(R1 |
; R2 MYSLQTSQ KAK PROSTRANSTWA KLASSOW \KWIWALENTNYH FUNKCIJ). |
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2. z A M E ^ A N I E. iME@T MESTO WKL@^ENIQ: C |
2 |
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. oTMETIM |
TAKVE PROSTOE, NO POLEZNOE UTWERVDENIE (!!): |
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3. dLQ f 2 R1: |
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2 f |
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x+2 |
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Z0 |
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2 R): |
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x157. tRIGONOMETRI^ESKIJ RQD fURXE
1. sISTEMA FUNKCIJ
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(1) |
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; p |
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cos x; p |
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sin x; : : : ; p |
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cos kx; p |
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sin kx; : : : |
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QWLQETSQ |
ORTONORMIROWANNOJ |
SISTEMOJ |
W UNITARNOM PROSTRANSTWE |
R2 [0; 2 ] |
(!!). tRIGONOMETRI^ESKIM RQDOM fURXE FUNKCII f |
2 R2 NA- |
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f |
ZOWEM• RQD fURXE FUNKCII f OTNOSITELXNO SISTEMY (1). |TOT RQD OBY^NO |
ZAPISYWAETSQ W WIDE |
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(2) |
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f (x) 2 + |
(ak coskx + bk sin kx); |
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k=1 |
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GDE |
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ak = |
1 |
Z02 f (t) coskt dt |
(k = 0; 1; 2; : : :), |
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(3) |
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bk = |
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Z02 f (t) sin kt dt |
(k = 1; 2; : : :): |
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nAPRIMER, ^LEN |
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RQDA |
fURXE, SOOTWETSTWU@]IJ FUNKCII |
p1 |
coskx |
(k 1), IMEET WID |
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Z0 |
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f (t)p |
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coskt dt p |
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coskx = ak cos kx: |
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z A M E ^ A N I Q. 2. eSLI f ABSOL@TNO INTEGRIRUEMA NA OTREZKE [0; 2 ], TO INTEGRALY (3) SHODQTSQ I, SLEDOWATELXNO, FORMALXNYJ RQD (1) MOVNO
SOPOSTAWITX FUNKCII IZ KLASSA R1 (A NE TOLXKO IZ KLASSA R2 ). |
f252 |
f |
3.mOVNO RASSMATRIWATX PERIODI^ESKIE FUNKCII S KAKIM-LIBO DRU-
GIM PERIODOM 2!. dELAQ PODSTANOWKU x = u!= , POLU^IM FUNKCI@ F (u) = f (u! ) 2 -PERIODI^ESKU@, ESLI f | 2!-PERIODI^ESKAQ. pO\TOMU W DALX- NEJ[EM OGRANI^IMSQ RASSMOTRENIEM 2 -PERIODI^ESKIH FUNKCIJ.
4.iDEQ PREDSTAWLENIQ FUNKCII f RQDOM fURXE PREDSTAWLQETSQ OSO- BENNO RAZUMNOJ, KOGDA ESTX OSNOWANIQ S^ITATX f(t) KOORDINATOJ KOLEB- L@]EJSQ TO^KI (t | WREMQ). rASSMOTRIM ^ASTNU@ SUMMU RQDA (1) | TRIGONOMETRI^ESKIJ POLINOM PORQDKA n:
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Ak cos(kt , 'k); |
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Sn (t) = |
2 |
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(ak coskt + bk sin kt) = 2 + |
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k=1 |
GDE Ak = (a2k +b2k )1=2; Ak cos 'k = ak; Ak sin 'k = bk. iTAK, KOLEBATELXNYJ PROCESS RASPADAETSQ W SUMMU GARMONIK S AMPLITUDAMI Ak I NA^ALXNYMI
FAZAMI 'k , SOOTWETSTWU@]IMI ^ASTOTAM k.
u P R A V N E N I Q. 5. pOKAVITE, ^TO ESLI f | ^•ETNAQ FUNKCIQ, TO PREDSTAWLENIE (2) PRIOBRETAET WID
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f(x) 2 |
ak = Z0 |
f(t) cos kt dt: |
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+ k=1 ak coskx; |
aNALOGI^NO, ESLI f | NE^•ETNAQ FUNKCIQ, TO
f (x) k1=1 bk sin kx; bk = |
2 Z0 f(t) sin kt dt: |
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6. eSLI NEKOTORYJ RQD PO SISTEME FUNKCIJ (1) SHODITSQ K FUNKCII f RAWNOMERNO NA OTREZKE [0; 2 ], TO ON QWLQETSQ EE• TRIGONOMETRI^ESKIM RQDOM fURXE.
x158. oSCILLQCIONNAQ LEMMA
pUSTX FUNKCIQ f ABSOL@TNO INTEGRIRUEMA NA R. eSLI RASSMOTRETX PROIZWEDENIE f(x) cos x, TO PRI BOLX[IH \TA FUNKCIQ SILXNO OSCIL- LIRUET, TAK ^TO PLO]ADI, OGRANI^ENNYE GRAFIKOM FUNKCII, LEVA]IE WY[E I NIVE OSI OX, KOMPENSIRU@TSQ. tO^NOE UTWERVDENIE TAKOWO:
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Z02 Dn (t , x)f (t)dt; |
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GDE Dn (s) = |
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n cos ks = |
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sin(n + 2)s |
| QDRO dIRIHLE PORQD- |
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kP=1 |
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sin |
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KA n (POSLEDNEE RAWENSTWO W EGO WYRAVENII MOVNO POLU^ITX METODOM, |
ISPOLXZOWANNYM W 141.3). zAMETIM, ^TO |
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(2) |
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Z0 |
Dn (s)ds = 1 + k=1 Z0 |
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cos ks ds = 1: |
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2. pOLU^IM TEPERX UDOBNOE WYRAVENIE DLQ OSTATKA APPROKSIMACII FUNKCII f EE• ^ASTNOJ SUMMOJ fURXE. iZ (1) I (2) IMEEM S U^•ETOM ^ETNOSTI•
QDRA dIRIHLE
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Sn(x) , f(x) = |
Z0 Dn (u)[f (x + u) , f(x)]du |
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Z0 |
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Dn (u) (f (x)) du; |
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u |
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GDE 2 (f(x)) |
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f(x + u) |
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2f (x) + f (x |
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u). tAKIM OBRAZOM, WOPROS |
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O SHODIMOSTI Sn(x) K f (x) SWODITSQ K IZU^ENI@ POWEDENIQ INTEGRALA
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Z0 |
Dn(u) u2 (f (x)) du. pREOBRAZUEM \TOT INTEGRAL. zAFIKSIRUEM |
Jn = |
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DLQ \TOGO ^ISLO (0 < < ). tOGDA |
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Jn |
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sin nu |
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cos nu |
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2 tg(u=2) |
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Z0 |
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sin nu |
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(f (x)) du + n(x): |
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R,1 |
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zDESX (WS@DU NIVE MY PI[EM |
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WMESTO |
+1) |
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n(x) |
= |
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Z1 |
cos nu |
h(u) 2u(f(x))du + |
sin nu |
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g(u) u2 |
(f (x)) du; |
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1Z |
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g(u) |
= |
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[ |
2 tg(u=2) , u |
] (0;) (u) + 2 |
tg(u=2) [;] (u); |
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h |
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[0;] |
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fUNKCII g; h | OGRANI^ENNYE I S KOMPAKTNYMI NOSITELQMI. w ^ASTNOSTI, g; h 2 R1(R). iZ OSCILLQCIONNOJ LEMMY TEPERX SLEDUET, ^TOn (x) ! 0 (n ! 1). nAPRIMER,
(3) |
Z |
sin nu |
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g(u) 2 |
(f(x))du = |
Z |
sin nu |
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g(u)f (x + u) du |
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, 2f (x)Z |
sin nu g(u) du + Z |
sin nu g(u)f(x , u) du. |
tAK KAK |
supp(g) = [0; ], |
FUNKCIQ |
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R |
PO PEREMENNOJ |
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g(u)f (x + u) 2 R1( ) ( |
u). pO\TOMU (158.2) |
Z |
sin nu g(u)f (x + u)du ! 0. aNALOGI^NO STREMQTSQ |
K NUL@ OSTALXNYE INTEGRALY W PRAWOJ ^ASTI (3). pODWEDEM• ITOG PRODE- |
LANNOJ RABOTY. |
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3. dLQ FUNKCII f 2 R1 IMEET MESTO PREDSTAWLENIE |
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1 |
sin nu |
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(4) |
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f(x) = Sn (x) |
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Z0 |
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u |
u(f (x))du |
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n(x); |
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PRI^•EM n (x) = o(1) (n ! 1).
4. z A M E ^ A N I E. w USLOWIQH P. 3 n (x) ! 0 (n ! 1) RAWNOMERNO NA KAVDOM OTREZKE [a; b], GDE FUNKCIQ f OGRANI^ENA. |TO OZNA^AET RAWNO- MERNU@ SHODIMOSTX RQDA fURXE NA TAKIH OTREZKAH. nIVE (SM. 164.3) MY DOKAVEM \TO UTWERVDENIE DLQ NEPRERYWNOJ KUSO^NO-GLADKOJ FUNKCII f .
x160. fUNKCII KLASSA Lip
1. wOPROS O SHODIMOSTI Sn (x) K f (x), KAK POKAZANO WY[E, SWODITSQ K IZU^ENI@ POWEDENIQ INTEGRALA W PRAWOJ ^ASTI (4) x159. mY WWEDEM• KLASS FUNKCIJ, DLQ KOTORYH ISSLEDUEMAQ ZADA^A POLU^AET IS^ERPYWA@- ]EE RE[ENIE. sKAVEM, ^TO FUNKCIQ f : [a; b] ! R PRINADLEVIT KLASSU Lip (0 < 1) | KLASSU lIP[ICA S POKAZATELEM , ESLI SU]ESTWUET KONSTANTA M > 0 TAKAQ, ^TO
( ) jf(x) , f(y)j Mjx , yj DLQ L@BYH x; y 2 [a; b]:
oTMETIM, ^TO Lip C [a; b] (!!).
p R I M E R Y. 2. eSLI f | NEPRERYWNAQ KUSO^NO-GLADKAQ NA OTREZKE [a; b], TO f 2 Lip 1. fpUSTX M TAKOWO, ^TO jf0(t)j M (a t b). tOGDA
jf (x) , f(y)j = jZ xf0(t)dtj Mjx , yj:g
y
= (tt ,,1)1
3. f(x) = jxj 2 Lip NA L@BOM OTREZKE [a; b]. DEL•ENNOSTI 0 < jyj < jxj I OBOZNA^IW t = jxy j, IMEEM
jjxj , jyj j jjxj , jyj j jx , yj jjxj , jyjj
pOSLEDNEE NERAWENSTWO W NAPISANNOJ CEPO^KE WERNO DLQ L@BOGO 2 (0; 1].g
4. pUSTX FUNKCIQ f 2 R PRINADLEVIT KLASSU Lip NA OTREZKE
[a; b]. tOGDA E•E TRIGONOMETRI^ESKIJ RQD fURXE SHODITSQ K f RAWNOMERNO
f1
NA KAVDOM OTREZKE [c; d] (a; b).
pRI DOSTATO^NO MALYH u TO^KI WIDA x u 2 [a; b] DLQ L@BYH x 2 [c; d], TAK ^TO
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j u2 (f (x))j jf (x + u) , f (x)j + jf (x) , f(x , u)j 2Mjuj ; |
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GDE M | KONSTANTA, FIGURIRU@]AQ W ( ). w SILU PREDSTAWLENIQ (4)x159 |
IMEEM |
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Sn(x) f (x) |
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sin nu |
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+ n (x) |
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jZ0 |
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(f (x)) du |
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j j |
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2M |
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2M |
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Z0 j uj |
du + j n (x)j |
= |
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+ j n (x)j: |
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pUSTX " > 0 PROIZWOLXNO. wYBEREM SNA^ALA > 0 TAK, ^TOBY |
2M |
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< |
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"=2, A ZATEM N TAK, ^TOBY |
n (x) < "=2 PRI n > N DLQ L@BOGO x |
2 |
[c; d] |
\TO MOVNO SDELATX W SILUj |
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j |
sLEDOWATELXNO |
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PRI |
n > N |
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( |
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159.4). |
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jSn(x) , |
f (x)j < " DLQ WSEH x 2 [c; d]: |
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5. eSLI f |
2 |
C | NEPRERYWNAQ KUSO^NO-GLADKAQ (NA [0; 2 ]) FUNKCIQ, |
TO E•E TRIGONOMETRI^ESKIJ RQD fURXE SHODITSQ K NEJ RAWNOMERNO. |
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pUSTX > 0 PROIZWOLXNO. w SILU P. 2 f 2 Lip 1 I OSTAETSQ• PRIMENITX |
P. 4 K OTREZKAM [a; b] = [, ; 2 + ]; [c; d] = [0; 2 ]: |
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u P R A V N E N I Q. 6. eSLI f |
2 C[a; b] I f0 (x) OGRANI^ENA NA (a; b), |
TO f 2 |
Lip 1. |
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7. kAKOW KLASS FUNKCIJ, UDOWLETWORQ@]IJ ( ) PRI > 1? |
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x161. pOLNOTA TRIGONOMETRI^ESKOJ SISTEMY FUNKCIJ tEPERX MOVNO DOKAZATX POLNOTU TRIGONOMETRI^ESKOJ SISTEMY FUNK-
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CIJ (1) x157 W UNITARNOM PROSTRANSTWE R2[0; 2 ]. |
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1. (A) sISTEMA FUNKCIJ f1; cosx; sin x; cos 2x; sin 2x; : : :g POLNA W PRO- |
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STRANSTWE C. |
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(B) |
sISTEMA FUNKCIJ |
f |
1; |
cos x; cos 2x; : : : |
g |
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POLNA W C |
[0; ], A TAKVE |
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W PODPROSTRANSTWE PROSTRANSTWA C, SOSTOQ]EM IZ ^•ETNYH FUNKCIJ. |
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(W) |
sISTEMA |
FUNKCIJ |
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sin x; sin 2x; : : : |
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POLNA |
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PROSTRANSTWE |
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C [0; ]jf (0) = f( ) = 0g, A TAKVE W PODPROSTRANSTWE PROSTRAN- |
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STWA C, SOSTOQ]EM IZ NE^•ETNYH FUNKCIJ. |
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dOKAVEM, |
NAPRIMER, (A). fUNKCIQ f |
2 |
C RAWNOMERNO NEPRERYWNA NA |
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e |
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POLIGON (TO ESTX NE- |
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[0; 2 ]. sLEDOWATELXNO, DLQ L@BOGO " > 0 NAJDETSQ• |
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PRERYWNAQ KUSO^NO-LINEJNAQ FUNKCIQ) |
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e |
C TAKOJ, ^TO |
max |
f (x) |
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< "=2. |
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2 |
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0 x 2 j |
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kAVDYJ POLIGON QWLQETSQ NEPRERYWNOJ KUSO^NO-GLADKOJ |
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j |
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I W SILU |
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e |
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PRI DOSTATO^NO |
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FUNKCIEJ |
, |
160.5 j |
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(x),Sn(x)j < "=2 (x |
2 [0; 2 ]) |
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BOLX[OM |
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ZDESX |
Sn (x) | |
^ASTNAQ SUMMA RQDA fURXE DLQ FUNKCII |
). |
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n ( |
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tOGDA DLQ L@BOGO x 2 [0; 2 ] |
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jf (x) , Sn(x)j jf (x) , (x)j + j (x) , Sn(x)j < ": |
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oTS@DA kf , Snk ", GDE k k | NORMA W C: |
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2. |
tRIGONOMETRI^ESKAQ SISTEMA FUNKCIJ |
(1) x157 |
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POLNA W |
R2, |
I SLE |
- |
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DOWATELXNO TRIGONOMETRI^ESKIJ RQD fURXE FUNKCII |
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SHODITSQ |
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K \TOJ FUNKCII PO NORME k k2. |
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f |
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2 R2 f |
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f |
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pUSTX |
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2 R2 . |
w SILU |
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153.6 |
SU]ESTWUET FUNKCIQ |
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' 2 |
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C |
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TAKAQ |
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2 |
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^TO |
Z0 j |
f (x) |
'(x) 2 dx < "2. w SILU P. |
1 |
MOVNO PODOBRATX TRIGONO- |
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, f |
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j |
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METRI^ESKIJ POLINOM Sn (x) TAKOJ, ^TO k' , Snk[0;2 ] < |
. zNA^IT, |
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2 |
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kf , Snk2 kf , 'k2 + k' , Snk2 < 2": |
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rAWENSTWO pARSEWALQ |
]. |
dLQ |
f 2 R2 |
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3. [ |
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1 |
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a02 |
f1 |
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2 |
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Z0 jf (x)j |
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dx = |
2 |
+ k=1(jakj |
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+ jbkj |
): |
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X |
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|TO SLEDSTWIE 155.7(G). >
z A M E ^ A N I E. 4. l. kARLESON DOKAZAL (1966), ^TO TRIGONOMETRI-
^ESKIE RQDY fURXE FUNKCIJ IZ R2 SHODQTSQ P.W. |
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f |
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5. p R I M E R. rASSMOTRIM 2 -PERIODI^ESKU@ FUNKCI@ f (x) TAKU@, |
^TO |
1; |
ESLI 0 < x < , |
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f(x) = 8 |
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,1; |
ESLI , < x < 0, |
k |
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2 |
2 |
< |
0; |
ESLI x = 0; |
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w SILU |
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= Z0 sin kx dx = , k ((,1) , |
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157.5 ak = 0 (k = 0; 1; : : :); bk |
1)(k |
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N). tAKIM OBRAZOM, f (x) |
= |
4 |
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1 |
sin(2k , 1)x (x |
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R). rQD W |
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2 |
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R |
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kP=1 |
2k , 1 |
2 |
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PRAWOJ ^ASTI SHODITSQ K f(x) W |
nf0; ; 2 ; : : :g SOGLASNO 160.5. w |
OSTALXNYH TO^KAH SHODIMOSTX RQDA K 0 | ZNA^ENI@ FUNKCII f (x) DLQ x = 0; ; 2 ; : : : | O^EWIDNA.
x162. pOLNOTA SISTEMY POLINOMOW W C [a; b]
1.sISTEMA FUNKCIJ f1; x; x2; : : :g POLNA W C [a; b].
2.oPREDELIM SNA^ALA WE]ESTWENNYE POLINOMY Qn (x) STEPENI n (ONI NAZYWA@TSQ POLINOMAMI ~EBY[EWA) RAWENSTWAMI cosnt = Qn (cost), TAK
^TO Qn(x) = cos(n arccos x) (n = 0; 1; 2; : : :). w ^ASTNOSTI, Q0 (x) = 1,
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Q1(x) = x; Q2(x) = 2x2 ,1. pRI PROIZWOLXNOM n DLQ POLU^ENIQ POLINOMA |
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Qn MOVNO WOSPOLXZOWATXSQ TOVDESTWOM |
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cosnt + i sin nt = |
eint = (eit)n = (cos t + i sint)n |
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n |
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= |
kP=0 in,k nk cosk t sinn,k t: |
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a0 |
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iTAK |
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KAVDYJ ^ETNYJ TRIGONOMETRI^ESKIJ |
POLINOM |
Tn(t) |
= |
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• |
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2 |
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+ n |
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ak cos kt PODSTANOWKOJ t |
= arccos x, KOTORAQ GOMEOMORFNO OTOBRA- |
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P |
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k=1 |
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VAET OTREZOK [0; ] NA OTREZOK [,1; 1], PREOBRAZUETSQ W ALGEBRAI^ESKIJ |
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POLINOM |
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Pn(x) = Tn (arccosx) = a0 |
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+ |
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ak Qk (x): |
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2 |
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k=1 |
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oBRATNO, L@BOJ WE]ESTWENNYJ POLINOM Pn(x) = a0 + a1x + : : : + anxn PODSTANOWKOJ x = cos t PREOBRAZUETSQ W ^•ETNYJ TRIGONOMETRI^ESKIJ PO-
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LINOM Tn (t) = Pn(cos t) = |
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k coskt. |TO QSNO, ESLI U^ESTX TOV- |
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2 |
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k=1 |
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DESTWO |
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cosk t = ( |
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(eit + e,it))k = 2,k |
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s!ei(2s,k)t = 2,k |
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s! cos(2s , k)t: |
2 |
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s=0 |
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3. pEREHODIM K DOKAZATELXSTWU P. 1. sLU^AJ 1: [a; b] = [,1; 1]. dLQ WSQKOJ f 2 C [,1; 1] FUNKCIQ f (cost) NEPRERYWNA NA [0; ] I SOGLASNO 161.1 SU]ESTWUET TRIGONOMETRI^ESKIJ POLINOM Tn(t) = Pn(cos t) TAKOJ, ^TO
jf(cos t) , Tn (t)j < " PRI WSEH t 2 [0; ]. sLEDOWATELXNO, jf(x) , Pn(x)j < " |
PRI WSEH jxj 1: |
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b ,2 |
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sLU^AJ 2 (OB]IJ). pREOBRAZOWANIE x = a + |
(z + 1) PEREWODIT |
[,1; 1] W [a; b], FUNKCIQ F (z) = f (a + |
b ,2 |
a |
(z + 1)) NEPRERYWNA NA [,1; 1], |
ESLI f 2 C [a; b]. pRI \TOM W SILU SLU^AQ 1 SU]ESTWUET ALGEBRAI^ESKIJ |
POLINOM Pn (x) TAKOJ, ^TO kF ,Pnk[,1;1] < ". oBRA]AQ PODSTANOWKU, IMEEM |
k |
f |
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Rn |
k |
[a;b] < "; GDE Rn (x) |
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Pn (2(x , a) |
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1): |
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b , a |
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x163. kOMPLEKSNAQ FORMA RQDA fURXE
1. wO MNOGIH OTNO[ENIQH UDOBNA KOMPLEKSNAQ FORMA TRIGONOMETRI- ^ESKOGO RQDA fURXE. ~TOBY POLU^ITX E•E, ZAMETIM, ^TO
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ak |
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+ e, |
ikx |
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bk |
ikx |
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ikx |
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ak cos kx + bk sin kx |
= |
2 |
(e |
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= ckeikx + c ke,ikx; |
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, |
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ikt |
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GDE ck = 2 |
(ak , ibk ) = 2 Z0 |
f (t)e, |
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dt (k 2 Z). eSLI f |
| WE]ESTWEN- |
NAQ FUNKCIQ, TO ck = c,k . iTAK, PREDSTAWLENIE (2) x157 PREOBRAZUETSQ K KOMPLEKSNOJ FORME RQDA fURXE
f(x) +1 ckeikx:
X
,1
rQD W PRAWOJ ^ASTI MOVNO RASSMATRIWATX KAK RQD fURXE FUNKCII f OTNOSITELXNO ORTONORMIROWANNOJ SISTEMY f(2 ),1=2 eikxgk2Z, POLNOJ W UNITARNOM PROSTRANSTWE R2.
f 260
