áî¤ã ­  T (á¬. ¯®¤à®¡­¥¥ [10]).
Ž¯¥à â®à W , ¡ã¤ãç¨ -¨­¢ à¨ ­â­ë¬, 㤮¢«¥â¢®àï¥â ᮮ⭮襭¨î:
W (~v ) , W (~v + ~u) = W (~v ) , W (~v +  T1 ~u) + W (~v) , W (~v +  T2 ~u), §­ ç¨â,
kW (~v) , W (~v + ~u)k2  kW (~v) , W (~v +  T1 ~u)k2 + kW (~v) , W (~v +  T2 ~u)k2 ;
â. ¥. ®¯¥à â®à W ï¥âáï H -®¯¥à â®à®¬ (¯à¨ H  I ).
„«ï  = 1 -¨­¢ à¨ ­â­ë© ®¯¥à â®à ­ §®¢¥¬ ¨­¢ à¨ ­â­ë¬ ®¯¥à â®à®¬. “á«®¢¨¥ (2) ¨­¢ à¨ ­â­®á⨠( = 1) ®¯¥à â®à  W ¨¬¥¥â
¢¨¤:
1

1

1

1

1

1

1

à¨¬¥ç ­¨¥ 1.1.

W (~v) , W (~v + 1T0 ~u) = 1T0
fW (~v ) , W (~v + ~u)g





~u; ~v 2 E1 (X ); T0  T :

(3)

Žâ®¡à ¦¥­¨¥  : E (X ) ! R + ­ §®¢¥¬  ¤¤¨â¨¢­®©
ä®à¬®©, ®¡ãá«®¢«¥­­®© F -­®à¬®© k
k ­  E (X ), ¥á«¨:
(a) ¤«ï «î¡ëå ~u; ~v 2 E (X ), supp ~u(t) \ supp ~v(t) = ?, ¢ë¯®«­ï¥âáï ãá«®¢¨¥
(~u + ~v ) = (~u) + (~v);
(b) (~xn) # 0 , k~xn k # 0 ¨ n ! 1;
(c) (~xn)   , k~xn k  k ; n 2 N ¨ (; k) 2 R 2 .
à¨¬¥à ¬¨  ¤¤¨â¨¢­ëå ä®à¬ ¤«ï ª®­ªà¥â­ëå ¯à®áâà ­á⢠¢¥ªâ®àä㭪権 f~(t) ¡ã¤ãâ ïâìáï ¨­â¥£à «ì­ë¥ ¬®¤ã«ïàë ¢¨¤ :
Ž¯à¥¤¥«¥­¨¥ 1.4.

(a) (f~) =
(b) (f~) =

Z
T

Z

T

kf~(t)kpX d(t); f~ 2 Lp (X );




' kf~(t)kX d(t); f~ 2 L' (X );

(4)
(5)

¥á«¨ '-äã­ªæ¨ï '(u) 㤮¢«¥â¢®àï¥â
2-ãá«®¢¨î.
F  -¯à®áâà ­á⢮ ¢¥ªâ®à-ä㭪権 E (X ) ­ §ë¢ ¥âáï
¯à®áâà ­á⢮¬ ⨯  L(X ), ¥á«¨:
(1) T 2 E (X );
(2) E (X ) ®¡« ¤ ¥â  ¡á®«îâ­® ­¥¯à¥à뢭®© F -­®à¬®©;
Ž¯à¥¤¥«¥­¨¥ 1.5.
1

1{46
‚. ƒ. ”¥â¨á®¢, . . ¥§ã£«®¢ 
(3) ¢ E (X ) áãé¥áâ¢ã¥â  ¤¤¨â¨¢­ ï ä®à¬  .
à¨¬¥ç ­¨¥ 1.2. à®áâà ­á⢮ Lp (X ) (0 < p <
) ï¥âáï ¯à®áâà ­á⢮¬ ⨯  E (X );  ­ «®£¨ç­® ¤«ï L' (X ), ¥á«¨ ' 㤮¢«¥â¢®àï¥â
2 -ãá«®¢¨î.
Š ª ¨§¢¥áâ­®, áà ¢­¥­¨¥ ᢮©á⢠¢¥ªâ®à-ä㭪権 ¨ ä㭪権 ®â ¤¢ãå ¯¥à¥¬¥­­ëå, á¢ï§ ­­ëå ¬¥¦¤ã ᮡ®© ä®à¬ã«®© (s; t) = [f~(t)](s), 㤮¡­® ¯à®¢®¤¨âì
¢ à ¬ª å ⥮ਨ ¯à®áâà ­á⢠ᮠᬥ蠭­®© ª¢ §¨­®à¬®© [8], â ª ª ª ¯®á«¥¤­¨¥
¯®§¢®«ïîâ ®¯¨á뢠âì ¯à¨­ ¤«¥¦­®áâì ¨­â¥£à «ì­ëå ®¯¥à â®à®¢ ­¥ª®â®àë¬
¢ ¦­ë¬ ª« áá ¬ ç¥à¥§ ᢮©á⢠ ¨å 拉à. ãáâì (T1 ; 1; 1) ¨ (T; ; ) | ¤¢ 
¯à®áâà ­á⢠ á ¬¥à ¬¨ 1 ¨  ᮮ⢥âá⢥­­®.
„«ï ¤ ­­ëå X | ˆ (= ¡ ­ å®¢  ¨¤¥ «ì­®£® ¯à®áâà ­á⢠) ­ 
(T1 ; 1; 1), E | Šˆ (= ª¢ §¨¡ ­ å®¢  ¨¤¥ «ì­®£® ¯à®áâà ­á⢠) ­  (T; ; )
ç¥à¥§ E [X ] ®¡®§­ ç¨¬ ¯à®áâà ­á⢮ ¢á¥å ¨§¬¥à¨¬ëå ä㭪権 (s; t) ­  T1 T ,
㤮¢«¥â¢®àïîé¨å ¤¢ã¬ ãá«®¢¨ï¬:
(1) ¯à¨ ¢á¥å t T äã­ªæ¨ï s (s; t) ¢å®¤¨â ¢ X ;
(2) äã­ªæ¨ï  = ( ; t) X ¢å®¤¨â ¢ E .
ˆ§¢¥áâ­®¥ ãá«®¢¨¥ (C ) (á¬. [8]) ¢ ¯à®áâà ­á⢥ X ®¡¥á¯¥ç¨¢ ¥â ¨§¬¥à¨¬®áâì ä㭪樨  . ‡­ ç¨â, E [X ] | «¨­¥©­®¥ ¬­®¦¥á⢮,  , á«¥¤®¢ â¥«ì­®,
¨ ¨¤¥ «ì­®¥ ª¢ §¨­®à¬¨à®¢ ­­®¥ ¯à®áâà ­á⢮ ­  ¯à®¨§¢¥¤¥­¨¨ T1 T ,   â ª
ª ª E | Šˆ, â® ä®à¬ã«   E[X ] =  E ¯à¥¢à é ¥â E [X ] ¢ Šˆ‘Š (á¬.
â ª¦¥ [3], £¤¥ ¤«ï ¡®«¥¥ ®¡é¨å á¨âã æ¨© ¨¬¥îâáï ¬®¤¥«ì­ë¥ ¯à¨¬¥àë Šˆ‘Š
L() , L( ) , Žà«¨ç  L(') ).
Žâ¢¥â ­  ¢®¯à®á, ª®£¤  ¨¬¥¥â ¬¥á⮠⮯®«®£¨ç¥áª®¥ ᮢ¯ ¤¥­¨¥ ¯à®áâà ­á⢠ ¢¥ªâ®à-ä㭪権 E (X ) á ¯à®áâà ­á⢮¬ ᮠᬥ蠭­®© ª¢ §¨­®à¬®©
E [X ], ¤ ¥â á«¥¤ãîé ï «¥¬¬ :
1



2

j

j

j

7!

k




k

j



k

k

kj

jk

‹¥¬¬  2.2. ‘«¥¤ãî騥 ãá«®¢¨ï íª¢¨¢ «¥­â­ë:

~
E (X ) = E [X ] ¯à¨ ª ­®­¨ç¥áª®¬ ¢«®¦¥­¨¨
 (s; t) = [f (t)](s); 
(2) X | ˆ á ãá«®¢¨¥¬ (€) : (xn # 0) ) kxn kX ! 0 ¯à¨ n ! 1 .
(1)

C „®ª § â¥«ìá⢮ «¥¬¬ë 2.2 ¯à®¢®¤¨âáï  ­ «®£¨ç­® ¤®ª § â¥«ìáâ¢ã
á«¥¤á⢨ï 2.3 à ¡®âë [9]. B
2. ¥ª®â®àë¥ á¢®©á⢠ ­¥«¨­¥©­ëå ®¯¥à â®à®¢
¢ ª¢ §¨­®à¬¨à®¢ ­­ëå ¯à®áâà ­á⢠å
¨§¬¥à¨¬ëå ¢¥ªâ®à-ä㭪権

E (X )

–¥«ì í⮣® ¯ à £à ä  | ­  ¥¤¨­®© ¬¥â®¤®«®£¨ç¥áª®© ®á­®¢¥ ¨áá«¥¤®¢ âì
­¥«¨­¥©­ë¥ ®¯¥à â®àë ⨯ : á㯥௮§¨æ¨¨, ¨­â¥£à «ì­®£® ®¯¥à â®à  “àëá®­ 
¢ ¯à®áâà ­á⢠å E (X ) ¨ ¤à.
’¥®à¥¬  2.1. ãáâì E1 (X ) | F -¯à®áâà ­á⢮, E2 (X ) | ¯à®áâà ­á⢮
⨯  L(X ),  |  ¤¤¨â¨¢­ ï ä®à¬ , ®¡ãá«®¢«¥­­ ï F -­®à¬®©
2 ,   ®¯¥à â®à
W : E1 (X ) E2 (X ) ¯®¤ç¨­ï¥âáï ãá«®¢¨ï¬:
k
k

!

1{47

‹®ª «ì­® ®£à ­¨ç¥­­ë¥ ¯à®áâà ­á⢠ ¢¥ªâ®à-ä㭪権

(a) W (~u) 0
(b) W (1T1 ~u1 + 1T2 ~u2 )
~u1 ; ~u2 E1 (X ); ~u1 ; ~u2 0


T , ¥á«¨ ~u 2 E1 (X ); ~u  0 ¯®ç⨠¢áî¤ã;
 1T1
W (~
u1 ) + 1T2
W (~u2 ) ¯®ç⨠¢áî¤ã ­  T , ¥á«¨
, ¯®ç⨠¢áî¤ã ­  T ¨ T1 ; T2 | ¨§¬¥à¨¬ë¥, T1 \ T2 = ?,

¯®ç⨠¢áî¤ã ­ 

2



T1  T; T2  T .

’®£¤  ¤«ï «î¡®£®  ¡á®«îâ­® ®£à ­¨ç¥­­®£® ¢

T (C ) |  ¡á®«îâ­® ®£à ­¨ç¥­ ¢ E2 (X ).

E1 (X )

¬­®¦¥á⢠ C ®¡à §

C Žâ ¯à®â¨¢­®£®. „®¯ãá⨬, çâ® áãé¥áâ¢ã¥â ¬­®¦¥á⢮ C ,  ¡á®«îâ­®
®£à ­¨ç¥­­®¥ ¢ E1 (X ) â ª®¥, çâ® ®¡à § T (C ) ¢ ¯à®áâà ­á⢥ E2 (X ) ­¥ ¡ã¤¥â
 ¡á®«îâ­® ®£à ­¨ç¥­­ë¬. â® §­ ç¨â, çâ® áãé¥áâ¢ãîâ "0 > 0, ¯®á«¥¤®¢ â¥«ì­®á⨠~uk (t) k=1 C ¨ Tk ; Tk T ( k) â ª¨¥, çâ® (Tk ) 0 ¯à¨ k
,
P
(Tk ) < ¨ 1Tk W (~uk ) 2 > "0 .
k=1
 áá㦤 ï ¯®  ­ «®£¨¨, ª ª ¢ [10], ¬®¦­® ã⢥ত âì, çâ® áãé¥áâ¢ã¥â
¯®á«¥¤®¢ â¥«ì­®áâì ¯®¤¬­®¦¥á⢠Tk k=1 T â ª ï, çâ® Ti Tj = ?; i = j ¨
f~ (t) n=1 E1 (X ), f~n (t) 0 ¯®ç⨠¢áî¤ã, n N, 㤮¢«¥â¢®àïî騥 ãá«®¢¨ï¬
Pn
n=1 1Tn f~n (t) 1 < + , ­® 1Tn W (f~n ) 2 > "0 > 0. ’®£¤  ­ ©¤¥âáï "0 > 0
â ª®¥, çâ® (1Tn W (f~n)) > "0 ; n N. Ž¡®§­ ç¨¬
( ~
fn (t); ¥á«¨ t Tn ;


~v (s) =
0; ¥á«¨ t T Sn=1 Tn = T0 :
Žç¥¢¨¤­®, çâ® ~v E1(X ) ¨, §­ ç¨â, ¯® ãá«®¢¨î W (~v) E2(X ). ‘ ¤à㣮©
áâ®à®­ë, ¨¬¥¥¬:
1

f

g



f

g 2



8

!

! 1

1

1

k




k

1

f

1

f

g

1



k

g





k

1

8

k

\

2

0

k

8

6

2

2

2

n

1

2

2

W (~v ) = W

1
X

n=1

~n
1Tn f


P

~
1
W
(
f
)
kW (~
v)k2 

1
n
n=1 Tn

®âªã¤ 
’ ª ª ª

2



â®

1
X

~n )
1Tn W (f



=



1
X



n=1

T W (f~n );

1 n

.
1
X

(1Tn W (f~n )) = 1;

n=1
n=1
n=1 1Tn W (f~n ) 2= E2 (X ); á«¥¤®¢ â¥«ì­®, W (~v) 2= E2 (X ).

P1

à®â¨¢®à¥ç¨¥. B

‘«¥¤á⢨¥ 2.1. ‚ ãá«®¢¨ïå ⥮६ë 2.1, ¥á«¨ ®¯¥à â®à

W

­¥¯à¥à뢥­ ¯®

¬¥à¥ ¢ â®çª¥ ~
u0 2 E1 (X ), â® ®¯¥à â®à W ­¥¯à¥à뢥­ ¯® F -­®à¬¥ ¯à®áâà ­á⢠
E1 (X ) ¢ â®çª¥ ~u0 .
à¨¬¥ç ­¨¥ 2.1. …᫨ W ¥áâì ¨­¢ à¨ ­â­ë© ®¯¥à â®à, ¯®¤ç¨­ïî騩áï ãá«®¢¨î ( ) ⥮६ë 2.1, ⮣¤  ®¯¥à â®à W ­¥¯à¥à뢥­ ¢ ª ¦¤®© â®çª¥
~u0 2 E1 (X ), £¤¥ W ­¥¯à¥à뢥­ ¯® ¬¥à¥ ¨ W (E1 (X ) ) = 1E2 (X ) , ( | ­®«ì

¯à®áâà ­á⢠).

1{48
Ei

‚. ƒ. ”¥â¨á®¢, . . ¥§ã£«®¢ 

‹¥¬¬  2.1.

ãáâì 1T





í«¥¬¥­â®¢, ¨¬¥îé¨å  ¡á®«îâ­® ­¥¯à¥à뢭ãî

®¯¥à â®à ¨§



2 E 1 (X ) ¨ 1T 2 E 2 (X ) (E i | ¯®¤¯à®áâà ­á⢠ ¢

E1 (X ) ¢ E2 (X ).

F -­®à¬ã), W

¯à®¨§¢®«ì­ë©

‘«¥¤ãî騥 ¯à¥¤«®¦¥­¨ï íª¢¨¢ «¥­â­ë:

~u0 2 E1 (X );
(2) lim ft; jW (~
u0 + ~z) , W (~u0 )j > g = 0, £¤¥  > 0, ~z | 䨪á¨à®¢ ­ë.
~u!
(1)

W

­¥¯à¥à뢥­ ¯® ¬¥à¥ ¢

’¥®à¥¬  2.2.

⨯ 

L(X ),

ãáâì

¯à¨ç¥¬ 1T

2

E1 (X ) | F -¯à®áâà ­á⢮, E2 (X ) | ¯à®áâà ­á⢮

E 1 (X ) ¨ 1T 2 E2 (X ),   W : E1 (X ) ! E2 (X ) |

¯à®¨§¢®«ì­ë© ®¯¥à â®à, ¯®¤ç¨­ïî騩áï ãá«®¢¨ï¬ ( ) ¨ (b) ⥮६ë 2.1. ’®£¤ 
á«¥¤ãî騥 ã⢥ত¥­¨ï íª¢¨¢ «¥­â­ë:

~u0 2 E1 (X );
~u0 2 E1 (X ).
1
C 2) ) 1). ãáâì f~un gn=1 ! ~u0 ¯® F -­®à¬¥ ¯à®áâà ­á⢠ E1 (X ), ⮣¤ 
f~un g1
n=1 ! ~u0 ¯® ¬¥à¥ ­  T ¯à¨ n ! 1. Ž¯¥à â®à W , ¡ã¤ãç¨ ­¥¯à¥à뢭ë¬
¯® ¬¥à¥ ¢ â®çª¥ ~u0 , ¤ ¥â W (~un ) ! W (~u0). € â ª ª ª ¯®á«¥¤®¢ â¥«ì­®áâì
f~un g1
n=1 ! ~u0  ¡á®«îâ­® ®£à ­¨ç¥­ , â® fW (~un )g  ¡á®«îâ­® ®£à ­¨ç¥­­®¥
¬­®¦¥á⢮ ¢ E2(X ) (ᮣ« á­® ⥮६¥ 2.1). ‡­ ç¨â, fW (~un )g1
n=1 ! W (~u0 ) ¯®
F -­®à¬¥ ¯à¨ n ! 1.
(1)

(2)

W
W

­¥¯à¥à뢥­ ¢ â®çª¥

­¥¯à¥à뢥­ ¯® ¬¥à¥ ¢ â®çª¥

(1) ) (2) „®ª § â¥«ìá⢮ ¯à¥¤®áâ ¢«ï¥¬ ç¨â â¥«î. B
à¨¬¥ç ­¨¥ 2.2. ‘ãé¥áâ¢ãîâ F -¯à®áâà ­á⢠ E1 (X ) ¨ E2 (X ), 1T 2

E 2 (X ), £¤¥ W : E1 (X ) ! E2 (X ) ¨­¢ à¨ ­â­ë© ®¯¥à â®à W (E1 (X ) ) = E2 (X ) ,
¤«ï ~a 2 E2(X ) ¨ ®â®¡à ¦¥­¨¥ : E1(X ) ! E2(X ) â ª¨¥, çâ®:
(1) jW (~u)(t)j  ~a(t) + (~u)(t) ¯®ç⨠¢áî¤ã ­  T , 8~u 2 E1 (X );
(2) ¤«ï «î¡®£® F -®£à ­¨ç¥­­®£® ¬­®¦¥á⢠ B 2 E1 (X ), (B ) ®£à ­¨ç¥­®
¢ E2 (X ).
à¨¬¥à ¬¨ ¬®£ãâ á«ã¦¨âì ¨§¢¥áâ­ë© ®¯¥à â®à á㯥௮§¨æ¨¨ W (~u)(t) =
N [t; ~u(t)] ¨ E1 (X ) = Lp1 (X ), E2 (X ) = Lp2 (X ) ¯à¨ ᮮ⢥âáâ¢ãîé¨å ®£à ­¨ç¥­¨ïå ­  p1 ; p2 > 0.


E1 (X ) ¨ E2 (X ) | ¤¢  F -¯à®áâà ­á⢠ ­  T , 1T 2 E 2 ,

W : E1 (X ) ! E2 (X ). ãáâì ¤«ï ~u0 2 E 1 (X ), ~u0 > 0 ­  T , áãé¥áâ¢ãîâ
+
+
'(u) =
¢®§à áâ îé ï äã­ªæ¨ï ' : R ! R , 㤮¢«¥â¢®àïîé ï ãá«®¢¨î lim
u!1 u
1, äã­ªæ¨ï ~a 2 E2(X ) ¨ ®â®¡à ¦¥­¨¥ : E1(X ) ! E2(X ), ¯¥à¥¢®¤ï饥 ¢á类¥
®£à ­¨ç¥­­®¥ ¯® F -­®à¬¥ ¬­®¦¥á⢮ B  E1 (X ) ¢ ®£à ­¨ç¥­­®¥ ¬­®¦¥á⢮
(B )  E2(X ), â ª¨¥, çâ®
 jW (f~)(t)j 
'
~u0 (t)  ~a(t) + (f~(t))
~u0 (t)
’¥®à¥¬  2.3.

ãáâì

1{49

‹®ª «ì­® ®£à ­¨ç¥­­ë¥ ¯à®áâà ­á⢠ ¢¥ªâ®à-ä㭪権
¯®ç⨠¢áî¤ã ­ 

T

~ ; r)) (è à 
W (B (
¦¥á⢮ ¢ E2 (X ).
0

¤«ï ¢á¥å

f~

á 業â஬ ¢

C ® ãá«®¢¨î, ~a

2

E1 (X ).

’®£¤  ¤«ï ª ¦¤®£®

r >

0,

®¡à §

 à ¤¨ãá  r) |  ¡á®«îâ­® ®£à ­¨ç¥­­®¥ ¬­®-

⮣¤ 
 ¤«ï " > 0, áãé¥áâ¢ã¥â 1 > 0 â ª®¥,
~ ; r) ®£à ­¨ç¥­­®¥, ⮣¤  áãé¥áâ¢ã¥â
çâ® 1~a(t) 2 < 2" . ® ãá«®¢¨î, B (
2 > 0 â ª®¥, çâ® 2 (f~) 2 < 2" ¤«ï «î¡®£® í«¥¬¥­â  f~ B (; r). ®«®¦¨¬
 = inf 1 ; 2 . Žâá:
k

2

E2 (X ),

8

0

k

k

f

k

2

0

g






' jW (f~)(t)j=~u (t)
~u (t)

0
0

2

~a(t)k2 + k (f~(t))k2 < ":

 k

~ ; r)) ¥áâì  ¡á®«îâ­® ®£à ­¨ç¥­­®¥ ¬­®â® ®§­ ç ¥â, çâ® ®¡à § W (B (
¦¥á⢮ ¢ E2(X ) (á¬. â ª¦¥ ⥮६ã 1.4.13 ¨§ [10]). B
0

1.
2.
3.
4.
5.
6.
7.
8.

‹¨â¥à âãà 

ˆ­â¥£à «ì­ë¥ ®¯¥à â®àë.|®¢®á¨¡¨àáª:  ãª , 1983.
Isomorphism between Lp-function spaces when p < 1 // J. Func.
Anal.|1981.|V. 42.|P. 299{337.
”¥â¨á®¢ ‚. ƒ. Ž¡ ®¯¥à â®à å ¢ ¨¤¥ «ì­ëå ª¢ §¨­®à¬¨à®¢ ­­ëå ¯à®áâà ­á⢠å ᮠᬥ蠭­®© ª¢ §¨­®à¬®© E (
) // ‘¥¢¥à®-Žá¥â¨­. £®áã­¨¢¥àá¨â¥â.|
„¥¯®­¨à. ¢ ‚ˆˆ’ˆ 22.05.90, ü 2784{‚90.
Rolewicz S. Metric linear spaces.|Warszawa: PWN, 1972.
Schwartz L. Un theoreme de la convergence dans les Lp , 0
p
// C. R.
Acad. Sci. Paris. Ser. A.|1969.|T. 268.|P. 704{706.
Matuszewska W., Orlicz W. A note on modular spaces IX // Bull. Acad.
Polon. Sci.|1968.|V. 16.|P. 801{807.
”¥â¨á®¢ ‚. ƒ. Ž ᢮©áâ¢ å ­¥«¨­¥©­ëå  | ¨­¢ à¨ ­â­ëå ®¯¥à â®à®¢ ¢
«®ª «ì­® ®£à ­¨ç¥­­ëå ¯à®áâà ­á⢠å // ƒà®§­¥­áª¨© £®áã­¨¢¥àá¨â¥â.|
’. 24.|ƒà®§­ë©.|1992.
Š®à®âª®¢ ‚. .

Kalton N. J.



 1

ã墠«®¢ €. ‚., Š®à®âª®¢ ‚. ., Šãáà ¥¢ €. ƒ., Šãâ â¥« ¤§¥ ‘. ‘., Œ ª -

‚¥ªâ®à­ë¥ à¥è¥âª¨ ¨ ¨­â¥£à «ì­ë¥ ®¯¥à â®àë.|®¢®á¨¡¨àáª:
 ãª , 1992.
9. ã墠«®¢ €. ‚. Ž ¯à®áâà ­á⢠å ᮠᬥ蠭­®© ­®à¬®© // ‚¥áâ­¨ª ‹ƒ“.|
1973.|ü 19.|‘. 5{12.
10. ”¥â¨á®¢ ‚. ƒ. Ž¯¥à â®àë ¨ ãà ¢­¥­¨ï ¢ F -ª¢ §¨­®à¬¨à®¢ ­­ëå ¯à®áâà ­á⢠å // „¨áá. ­  ᮨ᪠­¨¥ ãç. á⥯. ¤®ªâ. 䨧.-¬ â. ­ ãª, ˆ­-â
¬ â-ª¨. ‘Ž €, 1996.|280 á.
஢ . Œ.

£. ®á⮢

‘â âìï ¯®áâ㯨«  22 ¤¥ª ¡àï 2000 £.