Числовые и функциональные ряды (Лекции для ИУ), страница 4

eSLI VE ON RASHODITSQ PRI NEKOTOROM z = z2 6= θ,TO ON RASHODITSQ ∀ z ∈ C : |z| > |z1 |.dOKAZATELXSTWO.pUSTX RQD {ak z1k }k>0 SHODITSQ.tOGDA PO NEOBHODIMOMU PRIZNAKU∃ lim ak z1k = θ, TO ESTX (∀ ε > 0)(∃ N (ε) > 0) : |an z1n | < ε, ∀ n > N (ε). pUSTX z ∈ C I |z| < |z1 |.

tOGDAk→∞ n n z4 |z|nnn z = ε · q n , ∀ n > N (ε), GDE q =< 1 OTKUDA I SLEDUET|an · z | = an · z1 ·=|a·z|n1 z1z1|z1 |ÌÃÒÓÌÃÒÓkABSOL@TNAQ SHODIMOSTX RQDA {ak z k }k>0 PRI |z| < |z1 |.pUSTX TEPERX z∗ ∈ C I |z∗ | > |z2 |. eSLI BY RQD {ak z∗k }k>0 SHODILSQ , TO, SOGLASNO UVE DOKAZANNOMU,W TO^KE z2 ON BY SHODILSQ ABSOL@TNO, T.K. |z2 | < |z∗ |.

tAKIM OBRAZOM ISHODNAQ POSYLKA QWLQETSQLOVNOJ I W L@BOJ TO^KE z∗ ∈ C : |z| > |z2 | RASSMATRIWAEMYJ RQD RASHODITSQ.sLEDSTWIQ IZ TEOREMY aBELQ.ÔÍ-12k→∞ÔÍ-121. dLQ KAVDOGO KOMPLEKSNOGO STEPENNOGO RQDA {an z k }k>0 W C SU]ESTWUET KRUG S CENTROM W θ IRADIUSOM R, W KAVDOJ WNUTRENNEJ TO^KE KOTOROGO ISHODNYJ RQD SHODITSQ ABSOL@TNO I RASHODITSQ WKAVDOJ WNE[NEJ TO^KE.p2. R = lim |ak | |ak+1 | = lim 1 k |ak |.k→∞4ÌÃÒÓÌÃÒÓ3. eSLI HOTX W ODNOJ TO^KE OKRUVNOSTI lR = {z ∈ C : |z| = R} KRUGA SHODIMOSTI ISHODNYJSTEPENNOJ RQD {αk z k }k>0 SHODITSQ ABSOL@TNO, TO ON SHODITSQ ABSOL@TNO W KAVDOJ TO^KE OKRUVNOSTIlR .dOKAZATELXSTWO.

pUSTX z0 ∈ lR I RQD {ak z0k }k>0 SHODITSQ ABSOL@TNO, T.E. SHODITSQ ZNAKOPOLOVITELXNYJ ^ISLOWOJ RQD {|ak ||z0 |k }k>0 .nO TOGDA ∀ z ∈ lR IME@T MESTO RAWENSTWAkkkk|ak z | = |ak ||z| = |ak ||z0 | = |ak | R , OTKUDA I SLEDUET ISKOMYJ REZULXTAT.zk1√1. pUSTX αk (z) =, T.E. ak = √. tOGDA R = lim |ak | |ak+1 | = 2 Ik→∞2k · k k k>1k k · 2k1|zk |= 3/2 I ISHODNYJ RQD SHODITSQ ABSOL@TNOlR = {z ∈ C : |z| = 2}. eSLI z ∈ lR , TO |αk · z k | = √kkk k·2PRI |z| 6 2 I RASHODITSQ PRI |z| < 2.ÔÍ-12ÔÍ-12pRIMER4.

eSLI HOTX W ODNOJ TO^KE OKRUVNOSTI lR KRUGA SHODIMOSTI ISHODNYJ RQD RASHODITSQ PO NEOBHODIMOMU PRIZNAKU, TO ON RASHODITSQ W KAVDOJ TO^KE \TOJ OKRUVNOSTI – DOKAZATELXSTWO ANALOGI^NODOKAZATELXSTWU SLEDSTWIQ 3.pRIMERk→∞k{ak z0 }k>0ÌÃÒÓÌÃÒÓ2. rQD {z k }k>0 SHODITSQ ABSOL@TNO PRI |z| < 1 I RASHODITSQ PRI |z| > 1, T.K.lR = {z ∈ C : |z| = 1} I ∀ z ∈ lR ∃ lim |z|k = 1 6= 0.4. eSLI ∃ z0 ∈ lR I RQDSHODITSQ USLOWNO, TO W TO^KAH OKRUVNOSTI lR ISHODNYJ RQDMOVET KAK RASHODITXSQ, TAK I SHODITXSQ USLOWNO. kzRADIUS SHODIMOSTI R = 1, T.E. lR = {z ∈ C : |z| = 1}.

w L@BOJ3. dLQ RQDAk k>1|z|k1TO^KE z ∈ lR= , T.E. W L@BOJ TO^KE OKRUVNOSTI KRUGA SHODIMOSTI ISHODNYJ RQD ABSOL@TNOkkSHODITXSQ NE MOVET, NO NEOBHODIMYJ PRIZNAK WYPOLNQETSQ.z k 1w TO^KE z = +1 ∈ lR ISHODNYJ RQD RASHODITSQ, TO KAK= .k z=+1 kz k (−1)kw TO^KE z = −1 ∈ lR ISHODNYJ RQD SHODITSQ USLOWNO, TAK KAK=.k z=−1kÔÍ-12ÔÍ-12pRIMERÌÃÒÓÔÍ-1211ÌÃÒÓÔÍ-12ÌÃÒÓÌÃÒÓÌÃÒÓÔÍ-12ÔÍ-12ÌÃÒÓÔÍ-12oPREDELENIEÌÃÒÓrQDY fURXEfUNKCI@ y = f (x) NAZYWA@T INTEGRIRUEMOJ S KWADRATOM NA [a; b] ⊂ R,1.f 2 (x)dx<sOWOKUPNOSTX WSEH FUNKCIJ, INTEGRIRUEMYH S KWADRATOM NA+∞.ÌÃÒÓESLI ∃ZbÔÍ-125.

eSLI R – RADIUS SHODIMOSTI, TO W L@BOM KRUGE Kr = {z : |z| 6 r < R} ISHODNYJ STEPENNOJRQD SHODITSQ RAWNOMERNO.dOKAZATELXSTWO POLNOSTX@ ANALOGI^NO WE]ESTWENNOMU SLU^A@.a[a; b] ⊂ R1 ,OBOZNA^A@T L2 [a; b].sWOJSTWA FUNKCIJ IZ L [a; b].2ÔÍ-12eSLI FUNKCII f (x), ϕ(x)∈INTEGRIRUEMANA [a, b],T.K.aaaaÔÍ-12L2 [a, b], TO FUNKCIQ f (x) · ϕ(x) – ABSO|f (x) · ϕ(x)|60, 5{f 2 (x) + ϕ2 (x)} IL@TNOZ b ZbZbZbZb111222 f (x)ϕ(x)dx 6 |f (x)ϕ(x)|dx 6{f (x) + ϕ (x)}dx =f (x)dx +ϕ2 (x)dx < ∞.222(1).a(2). L [a, b] – LINEJNOE PROSTRANSTWO OTNOSITELXNO STANDARTNYH OPERACIJ SLOVENIQ FUNKCIJ I IHUMNOVENIQ NA ^ISLO.

dEJSTWITELXNO, DLQ L@BYH KONE^NYH WE]ESTWENNYH ^ISEL λ, µ I DLQ L@BYHf (x), ϕ(x) ∈ L2 [a; b] IMEEM:2ZbZbaÌÃÒÓ{λf (x)ϕ(x) + µϕ(x)} dx ≡|λ2 f 2 (x) + 2λµf (x)ϕ(x) + µ2 ϕ2 (x)|dx 6aZb6|λ|2 {f 2 (x) + 2|λµ||f (x)||ϕ(x)| + |µ|2 ψ 2 (x)}dx < +∞.ÌÃÒÓ2aÔÍ-12ÔÍ-12tAKIM OBRAZOM, ESLI |λ| < ∞, |µ| < ∞, f (x) ∈ L2 [a; b], ϕ(x) ∈ L2 [a; b], TO (λf (x) + µϕ(x)) ∈ L2 [a; b] IL2 [a; b] – LINEJNOE PROSTRANSTWO.oPREDELENIE2. dWE FUNKCII f (x) I ϕ(x) IZ L2 [a, b] NAZYWA@T \KWIWALENTNYMI I PI[UTZbf (x) ∼ ϕ(x), ESLI {f (x) − ϕ(x)}2 dx = 0.zAME^ANIQ K OPREDELENI@ 2.aÌÃÒÓÌÃÒÓ(1).

eSLI f (x), ϕ(x) ∈ L2 [a; b] I f (x) ∼ ϕ(x), TO ZNA^ENIQ \TIH FUNKCIJ MOGUT RAZLI^ATXSQ LI[X NAKONE^NOM MNOVESTWE TO^EK IZ [a; b].(2). eSLI W L2 [a; b] PONQTIE ”RAWENSTWO” ZAMENITX PONQTIEM ”\KWIWALENTNOSTX”, TO SKALQRNOE PROIZWEDENIE W \TOM LINEJNOM PROSTRANSTWE MOVET BYTX WWEDENO STANDARTNYMI SPOSOBOM, TAK KAK DLQZb2L@BYH f (x), ϕ(x) ∈ L [a, b] OPREDELENO WE]ESTWENNOE ^ISLO (f ; ϕ) =f (x)ϕ(x)dx I PRI \TOM:f 2 (x)dx > 0 I (f ; f ) = 0 ⇐⇒ f (x) ∼ 0;aZbf (x)ϕ(x)dx =a12ÌÃÒÓÔÍ-12aϕ(x)f (x)dx = (ϕ; f );ÔÍ-12β) (f ; ϕ) =ZbÌÃÒÓα) (f ; f ) =ÔÍ-12ÔÍ-12aZbÌÃÒÓÌÃÒÓZbλf (x)ϕ(x)dx = λaÔÍ-12ÌÃÒÓf (x)ϕ(x)dx = λ f (x); ϕ(x) ;aZbZb{f (x) + ϕ(x)}ψ(x)dx =δ) (f + ϕ; ψ) =aZbf (x)ψ(x)dx +aϕ(x)ψ(x)dx = (f ; ψ) + (ϕ; ψ).aÔÍ-12γ) (λf ; ϕ) =ÔÍ-12ÌÃÒÓÔÍ-12ZbÌÃÒÓÌÃÒÓ(3). tAK KAK LINEJNOE PROSTRANSTWO L2 [a, b] S WWEDENNYM SKALQRNYM PROIZWEDENIEM QWLQETSQ EWKLIDOWYM PROSTRANSTWOMs , TO W NEM STANDARTNYM OBRAZOM MOVET BYTX WWEDENA NORMA:Z bpf 2 (x)dx, ∀ f (x) ∈ L2 [a, b].kf k = (f ; f ) ≡apRI \TOM WSEpAKSIOMY NORMY:α) kf (x)k =(f ; f ) > 0 ∧ kf k = 0 ⇐⇒ f (x) ∼ 0;β) kλf (x)k = |λ| · kf (x)k;ÔÍ-12ÔÍ-12γ) kf (x) + ϕ(x)k 6 kf (x)k + kϕ(x)k WYPOLNQ@TSQ AWTOMATI^ESKI.(4).

w EWKLIDOWOM PROSTRANSTWE L2 [a, b] SO STANDARTNOJ NORMOJ METRIKU (RASSTOQNIE), KAK PRAWILO,WWODQT SLEDU@]IM OBRAZOM:qRp4bρ(f ; ϕ) = kf (x) − ϕ(x)k = (f − ϕ; f − ϕ) ≡{f (x) − ϕ(x)}2 dx.apRI \TOM WSE AKSIOMY METRIKI WYPOLNQ@TSQ AWTOMATI^ESKI:α) ρ(f ; ϕ) > 0 I (f ; ϕ) = 0 ⇐⇒ f (x) ∼ ϕ(x);ÌÃÒÓÌÃÒÓβ) ρ(f ; ϕ) = ρ(ϕ; f );γ) ρ(f ; ϕ) 6 ρ(f ; ψ) + ρ(ϕ; ψ).(5). w EWKLIDOWOM PROSTRANSTWE L2 [a, b] SO STANDARTNOJ NORMOJ I STANDARTNOJ METRIKOJ PONQTIE SHODIMOSTI TAK VE WWODQT STANDARTNYM SPOSOBOM: PUSTX f (x) ∈ L2 [a, b] I {fk (x)}k>1 ∈ L2 [a, b].

fUNKCI@f (x) NAZYWA@T PREDELOM POSLEDOWATELXNOSTI {fk (x)}k>1 I PI[UT lim fk (x) = f (x), ESLI SU]ESTWUETk→∞qRb{fk (x) − f (x)}2 dx = 0.lim ρ(fk , f ) = 0, TO ESTX, ESLI ∃ limaÔÍ-12k→∞ÔÍ-12k→∞zADA^A O NAILU^[EJ APPROKSIMACII W L [a, b]2{ψ (x)}∞∈ L2 [a, b] – NEKOTORAQ ORTONORMIROWANNAQ SISTEMA FUNKCIJ, T.E.k=1 knX0 ; k 6= m2(ϕk ; ϕm ) =, f (x) ∈ L [a, b] I Sn (x) =Ck ϕk (x). dLQ FIKSIROWANNOGO n NEOBHODIMO1 ; k=mpUSTXk=1NAJTI {Ck }nk=1 ∈ R1 TAKIE, ^TO ρ(f ; Sn ) ≡ kf (x) − Sn (x)k → min.ÌÃÒÓrE[ENIE.2ρ (f, Sn ) ≡ kf (x) − Sn (x)k ≡ (f (x) − Sn (x); f (x) − Sn (x)) ≡ f (x) −−2k=1m=1k=1Ck (f ; ϕk ) +nXk=1 m=1n nno XX222Ck ≡ kf k +(f ; ϕk ) − 2Ck (f ; ϕk ) + Ck −(f ; ϕk )2 ≡ kf k2 −k=1k=1−nX(f ; ϕk )2 +k=1ÔÍ-12m=1nXCk Cm (ϕk ; ϕm ) ≡ kf k2 −k=1nXÔÍ-12ÔÍ-12nnnXXXCm ϕm (x) ≡ (f ; f ) −Ck (ϕk ; f ) −Cm (f ; ϕm ) +2Ck − (f ; ϕk ) → min ⇐⇒ Ck ≡ (f ; ϕk ), ∀ k = 1 : n{Ck }k=113ÌÃÒÓnXCk ϕk (x); f (x)−k=1nXÔÍ-12−nXÌÃÒÓ2ÌÃÒÓ{Ck }ÌÃÒÓÌÃÒÓÌÃÒÓÔÍ-12oB]IE ZAME^ANIQ.ÌÃÒÓÔÍ-12ÔÍ-12tAK(2).KAKPRIk=1NAILU^[EJ0 6 ρ2 (f, Sn ) ≡ kf (x) − Sn (x)k2 ≡ kf k2 −nXAPPROKSIMACII(f ; ϕk )2 .TOtAKIM OBRAZOM ∀ n > 1 IMEET MESTOnXNERAWENSTWO bESSELQ.k=1ÌÃÒÓÌÃÒÓ(f ; ϕk ),k=1(f ; ϕk )2 , IZWESTNOE KAKk=1( n)X(3).

tAK KAK POSLEDOWATELXNOSTX(f ; ϕk )2NERAWENSTWO kf k >2≡CkÔÍ-12(1). eSLI {ϕk }nk=1 ∈ L2 [a, b] – ORTONORMIROWANNAQ SISTEMA FUNKCIJ, TO ∀ f (x) ∈ L2 [a, b] EE NAILU^[AQnXAPPROKSIMACIQ Sn (x) OPREDELQETSQ RAWENSTWOM: Sn (x) =(f ; ϕk ) · ϕk (x).QWLQETSQ NEUBYWA@]EJ I, SOGLASNO NERAWEN-n>1STWU bESSELQ, OGRANI^ENA SWERHU, TO PRI NALI^II S^ETNOJ ORTONORMIROWANNOJ SISTEMY FUNKCIJ∈ L [a, b] NERAWENSTWO bESSELQ DOPUSKAET OBOB]ENIE2k=1oPREDELENIE 3.oRTONORMIROWANNU@ SISTEMU FUNKCIJ∞X(f ; ϕk )2 6 kf k2 .k=1∞ϕk (x) k=1 ∈∞XzAME^ANIQ K OPREDELENI@ 3.L2 [a, b] NAZYWA@T ZAMKNUTOJ(f ; ϕk )2 ≡ kf k2 .(POLNOJ), ESLI ∀ f (x) ∈ L2 [a, b] IMEET MESTO RAWENSTWO pARSEWALQ:ÔÍ-12ϕk (x)∞ÔÍ-12k=1ÌÃÒÓÌÃÒÓ(1). wSQKAQ ZAMKNUTAQ W L2 [a, b] cISTEMA ORTONORMIROWANNYH FUNKCIJ OBRAZUET ORTONORMIROWANNYJ2BAZIS, T.K.

W PROTIWNOM SLU^AE ∃ ϕ0 (x) ∈ L [a, b] :kϕ0 (x)k = 1 ∧ (ϕ0 ; ϕk ) ≡ 0, ∀ k > 1 . nO∞X2(ϕ0 ; ϕk )2 ≡ 0 I MY PRIHODIM K PROTIWORE^I@.SOGLASNO RAWENSTWU pARSEWALQ 1 = kϕ0 k =1ÔÍ-12∞Xn→∞k=1n→∞k=14ÌÃÒÓk>1k=1(f ; ϕk )ϕk (x) NAZYWA@T SUMMOJ RQDA fURXEk=1tRIGONOMETRI^ESKIJ RQD fURXEpRIMER 1. sISTEMA TRIGONOMETRI^ESKIH FUNKCIJ 1; cos(kx); sin(kx)∞k=1ÌÃÒÓ(f ; ϕk )ϕk (x) ∼ f (x).

pRI \TOM FUNKCI@ Sf (x) =k=1no(f ; ϕk )ϕk (x)∞XÔÍ-12∞(2). eSLI ϕk (x) k=1 ∈ L2 [a, b] – ORTONORMIROWANNYJ BAZIS W L2 [a, b], TO ∀ f (x) ∈ L2 [a, b] SU]ESTWUET ∞S^ETNOE MNOVESTWO ^ISEL Ck k=1 , GDE Ck ≡ (f ; ϕk ), ∀ k > 1 – KO\FFICIENTY fURXE (KOORDINATYnX∞4Ck ϕk (x)FUNKCII f (x) W BAZISE ϕk (x) k=1 ) TAKOE, ^TO POSLEDOWATELXNOSTX {Sfn (x)}n>1 , Sfn (x) =k=1SHODITSQ K f (x) W L2 [a, b]. rQD (f ; ϕk ) · ϕk (x) k>1 NAZYWA@T RQDOM fURXE FUNKCII f (x) I PRI \TOM∞nnonXXX222(f ; ϕk )ϕk (x) ≡ lim f (x) −(f ; ϕk )ϕk (x) = lim kf (x)k −(f (x); ϕ(x)) = 0, T.E.f (x) −C+2πZC+2π 1 n o11 · cos(kx)dx = + sin(kx)≡sin k(C + 2π) − sin(kC) ≡ 0 ;kkC1; cos(kx) =C+2πo1n 11 · sin(kx)dx = − cos(kx)= − cos k(C + 2π) − cos(kC) ≡ 0 ;kkC1; sin(kx) =C14ÔÍ-12ÌÃÒÓÔÍ-12CC+2πZÌÃÒÓÔÍ-12ÔÍ-12QWLQETSQ ORTOGONALXNOJ NA OTREZKE [C; C + 2π] DLQ L@BOGO KONE^NOGO WE]ESTWENNOGO ^ISLA C, TAK KAKÌÃÒÓÌÃÒÓÌÃÒÓÔÍ-12ÔÍ-12(k 6= n) =⇒ sin(kx); sin(nx) =C+2πZcos(kx) cos(nx)dx =CC+2πZCC+2πZsin(kx) sin(nx)dx =C+2πZsin(kx) cos(nx)dx =sin (k + n)x + sin (n − k)x dx ≡ 0, (∀ k > 1) ∧ÌÃÒÓÌÃÒÓcos (k − n)x − cos (k + n)x dx ≡ 0 ;CCC+2πZsin(kx); cos(nx) =cos (k + n)x + cos (k − n)x dx ≡ 0 ;ÔÍ-12(k 6= n) =⇒ cos(kx); cos(nx) =ÌÃÒÓÔÍ-12C+2πZCC∧ (∀ n > 1).pRI \TOM k1k2 =C+2πZ12 dx = 2π ;Ck sin(kx)k2 =C+2πZ1cos (kx)dx =22CC+2πZsin2 (kx)dx =12CC+2πZC1 + cos(2kx) dx ≡ π ;ÔÍ-12ÔÍ-12k cos(kx)k2 =C+2πZ1 − cos(2kx) dx ≡ π .C∞1cos(kx) sin(kx)√ ; √tAKIM OBRAZOM; √– ORTONORMIROWANNAQ SISTEMA NA OTREZKEππ2πk=1[C; C + 2π], ∀ s ∈ R1 .