Directory UMM :Journals:Journal_of_mathematics:VMJ:

“„Š 517.98

‹ŽŠ€‹œŽ Žƒ€ˆ—…›… Ž‘’€‘’‚€ ‚…Š’Ž-”“Š–ˆ‰

ˆ …‹ˆ…‰›… Ž…€’Ž› ‚ˆ•

‚. ƒ. ”¥â¨á®¢, . . ¥§ã£«®¢ 

  ¥¤¨­®© ¬¥â®¤®«®£¨ç¥áª®© ®á­®¢¥ ¨áá«¥¤ãîâáï ­¥«¨­¥©­ë¥ ®¯¥à â®àë ⨯ 

áã-¯¥à¯®§¨æ¨¨, ¨­â¥£à «ì­®£® ®¯¥à â®à  “àëá®­  ¢ ¯à®áâà ­áâ¢ å ¨§¬¥à¨¬ëå

¢¥ªâ®à-ä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, ç⮢ᥠ â®¬ë¤¨áªà¥â­®©

ç á⨠T ïîâáï â®çª ¬¨. ãáâì () (ᮮ⢥âá⢥­­®

()) ¥áâì ª®«ìæ®

(ᮮ⢥âá⢥­­®-ª®«ìæ®)¬­®¦¥á⢨§,¨¬¥îé¨åª®­¥ç­ãî(ᮮ⢥âá⢥­­®

-ª®­¥ç­ãî) ¬¥àã. ‚áî¤ã ¢ ¤ «ì­¥©è¥¬ ¡ã¤¥¬ áç¨â âì, çâ®:

(a) ¥á«¨ AB2 ¨ (B)=0, â® A2 (¯®«­®â  ¬¥àë );

(b) ¥á«¨ ¤«ï «î¡®£® B2 ()¨¬¥¥¬ B\A2, â® A2;

(c) ¤«ï«î¡®£® A2 ¨¬¥¥¬ (A)=sup f(B):B A;B2()g;

(d) áãé¥áâ¢ãîâ ¤¨§êî­ªâ­ë¥ ¬­®¦¥á⢠ fT

i

g â ª¨¥, çâ® (Tn S

T

i

)=0 ¨

0<(T

i

)<+1 ¯à¨ «î¡®¬ i;

(e) ¤«ï «î¡®£® A2 ()áãé¥áâ¢ãîâ ¬­®¦¥á⢮ N ¬¥àë ­ã«ì ¨ ­¥ ¡®«¥¥,

祬 áç¥â­®¥, ¬­®¦¥á⢮J ¨­¤¥ªá®¢ i â ª¨¥, çâ® AnN = S

i2J

(A\T

i ).

Š ª¨§¢¥áâ­®[1],ãá«®¢¨ï( ){(¥)¢ë¯®«­¥­ë¤«ï«î¡®©¯®«­®©-ª®­¥ç­®©

¬¥à먤«ï¬¥àë,¯®à®¦¤¥­­®©áãé¥á⢥­­®¢¥àå­¨¬¨­â¥£à «®¬ ¬¥à될¤®­ 

­ «î¡®¬«®ª «ì­®ª®¬¯ ªâ­®¬ ¯à®áâà ­á⢥. ¥§ãé¥à¡ ¤«ï

­¥âਢ¨ «ì­®á-⨢ᥣ®¤ «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®©

-¬¥âਪ®© ¨ F-­®à¬®© k k

E

; ­ ¯à¨¬¥à, L p

; 0 < p < 1), X | ¡ ­ å®¢®

c

¨¤¥ «ì­®¥ ¯à®áâà ­á⢮. ‘¨¬¢®«®¬

L

0

(

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 ¨§¬¥-ਬëå ¢¥ªâ®à-ä㭪権

~f

:

T

!

X

â ª¨å, çâ® k

~f

k

E

(

X

)

= kk

~f

k

X

k

E

<

+1. Œë ®£à ­¨ç¨¢ ¥¬áï¢ á¢®¥¬¨§«®¦¥­¨¨ ¢®á­®¢­®¬¤¢ã¬ï ¬®¤¥«ì­ë¬¨ ¯à¨¬¥à ¬¨

E

(

X

),  ¨¬¥­­®:

(1) ç¥à¥§

L

p

(

X

) (¨«¨

L

p

(

T;X

) [2]) ®¡®§­ ç ¥âáï ¯à®áâà ­á⢮ ¢á¥å ¨§¬¥-ਬëå ¢¥ªâ®à-ä㭪権

~f

(

t

) â ª¨å, çâ®

F

-­®à¬  í«¥¬¥­â  ¢¢®¤¨âáï ä®à¬ã«®©:

k

~f

k

p

= 0

@ Z

T

k

~f

(

t

)k

p

X

d

(

t

)

1

A 1

p

<

+1 (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])

k

~f

k

'

=inf 8

<

:

" >

0: Z

T

'

(k

~f

(

t

)k

X

="

)

d

(

t

)

"

9

=

;

;

£¤¥

'

2(

L

)

:

Ž¯à¥¤¥«¥­¨¥1.1. ®á«¥¤®¢ â¥«ì­®áâì f

~f

n

(

t

)g 1

n

=1

í«¥¬¥­â®¢

¯à®áâà ­áâ-¢ 

E

(

X

)­ §ë¢ ¥âáï

C

-¯®á«¥¤®¢ â¥«ì­®áâìî,¥á«¨¤«ïª ¦¤®©ç¨á«®¢®© ¯®á«¥-¤®¢ â¥«ì­®á⨠f

n

g

1

n

=1

#0 (

n

2R), àï¤ 1

P

n

=1

n

~f

n

(

t

) á室¨âáï.

Ž¯à¥¤¥«¥­¨¥ 1.2. à®áâà ­á⢮ ¢¥ªâ®à-ä㭪権

E

(

X

) ­ §ë¢ ¥â-áï

C

-¯à®áâà ­á⢮¬, ¥á«¨ ¤«ï «î¡®©

C

-¯®á«¥¤®¢ â¥«ì­®á⨠¥£® í«¥¬¥­â®¢ f

~f

n

(

t

)g

1

n

=1

E

(

X

)àï¤ 1

P

n

=1

~f

n

(

t

)á室¨âáï.

‹¥¬¬  1.1. „«ï ⮣®, ç⮡믮᫥¤®¢ â¥«ì­®áâì í«¥¬¥­â®¢ f

~f

n

(

t

)g 1

n

=1

E

(

X

)﫠áì

C

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

¬­®-¦¥á⢮

A

0

= n

1

P

n

=1

a

n

~f

n

(

t

):j

a

n

j1 o

¡ë«® ®£à ­¨ç¥­­ë¬.

C „®áâ â®ç­®áâì. ãáâì ¨§¢¥áâ­®, çâ® ¬­®¦¥á⢮ í«¥¬¥­â®¢

A

0

=

n

1

P

n

=1

a

n

~f

n

(

t

) : j

a

n

j 1 o

ï¥âáï ®£à ­¨ç¥­­ë¬. ãáâì f

c

n

g 1

n

=1

# 0 |

¯à®-¨§¢®«ì­ ï ç¨á«®¢ ï ¯®á«¥¤®¢ â¥«ì­®áâì ¨

" >

0. ‚ ᨫ㠮£à ­¨ç¥­­®á⨠¬­®-¦¥á⢠

A

0

­ ©¤¥âáï ­®¬¥à

n

0

â ª®©, çâ® ¤«ï

n > n

0

, k

c

n

~f

n

(

t

)k

< "

¤«ï ¢á¥å í«¥¬¥­â®¢

~f

n

(

t

)2

A

0

, £¤¥ kk ®§­ ç ¥â

F

-­®à¬ã ¢¯à®áâà ­á⢥

E

(

X

). Žâá ¤«ï ­®¬¥à®¢

n;m;n

0

< n < m

¨¬¥¥¬:

m

X

i

=

n

+1

c

i

~f

i

(

t

)

=

c

k

X

m

i

=

n

+1

c

i

c

k

~f

i

(

t

)

1{44

£¤¥

c

k

= max

n

i

m

c

i

¨

~

f

(

t

) =

m

P

i

=

n

+1

c

i

c

k ~

f

i

(

t

)

2 A

0

. ‡­ ç¨â, ¯®á«¥¤®¢ â¥«ì­®áâì

f ~

f

n

(

t

)

g 1

n

=1

ï¥âáï

C

-¯®á«¥¤®¢ â¥«ì­®áâìî.

¥®¡å®¤¨¬®áâì.

à¥¤¯®«®¦¨¬, çâ® ¬­®¦¥á⢮

A

0

­¥ ï¥âáï

®£à ­¨ç¥­-­ë¬. ’®£¤  áãé¥áâ¢ãîâ á室ïé ïáï ç¨á«®¢ ï ¯®á«¥¤®¢ â¥«ì­®áâì

f

n

g 1

n

=1 #

0,

®£à ­¨ç¥­­ ï ç¨á«®¢ ï ¯®á«¥¤®¢ â¥«ì­®áâì

fa

n

g 1

n

=1

;ja

n

j

1, ¨

¯®¤¯®á«¥¤®¢ -⥫쭮á⨠­®¬¥à®¢

fn

k

g

,

fn 0

k

g

â ª¨¥, çâ® ¯®á«¥¤®¢ â¥«ì­®áâì

n

k

n

0 k

P

n

=

n

k

a

n

~

f

n

(

t

) :

k 2N o

­¥ á室¨âáï ª ­ã«î, £¤¥

n

k

<n 0

k

<n

k

+1

.

®« £ ¥¬:

c

n

=

k

a

k

;

¥á«¨

n

k

<nn 0

k

;

0

;

¤«ï ¢á¥å ®áâ «ì­ëå ­®¬¥à®¢.

Œ®¦­® ¢¨¤¥âì, çâ® ¯®á«¥¤®¢ â¥«ì­®áâì

fc

n

g 1

n

=1

#

0, ­® àï¤

1

P

n

=1 c

n

~

f

n

(

t

)

à á室¨âáï. Žâá ¢¨¤­®, çâ®

f ~

f

n

(

t

)

1

n

=1

g

­¥ ¡ã¤¥â

C

-¯®á«¥¤®¢ â¥«ì­®áâìî.

à®â¨¢®à¥ç¨¥.

B

‹¥¬¬  1.2 (€. . Š®«¬®£®à®¢ | €. Ÿ. •¨­ç¨­ | ‚. Žà«¨ç).

ãáâì ¤ ­  C-¯®á«¥¤®¢ â¥«ì­®áâì f ~

f

n

(

t

)

g 1

n

=1

¯à®áâà ­á⢠ L 0

(

X

)

. ’®£¤  ­ 

ª ¦¤®¬ ¬­®¦¥á⢥ T

0

ª®­¥ç­®© ¬¥àë àï¤ 1

P

n

=1 j

~

f

n

(

t

)

j 2

á室¨âáï -¯®ç⨠¢áî¤ã.

’¥®à¥¬  1.1 (‹. ˜¢ àæ  [5]). à®áâà ­á⢠ ¢¥ªâ®à-ä㭪権 L

p

(

X

)

¯à¨

0

p1 ïîâáï C-¯à®áâà ­á⢠¬¨.

‘«¥¤á⢨¥ 1.1

(á¬. [3]).

à®áâà ­á⢠ ¢¥ªâ®à-ä㭪権 L

'

(

X

)

ïîâáï C-¯à®áâà ­á⢠¬¨ ¯à¨ ãá«®¢¨¨, çâ® '-äã­ªæ¨ï ª« áá 

(

L

)

¯®¤ç¨­ï¥âáï

2

-ãá«®¢¨î ¯à¨ ¢á¥å u.

C

„®ª § â¥«ìá⢮ á«¥¤á⢨ï 1.1 ¢ë⥪ ¥â ¨§ ⥮६ë 1.1, ¥á«¨ ¯®«®¦¨âì

'

(

u

) =

u

p

; u 2R

+

.

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⢠

E 1

(

X

) ¨

E

2

(

X

) ¨ ¯à®¨§¢®«ì­ë© ®¯¥à â®à

W

:

E 1

(

X

)

! L 0

(

X

). Ž¯¥à â®à

W

­ §ë-¢ ¥âáï

-¨­¢ à¨ ­â­ë¬, ¥á«¨ ¤«ï ª ¦¤®£® ¨§¬¥à¨¬®£® ¯®¤¬­®¦¥á⢠

T 0

T

¢ë¯®«­ï¥âáï ãá«®¢¨¥:

W

(

~v

)

,W

(

~v

+

1

T

0

~

u

) =

1

T

0

fW

(

~v

)

,W

(

~v

+

~u

)

g

~

u;~v2E

1

(

X

)

¯®ç⨠¢áî¤ã ­  T.

‡¤¥áì 1

T

0

®¡®§­ ç ¥â å à ªâ¥à¨áâ¨ç¥áªãî äã­ªæ¨î ¨§¬¥à¨¬®£®

¯®¤¬­®-¦¥á⢠T

0 T.

Žç¥¢¨¤­®, -¨­¢ à¨ ­â­ë© ®¯¥à â®à ï¥âáï H

-®¯¥à â®à®¬.

„¥©á⢨-⥫쭮,W(~v),W(~v+~u ) =1

T

1

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

T

2

fW(~v),W(~v+~u)g

¤«ï «î¡ëå ¤¨§êî­ªâ­ëå ¨§¬¥à¨¬ëå ¯®¤¬­®¦¥á⢠T

1 \T

2

= ?, T

1 [T

2 = 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+1

T

1 ~

u)+W(~v),W(~v+1

T

2 ~

u), §­ ç¨â,

kW(~v),W(~v+~u)k

2

kW(~v),W(~v+1

T

1 ~u)k

2

+kW(~v),W(~v+1

T

2 ~ u)k

2 ;

â. ¥. ®¯¥à â®à W ï¥âáï H

-®¯¥à â®à®¬ (¯à¨ H I).

à¨¬¥ç ­¨¥ 1.1. „«ï =1 -¨­¢ à¨ ­â­ë© ®¯¥à â®à ­ §®¢¥¬

¨­¢ à¨- ­â­ë¬ ®¯¥à â®à®¬. “á«®¢¨¥ (2) ¨­¢ à¨ ­â­®á⨠( = 1) ®¯¥à â®à  W ¨¬¥¥â

¢¨¤:

W(~v),W(~v+1

T

0 ~ u) =1

T

0

fW(~v),W(~v+~u)g

~

u; ~v 2E

1

(X); T

0 T

: (3)

Ž¯à¥¤¥«¥­¨¥ 1.4. Žâ®¡à ¦¥­¨¥ : E(X) ! R +

­ §®¢¥¬  ¤¤¨â¨¢­®©

ä®à¬®©, ®¡ãá«®¢«¥­­®© F-­®à¬®© kk ­  E(X), ¥á«¨:

(a)¤«ï «î¡ëå~u;~v 2E(X), supp~u(t)\supp~v(t)=?, ¢ë¯®«­ï¥âáïãá«®¢¨¥

(~u+~v)=(~u)+(~v);

(b) (~x

n

)#0,k~x

n

k#0 ¨ n!1;

(c) (~x

n

),k~x

n kk

; n2N ¨ (;k

)2R

2

.

à¨¬¥à ¬¨  ¤¤¨â¨¢­ëå ä®à¬ ¤«ï ª®­ªà¥â­ëå ¯à®áâà ­áâ¢

¢¥ªâ®à-ä㭪権 ~

f(t)¡ã¤ãâ ïâìáï ¨­â¥£à «ì­ë¥ ¬®¤ã«ïàë ¢¨¤ :

(a) ( ~

f)= Z

T k

~

f(t)k p

X d(t);

~

f 2L p

(X); (4)

(b)( ~

f)= Z

T '

k ~

f(t)k

X

d(t); ~

f 2L

'

(X); (5)

¥á«¨ '-äã­ªæ¨ï '(u) 㤮¢«¥â¢®àï¥â

2

-ãá«®¢¨î.

Ž¯à¥¤¥«¥­¨¥ 1.5. F

-¯à®áâà ­á⢮ ¢¥ªâ®à-ä㭪権 E(X) ­ §ë¢ ¥âáï

¯à®áâà ­á⢮¬ ⨯ L(X), ¥á«¨:

(1) 1

T

2E(X);

(3) ¢ E(X)áãé¥áâ¢ã¥â  ¤¤¨â¨¢­ ï ä®à¬  .

à¨¬¥ç ­¨¥1.2. à®áâà ­á⢮L p

(X)(0<p<1)ï¥âáï

¯à®áâà ­áâ-¢®¬ ⨯  E(X); ­ «®£¨ç­® ¤«ïL

'

(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§ ­­ë嬥¦¤ãᮡ®©ä®à¬ã«®©(s;t)=[ ~

f(t)](s),㤮¡­®¯à®¢®¤¨âì

¢à ¬ª å⥮ਨ¯à®áâà ­áâ¢á®á¬¥è ­­®©ª¢ §¨­®à¬®© [8],â ªª 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 (T

1 ;

1 ;

1

) ¨ (T;;) | ¤¢ 

¯à®áâà ­á⢠ á ¬¥à ¬¨

1

¨ ᮮ⢥âá⢥­­®.

„«ï ¤ ­­ëå X | ˆ (= ¡ ­ å®¢  ¨¤¥ «ì­®£® ¯à®áâà ­á⢠) ­ 

(T

1 ;

1 ;

1

), E |Šˆ(=ª¢ §¨¡ ­ å®¢ ¨¤¥ «ì­®£® ¯à®áâà ­á⢠) ­ (T;;)

ç¥à¥§E[X]®¡®§­ ç¨¬¯à®áâà ­á⢮¢á¥å¨§¬¥à¨¬ëåä㭪権 (s;t)­ T

1 T,

㤮¢«¥â¢®àïîé¨å ¤¢ã¬ ãá«®¢¨ï¬:

(1) ¯à¨ ¢á¥å t2T äã­ªæ¨ï s 7!(s;t)¢å®¤¨â ¢ X;

(2) äã­ªæ¨ï jj=k(;t)k

X

¢å®¤¨â ¢ E.

ˆ§¢¥áâ­®¥ ãá«®¢¨¥ (C) (á¬. [8]) ¢ ¯à®áâà ­á⢥ X ®¡¥á¯¥ç¨¢ ¥â

¨§¬¥à¨-¬®áâì ä㭪樨 jj. ‡­ ç¨â, E[X] | «¨­¥©­®¥ ¬­®¦¥á⢮,  , á«¥¤®¢ â¥«ì­®,

¨ ¨¤¥ «ì­®¥ ª¢ §¨­®à¬¨à®¢ ­­®¥ ¯à®áâà ­á⢮ ­  ¯à®¨§¢¥¤¥­¨¨ T

1

T,   â ª

ª ª E | Šˆ,â® ä®à¬ã« kk

E[X]

=kjjk

E

¯à¥¢à é ¥â E[X] ¢Šˆ‘Š (á¬.

â ª¦¥[3],£¤¥¤«ï¡®«¥¥®¡é¨åá¨âã æ¨©¨¬¥îâáאַ¤¥«ì­ë¥¯à¨¬¥àë Šˆ‘Š

L

() , L

()

, Žà«¨ç  L

(') ).

Žâ¢¥â ­  ¢®¯à®á, ª®£¤  ¨¬¥¥â ¬¥á⮠⮯®«®£¨ç¥áª®¥ ᮢ¯ ¤¥­¨¥

¯à®áâ-à ­á⢠ ¢¥ªâ®à-ä㭪権 E(X) á ¯à®áâà ­á⢮¬ ᮠᬥ蠭­®© ª¢ §¨­®à¬®©

E[X], ¤ ¥âá«¥¤ãîé ï «¥¬¬ :

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

(1) E(X)=E[X] ¯à¨ ª ­®­¨ç¥áª®¬ ¢«®¦¥­¨¨(s;t)=[ ~

f(t)](s);

(2) X | ˆ á ãá«®¢¨¥¬ (€):(x

n

#0))

kx

n k

X

!0 ¯à¨ n!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 E(X)

¨§¬¥à¨¬ëå ¢¥ªâ®à-ä㭪権

–¥«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 E

1

(X) | F-¯à®áâà ­á⢮, E

2

(X) | ¯à®áâà ­á⢮

⨯  L(X), |  ¤¤¨â¨¢­ ï ä®à¬ ,®¡ãá«®¢«¥­­ ï F-­®à¬®©kk

2

,   ®¯¥à â®à

(a) W(~u)0 ¯®ç⨠¢áî¤ã ­  T, ¥á«¨~u2E

1

(X); ~u 0 ¯®ç⨠¢áî¤ã;

(b) W(1

T 1 ~ u 1 +1 T 2 ~ u 2 ) 1

T

1

W(~u

1 )+1

T

2

W(~u

2

) ¯®ç⨠¢áî¤ã ­  T, ¥á«¨

~ u

1 ;~u

2 2E

1

(X); ~u

1 ;~u

2

0,¯®ç⨢áî¤ã­ T ¨T

1 ;T

2

|¨§¬¥à¨¬ë¥,T

1 \T 2 =?, T 1 T; T 2 T.

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

1

(X) ¬­®¦¥á⢠ C ®¡à §

T(C) |  ¡á®«îâ­® ®£à ­¨ç¥­ ¢ E

2 (X).

C Žâ ¯à®â¨¢­®£®. „®¯ãá⨬, çâ® áãé¥áâ¢ã¥â ¬­®¦¥á⢮ C,  ¡á®«îâ­®

®£à ­¨ç¥­­®¥ ¢ E

1

(X) â ª®¥, çâ® ®¡à § T(C) ¢ ¯à®áâà ­á⢥ E

2

(X) ­¥ ¡ã¤¥â

 ¡á®«îâ­® ®£à ­¨ç¥­­ë¬. â® §­ ç¨â, çâ® áãé¥áâ¢ãîâ "

0

>0,

¯®á«¥¤®¢ â¥«ì-­®áâ¨f~u

k (t)g

1

k=1

C¨fT

k

g2; T

k

T (8k)â ª¨¥,çâ®(T

k

)!0¯à¨k !1,

1

P

k=1 (T

k

)<1 ¨ k1

T

k

W(~u

k )k 2 >" 0 .

 áá㦤 ï ¯®  ­ «®£¨¨, ª ª ¢ [10], ¬®¦­® ã⢥ত âì, çâ® áãé¥áâ¢ã¥â

¯®á«¥¤®¢ â¥«ì­®áâì ¯®¤¬­®¦¥á⢠fT

k g

1

k=1

T â ª ï, çâ® T

i \T

j

= ?; i 6= j ¨

f ~ f n (t)g 1 n=1 E 1 (X), ~ f n

(t)0¯®ç⨢áî¤ã,8n2N, 㤮¢«¥â¢®àïî騥ãá«®¢¨ï¬

P 1 n=1 k1 Tn ~ f n (t)k 1

< +1, ­® k1

Tn W( ~ f n )k 2 > " 0

> 0. ’®£¤  ­ ©¤¥âáï " 0

0 > 0

â ª®¥, çâ® (1

T n W( ~ f n ))>"

0

; 8n2N. Ž¡®§­ ç¨¬

~v(s) = (

~

f

n

(t); ¥á«¨ t 2T

n ;

0; ¥á«¨ t 2Tn S 1 n=1 T n =T 0 :

Žç¥¢¨¤­®, çâ® ~v 2E

1

(X) ¨, §­ ç¨â, ¯® ãá«®¢¨î W(~v)2 E

2

(X). ‘ ¤à㣮©

áâ®à®­ë, ¨¬¥¥¬:

W(~v)=W 1 X n=1 1 T n ~ f n 1 X n=1 1 T n W( ~ f n );

®âªã¤  kW(~v)k

2 P 1 n=1 1 T n W( ~ f n ) 2 . ’ ªª ª 1 X n=1 1 T n W( ~ f n ) = 1 X n=1 (1 T n W( ~ f n

))=1;

â® P 1 n=1 1 T n W( ~ f n )2= E

2

(X); á«¥¤®¢ â¥«ì­®, W(~v)2= E

2

(X). à®â¨¢®à¥ç¨¥. B

‘«¥¤á⢨¥2.1. ‚ãá«®¢¨ïå ⥮६ë2.1, ¥á«¨®¯¥à â®àW ­¥¯à¥à뢥­ ¯®

¬¥à¥¢ â®çª¥~u

0 2 E

1

(X),â® ®¯¥à â®àW ­¥¯à¥à뢥­ ¯® F-­®à¬¥¯à®áâà ­á⢠

E

1

(X) ¢ â®çª¥ ~u

0 .

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¥

~ u

0 2 E

1

(X), £¤¥ W ­¥¯à¥à뢥­ ¯® ¬¥à¥ ¨ W(

E

1 (X)

) = 1

E

2 (X)

, ( | ­®«ì

1{48

‹¥¬¬  2.1. ãáâì 1

T 2

E

1

(

X

)

¨ 1 T

2

E

2

(

X

)

(

E i | ¯®¤¯à®áâà ­á⢠ ¢ E i

í«¥¬¥­â®¢, ¨¬¥îé¨å  ¡á®«îâ­® ­¥¯à¥à뢭ãî F-­®à¬ã), W ¯à®¨§¢®«ì­ë©

®¯¥à â®à ¨§ E

1

(

X

)

¢ E 2

(

X

)

.

‘«¥¤ãî騥 ¯à¥¤«®¦¥­¨ï íª¢¨¢ «¥­â­ë:

(1) W ­¥¯à¥à뢥­ ¯® ¬¥à¥ ¢~u

0 2E

1

(

X

)

; (2)

lim

~ u!

ft;jW

(

~u 0

+

~

z

)

,W

(

~u 0

)

j>g

= 0

, £¤¥ >

0

, ~z | 䨪á¨à®¢ ­ë.

’¥®à¥¬  2.2. ãáâì E

1

(

X

)

| F-¯à®áâà ­á⢮, E 2

(

X

)

| ¯à®áâà ­á⢮

⨯  L

(

X

)

, ¯à¨ç¥¬ 1 T

2

E

1

(

X

)

¨ 1 T

2 E

2

(

X

)

,   W

:

E 1

(

X

)

! E 2

(

X

)

| ¯à®¨§¢®«ì­ë©®¯¥à â®à,¯®¤ç¨­ïî騩áïãá«®¢¨ï¬( )¨(b)⥮६ë2.1. ’®£¤ 

á«¥¤ãî騥 ã⢥ত¥­¨ï íª¢¨¢ «¥­â­ë:

(1) W ­¥¯à¥à뢥­ ¢ â®çª¥ ~u

0 2E

1

(

X

)

; (2) W ­¥¯à¥à뢥­ ¯® ¬¥à¥ ¢ â®çª¥ ~u

0 2E

1

(

X

)

. C

2)

)

1). ãáâì

f~u

n g

1

n=1 ! ~u

0

¯®

F

-­®à¬¥ ¯à®áâà ­á⢠

E 1

(

X

), ⮣¤ 

f~u

n g

1

n=1 ! ~u

0

¯® ¬¥à¥ ­ 

T

¯à¨

n ! 1

. Ž¯¥à â®à

W

, ¡ã¤ãç¨ ­¥¯à¥à뢭ë¬

¯® ¬¥à¥ ¢ â®çª¥

~u

0

, ¤ ¥â

W

(

~u

n

)

! W

(

~u

0

). € â ª ª ª ¯®á«¥¤®¢ â¥«ì­®áâì

f~u

n g

1

n=1 ! ~u

0

 ¡á®«îâ­® ®£à ­¨ç¥­ , â®

fW

(

~u n

)

g

 ¡á®«îâ­® ®£à ­¨ç¥­­®¥

¬­®¦¥á⢮ ¢

E 2

(

X

) (ᮣ« á­® ⥮६¥ 2.1). ‡­ ç¨â,

fW

(

~u n

)

g 1

n=1

! W

(

~u 0

) ¯®

F

-­®à¬¥ ¯à¨

n!1

.

(1)

)

(2) „®ª § â¥«ìá⢮ ¯à¥¤®áâ ¢«ï¥¬ ç¨â â¥«î.

B à¨¬¥ç ­¨¥ 2.2.

‘ãé¥áâ¢ãîâ

F

-¯à®áâà ­á⢠

E

1

(

X

) ¨

E 2

(

X

),

1 T

2

E

2

(

X

), £¤¥

W

:

E 1

(

X

)

!E 2

(

X

) ¨­¢ à¨ ­â­ë© ®¯¥à â®à

W

(

E

1

(X)

) =

E 2

(X)

,

¤«ï

~a2E 2

(

X

) ¨ ®â®¡à ¦¥­¨¥ :

E 1

(

X

)

!E 2

(

X

) â ª¨¥, çâ®:

(1)

jW

(

~u

)(

t

)

j~a

(

t

) + (

~u

)(

t

) ¯®ç⨠¢áî¤ã ­ 

T

,

8~u2E 1

(

X

);

(2) ¤«ï «î¡®£®

F

-®£à ­¨ç¥­­®£® ¬­®¦¥á⢠

B 2E 1

(

X

), (

B

) ®£à ­¨ç¥­®

¢

E 2

(

X

).

à¨¬¥à ¬¨ ¬®£ãâ á«ã¦¨âì ¨§¢¥áâ­ë© ®¯¥à â®à á㯥௮§¨æ¨¨

W

(

~u

)(

t

) =

N

[

t;~u

(

t

)] ¨

E

1

(

X

) =

L p

1

(

X

),

E

2

(

X

) =

L p

2

(

X

) ¯à¨ ᮮ⢥âáâ¢ãîé¨å

®£à ­¨ç¥-­¨ïå ­ 

p 1

;p

2 >

0.

’¥®à¥¬ 2.3. ãáâìE

1

(

X

)

¨E 2

(

X

)

|¤¢ F-¯à®áâà ­á⢠­ T,1 T 2 E 2 ,

W

:

E 1

(

X

)

! E 2

(

X

)

. ãáâì ¤«ï ~u 0

2

E

1

(

X

)

, ~u

0

>

0

­  T, áãé¥áâ¢ãîâ ¢®§à áâ îé ïäã­ªæ¨ï '

:

R

+

!R +

, 㤮¢«¥â¢®àïîé ï ãá«®¢¨î

lim

u!1

'(u)

u

=

1,äã­ªæ¨ï~a2E

2

(

X

)

¨®â®¡à ¦¥­¨¥

:

E 1

(

X

)

!E 2

(

X

)

,¯¥à¥¢®¤ï饥¢á类¥ ®£à ­¨ç¥­­®¥ ¯® F-­®à¬¥ ¬­®¦¥á⢮ B E

1

(

X

)

¢ ®£à ­¨ç¥­­®¥ ¬­®¦¥á⢮

(

B

)

E 2

(

X

)

, â ª¨¥, çâ®

'

jW

(

~

f

)(

t

)

j ~ u 0

(

t

)

~ u 0

(

t

)

~a

(

t

) + (

~

1{49

¯®ç⨠¢áî¤ã ­  T ¤«ï ¢á¥å ~

f 2 E

1

(

X

)

. ’®£¤  ¤«ï ª ¦¤®£® r >

0

, ®¡à § W

(

B

0

(

~

;r

))

(è à  á 業â஬ ¢

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

2

(

X

)

.

C

® ãá«®¢¨î,

~a 2 E 2

(

X

), ⮣¤  ¤«ï

8" >

0, áãé¥áâ¢ã¥â

1

>

0 â ª®¥,

çâ®

k 1

~a

(

t

)

k 2

< "

2

. ® ãá«®¢¨î,

B 0

(

~

;r

)

®£à ­¨ç¥­­®¥, ⮣¤  áãé¥áâ¢ã¥â

2

>

0 â ª®¥, çâ®

k 2

(

~

f

)

k 2

< "

2

¤«ï «î¡®£® í«¥¬¥­â 

~

f 2 B 0

(

;r

). ®«®¦¨¬

= inf

f

1 ;

2

g

. Žâá:

'

jW

(

~

f

)(

t

)

j=~u 0

(

t

)

~u

0

(

t

)

2

k~a

(

t

)

k 2

+

k

(

~

f

(

t

))

k 2

<":

â® ®§­ ç ¥â, çâ® ®¡à §

W

(

B 0

(

~

;r

)) ¥áâì  ¡á®«îâ­® ®£à ­¨ç¥­­®¥

¬­®-¦¥á⢮ ¢

E 2

(

X

) (á¬. â ª¦¥ ⥮६ã 1.4.13 ¨§ [10]).

B

‹¨â¥à âãà 

1.

Š®à®âª®¢ ‚. .

ˆ­â¥£à «ì­ë¥ ®¯¥à â®àë.|®¢®á¨¡¨àáª:  ãª , 1983.

2.

Kalton N.J.

Isomorphism between

L

p

-function spaces when

p<

1 // J. Func.

Anal.|1981.|V. 42.|P. 299{337.

3.

”¥â¨á®¢ ‚. ƒ.

Ž¡ ®¯¥à â®à å ¢ ¨¤¥ «ì­ëå ª¢ §¨­®à¬¨à®¢ ­­ëå

¯à®áâ-à ­á⢠å ᮠᬥ蠭­®© ª¢ §¨­®à¬®©

E

() // ‘¥¢¥à®-Žá¥â¨­. £®áã­¨¢¥àá¨â¥â.|

„¥¯®­¨à. ¢ ‚ˆˆ’ˆ 22.05.90, ü 2784{‚90.

4.

Rolewicz S.

Metric linear spaces.|Warszawa: PWN, 1972.

5.

Schwartz L.

Un theoreme de la convergence dans les

L p

, 0

p 1

// C. R.

Acad. Sci. Paris. Ser. A.|1969.|T. 268.|P. 704{706.

6.

Matuszewska W., Orlicz W.

A note on modular spaces IX // Bull. Acad.

Polon. Sci.|1968.|V. 16.|P. 801{807.

7.

”¥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.

8.

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

‚¥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®áâ-à ­á⢠å // „¨áá. ­  ᮨ᪠­¨¥ ãç. á⥯. ¤®ªâ. 䨧.-¬ â. ­ ãª, ˆ­-â

¬ â-ª¨. ‘Ž €, 1996.|280 á.