Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 38
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JJ:oKa3aTh,Xx EX HaHJiyqwttH:TaKOH, qTo seKTOpqrn ecJJ11X,B YHHTapHOM npocrpaHCTBeF-5IBJrneTc5I JJHHeHHhIM3JieMeHTx - PF(x)PF(x))J.JIRxBoproroHaJieH mo-3aMKHyrne BhmyKJioe MHO)l(ecrnoTO MHO)J(ecrnoF5IBJI5IeTC5l MHO)l(eCT-BOM cyw:ecTBOBaHHJl H e)J.HHCTBeHHOCTH.17.20.
IlycTb x 1•• •••x,, -BeKTOpbI yHttTapHoro npocipaHCTBaX.Mampuiieii I'paMa cttcTeMhI 3THX seKTopos Ha3hrnaerc5l Marpm:wG(xr. ... , x,,) = (11 1) , rnKaJl, qrn T]I} = (x;, xj), iJ = 1, 2, ... , n. JJ:oKa3aTh,qro eCJIH CHCTeMa BeKTOpOBmo6oro seKTopaXr, ... , x,, JIHHeHHO He3aBHCHMa, TO )J.JIJ[x E X cnpase.nJIHBO_ ))£( x, L( x, , ... ,.xii17.21.1)JJ:oKa3aTb, qrn ecJIHX= det G(x, x" ... , x,,) .det G(xp ... , x,,)- ytt11rapttoe npocrpaHcrno, rn:CHCTeMa BeKTOpOB Xr, ...
,x,,JIHHeHHO He3aBHCHMa TOf)J.a I1TOJibKO TOr.na, Kor.nadet2)G(x 1,. • •,x,,) += O;)J.JIJ[ mo6oH: CHCTeMhl BeKTOpOB Xr. ... ,detx,, ExG(xr. ... , x,,) :'.'.: 0,3H3K paBeHCTBa HMeeT MeCTO TOf)J.a I1 TOJibKO Tor.na, KOf)J.a CHCTeMaBeKTOpOByx, , ... , x,, JlHHei:IHO 33BHCHMa.K333HHOe HepaBeHCTBO 5lBJrneTC5I 0606iljeHUeJ11Hepa8e11cmeaKoiuu-EyHRKOBCK020;3))J.Jl5I mo6oH: JlHHeHHO He33BHCHMOH CHCTeMhI BeKTOpOBx,, ... , x" E Xtt .UJI5I mo6oro 111,1 :::;/11 :::;n,I'lla6a 17248det G(x 1,••• ,x,,):; det G(x 1,••• ,x 111 ) det G(x111+1, ...
, x,,).[(oKa3aTb, qTo 3HaK paBeHCTBa 3)leCb HMeeT MeCTO TOr)la 11 TOJlbKOTOr)la, Kor.na BC5IKl1H BeKTOp Xk,rnpy x 1, mk :;111, opTOroHaJieH BC5IKOMY BeK-< l :; n;4) )lJ15! mo6oH: CHCTeMbI BeKTOpOB Xi, ... , x,,detExG(xi, ... , x,,) :; (xi, x 1)(x2, x2) ... (x,,, x,,).17.22. IlycTb A =(a;) -KBa,uparna5! MaTJJm~a nop5!,uKan ENcKOMnJieKCHbIMH 3JieMeHTaM11 .
.[(OKaJaTb, qTo2222I det A 1 :::; L~=1 I a,k 1 L~=1 I a2k 1 ···L~J a,,k 1 ·.[(oKa3aTb, qTo 3HaK paBeHCTBa 3,UeCb MMeeT MeCTO TOr.ua H TOJlbKOTor.ua, Kor,ua )lJI5I mo6hIX17 .23. IlycTb a, b -i,j = 1, .. ., n, i:;;e j,cpttKCHposaHHhie .ueifcrnmeJibHbie q11c11a,nEN.Cpe.n11 scex TJJl1rDHOMeTp11qecK11x noJIHHOMOBT,, (x)= a,, cos 1zx + b,, sin nx + a,,_1 cos (11 - l)x + b,, _ 1 sin (n - l)x +... +a 1 cosx+b 1 sinx+a0Hai1Ttt TOT, .UJI5! KOToporo HHTerpaJIf:JT IT,, (x) IPdx, p~ 1,npHHttMa-eT Ha11MeHblllee 3HaqeHHe ..[(pyrnM11 c110BaMM 3TY 3a,uaqy MO)KHO ccpopMy11ttpoBaTb TaK. Hai1TMH311JIYqllJee npH6JIH)KeH11e ¢YHKUH11a cos11.X+ b sin llX cpyHKUM5IMH H3rro,unpOCTJJ3HCTB3 TJJl1fOHOMeTJJ11qeCK11X IlOJIMHOMOB BH)lat,,_1(x)=a,,_ 1cos (n -l)x + b,,_1sin (n - l)x + ... + a 1cos x + b 1sin x + a0B MeTpHKe Lp[O, 2n], p :'.'.:l.17.24.
IlycTb q, pi, .. , p,, -pa3JIHqHbie KOMnJieKCHbie lJHCJia,Re q > - V2, Re Pk > - Y2, k = l, ... , n. Hai1THinft1 •.•••1flt"-a,,eC 0(a/''+ ... +2a,,t"") 1 dt .CcpopMym1posaTb 3TY 3a_naqy KaK 3a,uaqy on1cKaH115! Ha1111y<Jlllero rrp1161111)KeHH5!.f17.25 . .[(OKa33Tb, •no w(j, t)x - MOJJ:YJlb HenpepbIBHOCTHEX (cM. sse,neH11e) 0611a.naeT c11e,nyIOllJMMM csoH:crnaM11:cpyHKUHM3JIEMEHTbl TEOPHH ITPHEJIH)f{EHHM1)2)3)m(j,m(j,O)x = O;t)x, t 2: 0,He y6hmaeT;MOLlYJib Henpepb!BHOCTH eCTb IIOJiyaL(L(I-ITHBHaH cpyHKUHH,T. e. L(JIH mo6h1x ti, t22: 0:Sw(j, t 1 + t2) x4) w(j, t)x, t 2: 0, -17.26 . .UoKa3aTh,B249w(j, t 1) x + w(j, t2)x;HenpepbIBHaH cpyHKUHH.qrn m(j,t)x=w(j, n:)xL(JIH mo6orot 2: n:.qacTHOCTH, 0Tc10na CJienyeT, qTo MOLlYJih ttenpepbIBHOCTHw(j, t)x npH t 2: nB03paCTaTb He MOJKeT.17.27.
IIycTh Xw(t), t 2: 0,= C[O,2n:] ..UoKa3aTh, qrn L(JUI JII06ow cpyHKUHH1)-4)o6JiaL(aIOmew CBOMCTBaMHKOM, qm m(t)(CM. 3aL(aqy17.25)H Ta-= w(n) npH t 2: n, HaML!eTcH <l>YHKUHRf E 2n], TaKaH, qrnw(j, t)c= w(t).TaKHe cpyHKU:HH Ha3hIBaIOT 111ooy;15/Mu Henpepbl6Hocmu.17.28 . .UoKa3aTh,'-I.TO eCJIH cpyHKU:HHr 1w(t), t > 0, He B03pacTaeT, TO w(t) 17.29.<l>yHKUHIOJI106h1x ti, t22: 0w(t), t 2: 0,H JII06oro a Eaw(t1) + (1 -a)w(t)TaKOBa, qTQ cpyttKUHHnoJiya)l)(HTHBHaH cpyHKU:HH.Ha3h!BaIOT 6bl11YKllOU eeepx, eCJIH L(JIH[O, 1]w(t2):Sm(a t 1 +(l -a) t 2 ) ..UoKa3aTb, qTo BCHKaH HenpepbIBHaH ttey6hrna10maH BbIIIYKJiaH BBepxHa[O, +co)<l>YHKU:HHw(t), w(O)= 0, w(t) =w(n:), IIpH t 2: n: HBJIHeTCH MO-cyJieM HenpepbIBHOCTH.17.30 .
.UoKa3aTb,qTQ cpyHKUHH1) w(t) =Kt, t2: 0, 0 <a :S 1, K>O;2) UJ(t) =Kta,0:::; t:::; c5,0<a:::;1, K > 0,{ Kc5a,t > c5HBJIHIOTCH MOL!YJIHMH ttenpephIBHOCTH. EcJIH cpyttKUHH'ITOw(j, t) = Kta, 0 <a :S 1, K> 0, TO roBopHT,17.31. .UoKa3aTb,qTQ eCJIHf(x)TaKOBa,qmj{x) E LipKaw(t) - BhIIIYKJib!H BBepx MOllYJib He-npepbIBHOCTH, TO:1)L!JIH n106orot>0cymecrnyIOT HeBo3pacrn10mHe HeoTpHua-TeJibHbie IIpOH3BOL(Hhie m'_(t),w'+(t),IIpH'IeMw'+(t):Sw'_(t);250I'.wea 172) w(t) - a6comorno HenpepbIBHa51 cpyHKUH51 Ha (0, oo).17.32 .
.LI:oKa3aTb,'-ITO .UJUI mo6oro MO.UYJI51 Henpepb!BHOCTHw(t),OTJIHY:HOfO OT TO)K.lJ.eCTBeHHOfO HYJI51, cyruecrnyeT BbinYKJiblH BBepxMO.UYJib ttenpepb!BHOCTHw.(t),t :::: 0TaKOH, '-ITO mm mo6orow(t) ::; w.(t) < 2 w(t) ..IJ:oKa3aTb, '-ITO IlOCTO.iIHHYIO2B 110CJie.utteM ttepaBeHCTBe yMeHb-ll111Tb tteJih3.si:. TaK11e MO.UyJI11 tterrpepbIBHOCTl1--w.(t)tta3bIBaIOT Bbl-ny1owu Ma:J1Copw1.1nou.17 .33. ITycTb w(t) = w(j, t) .t>0.LI:oKa3aTb, '-ITO .UJI.sI mo6oroA. > 011.UJI51 JII06orow(A.t) ::; (A.+ 1) w(t).17.34 . .LI:oKa3aTb,'-ITO eCJil1Jim w(f ,t) = Ott-->+0'TO cpyHKU115If(x) eCTb 110CT05IHHa51 BeJil1'-ll1Ha.17.35.
ITycThOJEcJIH.(f ,t) =(x, 2+ x,+ f(x 2 )-2f - - JI .w'(t) ::; Mt, M > 0, t > 0,lim o/(f,t) =0,1-->+0Isup< f(x 1)1.1,-.,,1_1TO rosop.si:T, '-ITOTO fOBOp51T, Y:TOfEt.IJ:oKa3aTb, '-ITO ecJIH f(x) E LipfEA·;ecJIH )KeA.*.1, TO f(x)EA•. ilp11secrnrrp11Mep,flOKa3bJBaIOIU.HH, '-ITO o6paTHOe yrnep)f(.Uett11e tteseptto.17.36. ilyCTbcpyHKUI151 f(x) paBHOMeptto HerrpepbIBHa .UJI51pacTeT Ha 6eCKOHeY:HOCTl1 He 6h1crpee, Y:eM1x1 .
.IJ:oKa3aTb,If(x) - f(a)I :S Clx - al + 1) w(j, t)17.37. ITycTb f(x) -rjJyHKLfUflfr.ue0 <a< 1, b -(x) =Beilepuunpacca2..:;; ak cos( bk x),1ttelfeTHOe 411CJIO, 6oJihillee 1.)loKa3aTb, '-ITO:1) f(x) - tterrpepbIBHa51 27!-nep110.u11qecKa5! cpyHKU115!;'-ITOx:::: a113JIEMEHTbl TEOPl111 flPl1EJll1)!{EHl1H2512) rrptt ab> 1 .fCx) E Lip a, a= ln(l/a)/lnb;3) rrptt ab= 1 w(j, t):SKt ln(llt), 0 < t:S Yi;ri Lip 1.4) rrptt ab= 1 f(x) E A*,f(x)17.38.
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c 3aJJ,aYei1 17 .35). CoorneTcTBeHHO1( J;1'1 .f(x)- fi .1 (x) I" dx )" ':::;,1 1::S: ~sup ( { ''1f (x -!) + f (x + t)- 2f (x) 11' dx )2 OStSII11/1,(£ !T lf(x)-lr,_"(x)l l' d.xf''~2I::S:1suph OStS/J(IO;rlj(x-t)+f(x+l)-2f(x)ll'dx17.40. nycTb JJ,aHO npocTpaHCTBOx = C[a,)I/ ,flb]. nycTbfE X,l!fllc= max If (t) I, MHO)l(ecrno F =Fil - noJJ,npocTpattcrno anre6pa110'5:r~"yecK11x noJJ11HoMoB, cTeneHb KOTOpblX He npeBocxoJJ,11T11E N.
,Qo-Ka3aTb, 4TO MHm1<eCTBO F S1BJJS1eTcsi MHO)!<ecTBOM cyuteCTBOBaH11S1(cM. BBeJJ,eH11e). ,QoKa3aTb meope.wy BeuepUlmpacca, T. e. ecJJ11 f(x)HenpepblBHa Ha 0Tpe31<e [a , b] , TO )J,JlSI Jll06oro £ > 0 H3HJJ,eTCSI anre6pa11YeCKl1H noJJ11HOM p(x) E Fil, rn1<oi1 , YTO l!f(x) - p(x)llc17.41. nycTb JJ,aHO npocTpaHCTBOl!f(x)llx = C[O,= max I j(t) I, MHm1<ecrno F =Fil o::;,::;2n-HOMeTp114eCKl1X nOJl11HOMOB Bl1)J,3:S2rr].
nycTbf£.E X,no.D,npocTpaHcTBo Tp11ro-')J ll Ml.In bl rI OPl-111111'1"16Jll-1)1{EHV1M253a0 + L ~ 1 (a. coskx+b. sinkx).F)J,oKa3aTb, LJTO MHO)l<eCTBOHBJrneTcH MHO)l(eCTBOM cyUlecrno-C1aH115I (cM. onpe.ue11eH11e B HaLJaJle rnaBbl).• )J,oKa3aTb TeopeMy Bei.:tepwTpacca, T. e. ec1111j(x) -HaH, 2n-nep110.U114eCKa51 qiyHKUl151, TO .UJl51 mo6oro £ >Tp11roH0MeTp114ecK11i.:t no1111HOMt(x) El)/(x) - t(x)llc17.42.F11 ,Henpepb1B-0Hai.:t.ueTC51TaKOH, LJTO:s; £.cqiopMy1111poBaTb 11 .il.OKa3aTb YTBep)l(,UeHl151, aH8Jl0rl14HbleTeM, l<OTOpbte ccliopMy 11 11poBaHbl B 3a,UaLJax 17.25 11 17 .26 ,X = L1,[a, b], p17.43.~I, 11X = L1,[0.x = C[O,nycTb2n],I]HO LlJl51p ~ I.111111x = L2[0,I]COOTBeTCTBeHHO c{pd,np11 KOTOpb1X c11c-HOpMaMl1fE C[O, I], l!fllc= suplf(t)I,OStSIfE L2[0, I], 1)/112 = ( .f'.1 f (I) f dt (nycTbp1.p2, ..
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