d( ) = 0g |
| KONE^NO, LIBO 1 = Br( 0 ). wTOROE, ODNAKO, NEWOZMOVNO. |
f|TO NEWOZMOVNO PRI 0 = 0, IBO INA^E URAWNENIE (2) IMEET NENULEWOE |
RE[ENIE PRI = 0, A ZNA^IT (SM. (ii)), PRI = 0 URAWNENIE (1) IMEET |
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6 |
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NENULEWOE RE[ENIE. pRI = 0 ISPOLXZUJTE SWQZNOSTX C |
(!!). |
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(iv). pOKAVEM, ^TO d( ) = 0 (T. E. |
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Br ( 0 ) 1) OZNA^AET, ^TO URAW- |
NENIE |
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(I , C ( ))' = |
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ODNOZNA^NO RAZRE[IMO PRI L@BOM |
. pOLOVIM ' = |
+ N , GDE N | |
RE[ENIE URAWNENIQ
(IK , C( )) N = C( ) :
|TO POSLEDNEE URAWNENIE | URAWNENIE W K, I ONO RAZRE[IMO, TAK KAK det[I , C ( )] = d( ) 6= 0. tOGDA
[I , C ( )]( + N ) = , C ( ) + C( ) = ;
^TO I TREBOWALOSX. tAKIM OBRAZOM, I , C ( ) OBRATIM, IBO [I , C( )],1 OPREDELEN• WS@DU W H (I ZAMKNUT). >
x253. sPEKTRALXNAQ TEOREMA DLQ SAMOSOPRQV•ENNOGO
KOMPAKTNOGO OPERATORA
1.[tEOREMA rISSA-{AUDERA]. pUSTX A 2 C(H ). tOGDA DLQ L@BOGO
"> 0 MNOVESTWO (A)nB"(0) KONE^NO, PRI^•EM ESLI 0 6= 2 (A), TO | SOBSTWENNOE ZNA^ENIE OPERATORA A KONE^NOJ KRATNOSTI.
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pUSTX | DISKRETNOE MNOVESTWO IZ TEOREMY x252. pRI \TOM |
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2 (A)nB"(0) |
TTOGDA |
1= 2 |
B1="[0]. |
iZ DISKRETNOSTI |
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OTS@DA |
SLEDUET, ^TO (A)nB"(0) KONE^NO.TdALEE, ESLI K | PODPROSTRANSTWO |
WSEH SOBSTWENNYH WEKTOROW IZ H, |
PRINADLEVA]IH SOBSTWENNOMU ZNA^E- |
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NI@ = 0; TO OGRANI^ENIE NA K KOMPAKTNOGO OPERATORA 1 A QWLQETSQ
TOVDESTWENNYM I KOMPAKTNYM OPERATOROM W K. iZ 244.4 SLEDUET, ^TO K KONE^NOMERNO. >
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2. eSLI A 2 C(H ) | SAMOSOPRQV•ENNYJ OPERATOR I (A) = f0g, TO |
A = 0. |
k |
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kfk=1 jh |
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ij 6 |
k k |
kfk=1h |
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pUSTX |
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= sup |
Af; f |
= 0 I DLQ OPREDELENNOSTI• |
A |
= sup |
Af; f |
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(MY POMNIM, ^TO KWADRATI^NAQ FORMA hAf; fi W DANNOM SLU^AE PRINIMAET WE]ESTWENNYE ZNA^ENIQ). tOGDA SU]ESTWUET POSLEDOWATELXNOSTX (gn ),
TAKAQ, ^TO kgnk |
= 1 I hAgn; gni ! kAk. tAK KAK A 2 C(H ); (Agn) OBLA- |
DAET SHODQ]EJSQ PODPOSLEDOWATELXNOSTX@ |
OBOZNA^AEMOJ TAKVE |
(Agn)) : |
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Agn ! h. |
iMEEM |
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kAgn , kAkgn k2 = |
kAgnk2 |
, 2kAkhAgn; gni + kAk2 |
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kAk |
k k |
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2[kAk2 |
, kAkhAgn; gni] ! 0 (n ! 1): |
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! kAk |
6 |
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oTS@DA gn = |
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( A |
gn |
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Agn +Agn) |
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1 |
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h = . sLEDOWATELXNO, Ah = |
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k k |
n |
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limAgn = |
A h |
WLE^ET |
^TO |
h | |
SOBSTWENNYJ WEKTOR OPERATORA |
A, |
A |
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• , |
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kAk | OTWE^A@]EE EMU SOBSTWENNOE ZNA^ENIE. pO\TOMU kAk 2 (A): >
3.z A M E ^ A N I E. eSLI dim H = 1, TO 0 2 (A) DLQ L@BOGO A 2 C(H )
(!!).
4.[tEOREMA gILXBERTA-{MIDTA (SPEKTRALXNAQ TEOREMA DLQ SAMOSOPRQV•ENNOGO KOMPAKTNOGO OPERATORA)]. pUSTX A | SAMOSOPRQV•ENNYJ KOM-
PAKTNYJ OPERATOR W SEPARABELXNOM GILXBERTOWOM PROSTRANSTWE H. tOGDA SU]ESTWUET ORTONORMIROWANNYJ BAZIS (fn ) W H TAKOJ, ^TO
A = X nh ; fnifn; n 2 R; n ! 0:
n
pUSTX k | NENULEWYE TO^KI SPEKTRA A. w SOOTWETSTWII S 251.7 I PP.
1,3 (A) = f0g [ f 1; 2; : : :g; k 2 R. kAVDOE SOBSTWENNOE ZNA^ENIE k IMEET KONE^NU@ KRATNOSTX:
dim Hk < +1; GDE Hk = ff 2 H j Af = kfg:
w KAVDOM Hk WYBEREM ORTONORMIROWANNYJ BAZIS I RASSMOTRIM SISTEMU (fn ) | OB_EDINENIE \TIH BAZISOW. |TO ORTONORMIROWANNAQ SISTEMA, TAK KAK SOBSTWENNYE WEKTORY, OTWE^A@]IE RAZLI^NYM SOBSTWENNYM ZNA^E- NIQM ORTOGONALXNY (SM. 251.9). pUSTX K | ZAMYKANIE LINEJNOJ OBOLO^-
KI SISTEMY (fn ). tOGDA AK |
K; AK? |
K? (!!). pO\TOMU OPERATOR |
A |
AjK? | SAMOSOPRQV•ENNYJ KOMPAKTNYJ OPERATOR (W K? ). sNOWA W |
SILU P. 1 0 = |
2 |
(A) OZNA^AET, ^TO | SOBSTWENNOE ZNA^ENIE OPERA- |
e |
6 |
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A. nO PO POSTROENI@ NE SU]ESTWUET NI |
TORA A, A ZNA^IT, I OPERATORA |
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e |
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ODNOGO NENULEWOGO SOBSTWENNOGO ZNA^ENIQ, NE PRINADLEVA]EGO SEMEJSTWU |
f 1; 2e; : : :g. pO\TOMU (A) = f0g I SOGLASNO P. 2 A = 0. |
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442 |
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dOPOLNIM ORTONORMIROWANNU@ SISTEMU (fn) DO ORTONORMIROWANNOGO BAZISA W H (T. E. PRISOEDINIM K (fn) BAZIS W K?). oBRAZUEM TEPERX POSLE- DOWATELXNOSTX: 1; 1; : : : ; 1; 2; : : : , W KOTOROJ KAVDOE SOBSTWENNOE ^ISLOk DUBLIRUETSQ STOLXKO RAZ, KAKOWA EGO KRATNOSTX, I, ZANOWO PERENUMERO- WYWAQ POLU^ENNU@ POSLEDOWATELXNOSTX, MY POLU^IM POSLEDOWATELXNOSTX
( n ), |
PRI^EM |
0. |
|TA POSLEDOWATELXNOSTX ISKOMAQ |
. |
dEJSTWITELXNO |
, |
• n ! |
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PREDSTAWIW L@BOJ WEKTOR |
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W WIDE |
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RAZLOVENIE |
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W |
RQD fURXE), IMEEM |
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f 2 H |
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f = Phf; fnifn ( |
f |
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X |
hf; fnifn) = |
X |
hf; fniAfn = |
X |
nhf; fnifn: |
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Af = A( |
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5. [kANONI^ESKAQ FORMA KOMPAKTNOGO OPERATORA]. pUSTX A | KOM-
PAKTNYJ OPERATOR W SEPARABELXNOM GILXBERTOWOM PROSTRANSTWE H.
tOGDA SU]ESTWU@T ORTONORMIROWANNYE SISTEMY (fn); (gn ) I ^ISLA n |
0 TAKIE, ^TO |
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P |
h i |
RQD SHODITSQ PO OPERATORNOJ NORME |
). |
A = n n ; gn fn ( |
uTWERVDENIE O^EWIDNO, ESLI A = 0. pUSTX A | KOMPAKTNYJ OPERATOR, A 6= 0. tOGDA A A | SAMOSOPRQV•ENNYJ KOMPAKTNYJ OPERATOR, I PO TEO- REME gILXBERTA-{MIDTA SU]ESTWUET ORTONORMIROWANNAQ SISTEMA (gn ) I ^ISLA n > 0 TAKIE, ^TO
A A = |
X |
nh ; gnign; n ! 0: |
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n |
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fw PREDSTAWLENII OPERATORA A A SOGLASNO P. 4 MY OSTAWLQEM LI[X NE- NULEWYE SLAGAEMYE; n > 0 W SILU WYKLADKI:
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n = nkgnk2 = h ngn; gni = hA Agn; gni = kAgnk2:g |
oTMETIM DALEE, ^TO Ker (A A) = fg1; g2; : : :g? . pOLOVIM n = p |
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; fn = |
n |
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Agn |
. pOLU^ENNAQ SISTEMA WEKTOROW (fn ) | ORTONORMIROWANNAQ: |
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n |
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n2 |
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hfn ; fmi = n m hAgn; Agmi = |
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n m hA Agn ; gmi |
= n m hgn; gmi |
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= nm: |
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pROIZWOLXNYJ WEKTOR f 2 |
H PREDSTAWIM W WIDE f = n hf; gnign + h, GDE |
h 2 Ker (A A). nO TOGDA h |
2 Ker(A), OTKUDA |
P |
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Af = |
X |
hf; gniAgn = |
X |
nhf; gnifn (f 2 H ): |
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iSKOMOE PREDSTAWLENIE A POLU^ENO. pRI \TOM n ! 0 WLE^ET• SHODIMOSTX POLU^ENNOGO RQDA PO OPERATORNOJ NORME (!!). >
x254. pRILOVENIQ K LINEJNYM INTEGRALXNYM URAWNENIQM
1. bUDEM RASSMATRIWATX INTEGRALXNYE OPERATORY WIDA (T f)(t) = Z K (t; s)f (s) (ds) W PROSTRANSTWE H = L2(M; ). pRI \TOM PREDPOLAGA-
M
ETSQ, ^TO K 2 L2(M M; ). uRAWNENIE
Z K(t; s)f(s) (ds) = g(t)
M
(OTNOSITELXNO f ) NAZYWAETSQ URAWNENIEM fREDGOLXMA 1-GO RODA. fUNK- CIQ K (t; s) NAZYWAETSQ QDROM INTEGRALXNOGO URAWNENIQ (QDRO gILXBERTA- {MIDTA), A OPERATOR T (ON QWLQETSQ KOMPAKTNYM) NAZYWAETSQ OPERATO- ROM gILXBERTA-{MIDTA. pRI \TOM (SM. 246.5)
(T f )(t) = Z K(s; t)f(s) (ds):
M
2. z A M E ^ A N I E. bOLEE OB]IM OBRAZOM MOVNO RASSMATRIWATX OPERATORNOE URAWNENIE
GDE T | NEKOTORYJ KOMPAKTNYJ OPERATOR. oNO NAZYWAETSQ OPERATORNYM URAWNENIEM fREDGOLXMA 1-GO RODA. uRAWNENIE (1) NE KORREKTNO W SLEDU@- ]EM SMYSLE: ESLI g1 ; g2 | BLIZKIE (PO NORME) PRAWYE ^ASTI, TO SOOTWET- STWU@]IE RE[ENIQ f1; f2 (ESLI ONI SU]ESTWU@T) MOGUT BYTX DALEKIMI• DRUG OT DRUGA. dEJSTWITELXNO, OPERATOR T ZAWEDOMO NE OBRATIM (T.K. 0 2 (T )), I DAVE ESLI T ,1 OPREDELEN• , ON QWLQETSQ ZAMKNUTYM, NO NE NE- PRERYWNYM OPERATOROM. pO\TOMU SU]ESTWUET POSLEDOWATELXNOSTX gn ! TAKAQ, ^TO T,1gn NE STREMITSQ K .
3. uRAWNENIE WIDA
f (t) , Z K (t; s)f (s) (ds) = g(t)
M
(OTNOSITELXNO NEIZWESTNOJ FUNKCII f) NAZYWAETSQ URAWNENIEM fRED- GOLXMA 2-GO RODA. mY BUDEM ZAPISYWATX EGO W OPERATORNOJ FORME:
GDE T | KOMPAKTNYJ OPERATOR. eSLI g 6= , TO URAWNENIE (2) NAZYWAETSQ NEODNORODNYM; ESLI g = , | ODNORODNYM:
(3) |
(I , T )f = : |
uRAWNENIE |
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(4) |
(I , T )f = |
NAZYWAETSQ SOPRQVENNYM• ODNORODNYM URAWNENIEM.
oSNOWNYE REZULXTATY, KASA@]IESQ RAZRE[IMOSTI URAWNENIJ fRED- GOLXMA 2-GO RODA, SOBRANY W SLEDU@]IH TREH TEOREMAH (TAKVE NAZYWAE- MYH TEOREMAMI fREDGOLXMA):
4. uRAWNENIE (2) RAZRE[IMO TTOGDA g ORTOGONALXNO KAVDOMU RE- [ENI@ SOPRQV•ENNOGO ODNORODNOGO URAWNENIQ (4).
5. [aLXTERNATIWA fREDGOLXMA]. lIBO URAWNENIE (2) IMEET PRI L@BOM g EDINSTWENNOE RE[ENIE, LIBO ODNORODNOE URAWNENIE (3) IMEET NENULE- WOE RE[ENIE.
6. oDNORODNYE URAWNENIQ (3), (4) IME@T ODNO I TO VE KONE^NOE ^ISLO LINEJNO NEZAWISIMYH RE[ENIJ.
4. sLEDUET NEMEDLENNO IZ PREDSTAWLENIQ
(5) |
H = Ker (I , T ) R(I , T) |
(SM. 248.4(iv)), W KOTOROM U^TENO, ^TO R(I , T ) ZAMKNUTO (SM. 245.7). |
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5. |TO SLEDSTWIE TEOREMY x252 PRI = 1. |
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6. pUSTX ff1; : : : ; fmg; fh1; : : : ; hng | ORTONORMIROWANNYE BAZISY IZ |
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RE[ENIJ URAWNENIJ (3) I (4) SOOTWETSTWENNO I PUSTX, NAPROTIW, n = m. |
rASSMOTRIM DLQ OPREDELENNOSTI• |
SLU^AJ n > m (SLU^AJ n < m RASSMAT- |
RIWAETSQ ANALOGI^NO). oPREDELIM KOMPAKTNYJ OPERATOR S RAWENSTWOM
m
S = T , kX=1h ; fkihk
I ZAMETIM, ^TO URAWNENIE Sf = f IMEET LI[X TRIWIALXNOE RE[ENIE. fdEJSTWITELXNO, ESLI f | RE[ENIE \TOGO URAWNENIQ, TO WSE SLAGAEMYE W LEWOJ ^ASTI RAWENSTWA
m
(I , T )f + Xhf; fkihk =
k=1
POPARNO ORTOGONALXNY (SM. (5)). w SILU 152.10
(I , T )f = ; hf; fki = 0 (1 k m):
tOGDA IZ PERWOGO RAWENSTWA SLEDUET, ^TO WEKTOR f | LINEJNAQ KOMBINA- CIQ WEKTOROW (fk ), A IZ OSTALXNYH, | ^TO f = .g
sOGLASNO ALXTERNATIWE fREDGOLXMA ZAKL@^AEM, ^TO URAWNENIE (I , S)f = hm+1 ODNOZNA^NO RAZRE[IMO. uMNOVAQ OBE ^ASTI \TOGO URAW- NENIQ SKALQRNO NA WEKTOR hm+1 , POLU^AEM (SNOWA SM. (5)) PROTIWORE^IE:
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m |
1 = |
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; hm+1 |
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(I , S)f; hm+1i |
= |
h(I , T )f; hm+1i |
+ k=1hf; fkihhk; hm+1i = 0: |
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x255. sLU^AJ SIMMETRI^NYH I WYROVDENNYH QDER
1. rASSMOTRIM INTEGRALXNOE URAWNENIE fREDGOLXMA 2-GO RODA S SIM- METRI^NYM QDROM gILXBERTA-{MIDTA K (t; s)(= K(s; t)):
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f(t) , Z K(t; s)f(s) (ds) = g(t): |
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M |
w SOOTWETSTWII S x254 \TO URAWNENIE S KOMPAKTNYM SAMOSOPRQVENNYM• |
OPERATOROM (T f )(t) Z K (t; s)f (s) (ds): |
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M |
( ) |
(I , T )f = g: |
oTMETIM SLEDSTWIQ TEOREM fREDGOLXMA PRIMENITELXNO K DANNOMU SLU- ^A@:
2. (i) eSLI ^ISLO 1 NE ESTX SOBSTWENNOE ZNA^ENIE OPERATORA T , TO URAWNENIE ( ) ODNOZNA^NO RAZRE[IMO.
(ii) eSLI 1 | SOBSTWENNOE ZNA^ENIE T , TO URAWNENIE ( ) RAZRE[IMO, ESLI FUNKCIQ g ORTOGONALXNA WSEM SOBSTWENNYM FUNKCIQM, PRINADLE- VA]IM SOBSTWENNOMU ZNA^ENI@ 1.
3. pOLU^IM RE[ENIE URAWNENIQ ( ), ISPOLXZUQ SPEKTRALXNU@ TEOREMU DLQ KOMPAKTNOGO SAMOSOPRQVENNOGO• OPERATORA. pUSTX (fn) | ORTONORMI- ROWANNYJ BAZIS IZ SOBSTWENNYH WEKTOROW OPERATORA T I
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T = |
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nh ; fnifn; n 2 R; n ! 0; |
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| EGO PREDSTAWLENIE PO SPEKTRALXNOJ TEOREME 253.4. pUSTX |
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0 = |
fn |
2 N j n = 0g; 1 = fn 2 N j n = 1g; |
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= |
Nn( 0 [ 1 ): |
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rE[ENIE URAWNENIQ ( ) I]EM W WIDE f = |
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nfn , GDE n = |
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WESTNYE KO\FFICIENTY fURXE WEKTORA |
f . |
tOGDA RAWENSTWO |
(I T ) n nfn = |
Ph |
i |
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n |
g; fn |
fn PEREPI[ETSQ W WIDE |
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(I , T ) n n fn = |
n 0 n(I , T)fn + n 1 n (I , T )fn |
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2 |
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nfn + n n (1 , n)fn |
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+ |
n n (I , T)fn |
= n 0 |
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= |
n 0hg; fnifn + n 1hg; fnifn + n hg; fnifn: |
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s U^ETOM• |
EDINSTWENNOSTI PREDSTAWLENIQ \LEMENTA RQDOM fURXE POLU^A- |
EM W SLU^AE 2(i) (TOGDA 1 = ;): |
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g; fn |
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ESLI n |
0, |
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n = 8 hhg; fnii ; |
ESLI n |
2 . |
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< |
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iSKOMOE RE[ENIE IMEET WID: |
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f = |
X2 |
hg; fnifn |
+ |
(1 , n ),1hg; fnifn: |
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X2 |
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0 |
6 ; |
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w SLU^AE 2(ii) (TOGDA 1 = |
I NEOBHODIMO |
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g; fn |
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= 0 (n |
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1)) ISKOMOE |
RE[ENIE IMEET WID |
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f = |
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+ |
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, n ),1hg; fnifn; |
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GDE n (n 2 1 ) | PROIZWOLXNYE KONSTANTY.
4. rASSMOTRIM W ZAKL@^ENIE SLU^AJ WYROVDENNOGO QDRA. iMENNO,
n
PUSTX K (t; s) = P Pj (t)Qj (s), GDE fPj g; fQjg | NABORY LINEJNO NEZA-
j=1
WISIMYH FUNKCIJ. tOGDA
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(T f )(t) = |
[ Pj (t)Qj (s)]f (s) (ds) = |
j=1h |
f; Qj |
Pj (t): |
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Z |
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M X |
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iTAK, T | KONE^NOMERNYJ OPERATOR. oBOZNA^IM |
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xj = hf; |
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i; aij = Z Pi(t)Qj (t) (dt) = hPi; |
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i: |
Qj |
Qj |
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tOGDA ( ) PREWRATITSQ W URAWNENIE f(t) = g(t) + j |
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=1 xj Pj (t). sNOWA POD- |
STAWLQQ f(t) W ( ), POLU^IM |
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g(t) + |
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iPj (t) = g(t); |
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ILI j=1fxj , bj , i=1 aijxigPj (t) = 0. tAK KAK fPj g | LINEJNO NEZAWISI- |
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MYE FUNKCII, PRIHODIM K SLEDU@]EJ SISTEME URAWNENIJ OTNOSITELXNO |
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j n). tAKIM OBRAZOM, RE[ENIE |
NEIZWESTNYH xj : |
xj , i=1 aijxi = bj (1 |
INTEGRALXNOGO URAWNENIQP S WYROVDENNYM QDROM SWEDENO K RE[ENI@ SIS- TEMY LINEJNYH ALGEBRAI^ESKIH URAWNENIJ, USLOWIQ RAZRE[IMOSTI KOTO- ROJ HORO[O IZWESTNY IZ KURSA LINEJNOJ ALGEBRY.
|lementy nelinejnogo analiza w normirowannyh prostranstwah
zAKL@^ITELXNYJ RAZDEL KURSA MOVNO RASSMATRIWATX KAK WOZWRA]E- NIE K EGO NA^ALU W KONTEKSTE NORMIROWANNYH PROSTRANSTW. pO SU]ESTWU RE^X IDET• O LOKALXNOM IZU^ENII NELINEJNYH OTOBRAVENIJ POSREDSTWOM OTOBRAVENIJ LINEJNYH. w \TOM SMYSLE \TOT ZAKL@^ITELXNYJ RAZDEL MO- VET SLUVITX OTPRAWNOJ TO^KOJ DLQ NELINEJNOGO FUNKCIONALXNOGO ANA- LIZA, WKL@^A@]EGO W SEBQ, S ODNOJ STORONY, KLASSI^ESKOE WARIACIONNOE IS^ISLENIE, WOSHODQ]EE K TRUDAM |JLERA I lAGRANVA, S DRUGOJ STORONY, \TO | SOWREMENNYE RAZDELY FUNKCIONALXNOGO ANALIZA, INTENSIWNO RAZ- WIWA@]IESQ I DALEKO E]•E NE ZAWER[ENNYE• . zDESX MY OGRANI^IMSQ LI[X SAMYMI PERWONA^ALXNYMI SWEDENIQMI.
x256. pROIZWODNAQ fRE[E I E•E SWOJSTWA
1. pUSTX E; F | NORMIROWANNYE PROSTRANSTWA NAD POLEM (= C ILI
R); U( E) | OTKRYTO. oTOBRAVENIE A : U ! F NAZYWAETSQ DIFFE- |
RENCIRUEMYM W TO^KE x 2 U , ESLI SU]ESTWUET OGRANI^ENNYJ LINEJNYJ |
OPERATOR Lx 2 L(E; F ) TAKOJ, ^TO SPRAWEDLIWO ASIMPTOTI^ESKOE RAWEN- |
STWO |
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A(x + h) , A(x) = Lxh + o(h) (h ! ): |
fzDESX, KAK OBY^NO, RAWENSTWO r(h) = |
o(h) (h ! ) OZNA^AET, ^TO |
h! |
kkhkk |
g |
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lim |
r(h) |
= 0, SM. 103.1. oPERATOR Lx |
NAZYWAETSQ PROIZWODNOJ fRE- |
[E OTOBRAVENIQ A I OBOZNA^AETSQ TAKVE A0(x); Lxh | DIFFERENCIAL fRE[E OTOBRAVENIQ A W TO^KE x.
oTMETIM \LEMENTARNYE SWOJSTWA PROIZWODNOJ fRE[E.
2.eSLI A DIFFERENCIRUEMO W TO^KE x, TO PROIZWODNAQ fRE[E Lx OPREDELENA ODNOZNA^NO.
3.eSLI OTOBRAVENIE A DIFFERENCIRUEMO W TO^KE x, TO ONO NEPRE- RYWNO W \TOJ TO^KE.
4. eSLI OTOBRAVENIE A POSTOQNNO, TO EGO PROIZWODNAQ fRE[E RAWNA NUL@ (TO ESTX NULEWOMU LINEJNOMU OPERATORU).
IA0(x) = A. 2 L(E; F ), TO A DIFFERENCIRUEMO W KAVDOJ TO^KE x 2 E
6.eSLI A; B : U ! F DIFFERENCIRUEMY W TO^KE x 2 U, TO W \TOJ TO^KE DIFFERENCIRUEMY OTOBRAVENIQ A + B; A ( 2 ), PRI^•EM
(A + B )0(x) = A0(x) + B0(x); ( A)0(x) = A0(x):
7. pUSTX E; F; G | NORMIROWANNYE PROSTRANSTWA U ( E); V ( F ) OTKRYTY, A : U ! F DIFFERENCIRUEMO W TO^KE x 2 U; A(U ) V I B : V ! G DIFFERENCIRUEMO W TO^KE A(x). tOGDA W TO^KE x DIFFEREN- CIRUEMO OTOBRAVENIE B A, PRI^•EM (B A)0(x) = B0(A(x))A0(x).
dOKAZATELXSTWO UKAZANNYH UTWERVDENIJ PROWODITSQ PO IZWESTNYM SHEMAM (SM. x75) BEZ KAKIH-LIBO PRINCIPIALXNYH IZMENENIJ. tEM NE ME- NEE, REKOMENDUETSQ PROWESTI \TI DOKAZATELXSTWA W KONTEKSTE uPRAVNE- NIQ 10 (SM. NIVE).
p R I M E R Y. 8. pUSTX f (u; v) | NEPRERYWNAQ FUNKCIQ DWUH PEREMEN- NYH, OBLADA@]AQ NEPRERYWNOJ ^ASTNOJ PROIZWODNOJ fv0 (u; v). iSSLEDUEM
NA DIFFERENCIRUEMOSTX FUNKCIONAL : C [a; b] ! R, ZADANNYJ INTEGRA-
b
LOM (x) = Za f(t; x(t)) dt. iMEEM
(x + h) , (x) = Zabb |
[f (t; x(t) + h(t)) , f(t; x(t))] dt |
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= |
Za |
[f0 |
(t; x(t))h(t) + o(h(t))] dt (h |
! |
): |
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v |
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kROME TOGO, IZ RAWENSTWA |
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! |
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f (t; x(t) + h(t)) |
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f(t; x(t)) = f0(t; x(t))h(t) + o(h(t)) (h |
) |
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v |
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SLEDUET, ^TO OSTATOK o(h(t)) | NEPRERYWNAQ FUNKCIQ I PO\TOMU INTEGRAL
Z bo(h(t))dt KORREKTNO OPREDEL•EN. pRI \TOM
a
lim |
1 |
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bo(h(t))dt |
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lim |
b |
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o(h(t)) |
h(t) |
dt |
khk |
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Za |
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jh(t)j |
j jkhkj |
h! |
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h! Zab j |
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lim |
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jo(h(t))j dt = 0; |
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h! Za |
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jh(t)j |
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