Домрина. Лекции (2009) v4.0 (1118430), страница 3
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sgn u = sgn τ è 1 + θτ > 0})2 ¸¸ ·¸ ··¸·uτu = 1+θτ ⇒ (1 + θτ )u = τ ⇒ τ (1 − θu) = u ⇒ τ = 1 − θu (τ 6= 0)dτÏîäñòàâèì ýòî â (∗∗) : du= u + 1 − θu = 1 + (1 − θ)u ( ïðè τ 6= 0)dudτÏðè τ = 0 dτ = 1 ⇒ du = 1 ⇒ ôîðìóëà ñïðàâåäëèâà è ïðè τ = 0.dτÈòîã: îöåíèëè du= 1 + (1 − θ)u 4 (íå çàáûâàåì, ÷òî θ - çàâèñèò îò τ, u)2. Èòàê,dτdu= 1 + (1 − θ)u, ïðè −∞ < u < +∞. Òåïåðü äåëàåì îöåíêó:R∞R∞ −x u2e 2 [1 + (1 − θ)u]du =Γ(x + 1) = ( xe )x x e−x(τ −ln(τ +1)) dτ = ( xe )x x−11)+∞Re−∞+∞R2) |−∞+∞R−xu22edu = {s = u−xu22qpx2} =(1 − θ)udu| ≤+∞Re−∞2x−xu22+∞Re−s2−∞ds =q √2x π|u|du =−∞R∞22d u2 = { xu2 = s} = x2 e−s ds = x2000√= (ïðîâåðüòå ñàìîñòîÿòåëüíî) = 2πx( xe )x (1 + α(x)), |α(x)| ≤=2e−xu22udu = 2+∞Re−xu223Îñîçíàéòå,4îòêóäà áåðóòñÿ ýòè ïðåäåëû è çà÷åì ìû èññëåäîâàëè ïîâåäåíèå φ(τ )Òåïåðü âñïîìíèòå, ÷òî ìû äîêàçûâàëè, ÷òî u0 (τ ) > 0.
Ãäå ìû ýòî èñïîëüçîâàëè?15√2 √12π x→ 0, x → ∞×àñòü 2. Ðÿäû ÔóðüåÏîíÿòèå ðÿäà Ôóðüå â åâêëèäîâîì è ïî÷òè åâêëèäîâîìïðîñòðàíñòâàõÏóñòü L - ëèíåéíîå ïðîñòðàíñòâî íàä ïîëåì R. L íàçûâàåòñÿ åâêëèäîâûì,åñëè ∀ x, y ∈ L → (x, y) ∈ R:1) (x, y) = (y, x) ∀ x, y ∈ L (ñèììåòðèÿ)2) (αx + βy, z) = α(x, z) + β(y, z) ∀ x, y, z ∈ L, ∀α, β ∈ R (ëèíåéíîñòü)3) (x, x) > 0 ∀ x ∈ L4) (x, x) = 0 ⇔ x = 0Îïðåäåëåíèå.Îïðåäåëåíèå. Ëèíåéíîå ïðîñòðàíñòâî L íàä ïîëåì R íàçûâàåòñÿ ïî÷òè åâêëèäîâûì,åñëè ∀ x, y ∈ L → (x, y) ∈ R è (x, y) óäîâëåòâîðÿåò 1), 2), 3).Çàìå÷àíèå: Ëþáîå åâêëèäîâî ïðîñòðàíñòâî ÿâëÿåòñÿ ïî÷òè åâêëèäîâûì.Ïðèìåð: L2R [a, b] - ïðîñòðàíñòâî ôóíêöèé, èíòåãðèðóåìûõ â ñîáñòâåííîì ñìûñëå ïî Ðèìàíóíà îòðåçêå [a, b] ñî ñêàëÿðíûì ïðîèçâåäåíèåìZb(x, y) =x(t)y(t)dt (∗)a1), 2), 3), ñëåäóþòèç ñâîéñòâ èíòåãðàëà Ðèìàíà, 4) íå âûïîëíåíî:(Rb 20, x ∈ (a, b]Ïóñòü x(t) =⇒x (t)dt = 0, ïðè ýòîì x 6= 01, t = aaò.å.
L2R [a, b] ïî÷òè åâêëèäîâî.Ïðèìåð: Ĉ[a, b] - ïð-âî êóñî÷íî-íåïð. ôóíêöèé íà [a, b] ñî ñêàëÿðíûì ïðîèçâåäåíèåì (∗)x ∈ Ĉ[a, b] ⇔ x(t) íåïðåðûâíà íà [a, b], çà èñêëþ÷åíèåì êîíå÷íîãî ÷èñëà òî÷åê,ïðè ýòîì:∀t ∈ (a, b) ∃ x(t − 0), x(t + 0); ∃ x(a + 0), x(b − 0).(x(t) = 21 [x(t − 0) + x(t + 0)] , t ∈ (a, b)x(a) = x(b) = 12 [x(a + 0) + x(b − 0)]Íàïðèìåð sgn(sin t) ∈ Ĉ[−π, π]Ïîêàæåì, ÷òî Ĉ[a, b] - åâêëèäîâî. Äîñòàòî÷íî ïðîâåðèòü âûïîëíåíèå ÷åòâåðòîãî ñâîéñòâà:tkn RRbPx2 (t)dt0 = (x, x) = x2 (t)dt = {a = t0 < ...
< tn = b; x(t) ∈ C(tk−1 , tk ), 1 ≤ k ≤ n} =a⇒ ∀ k = 1, ..., nk=1 tk−1Rtktk−1x2 (t)dt = 0⇒ x(t) = 0 ïðè t ∈ (tk−1 , tk ) ⇒x(tk−1 +0) = x(tk −0) = 0Òàêèì îáðàçîì, x(t1 ) = ... = x(tn ) = x(a) = x(b) = 0 ⇒ x(t) ≡ 0, t ∈ [a, b] ⇒ Ĉ[a, b] −åâêëèäîâî.Çàìå÷àíèå: Ĉ[a, b] ⊂ L2R [a, b] - ïîòîì ìû âîñïîëüçóåìñÿ ýòèì ôàêòîì16Îïðåäåëåíèå.L - Ëèíåéíîå ïðîñòðàíñòâî íàä R íàçûâàåòñÿ íîðìèðîâàííûì, åñëè∀ x ∈ L → kxk ∈ R:1) kλxk = |λ| · kxk, λ ∈ R, x ∈ L2) kx + yk ≤ kxk + kyk, x, y ∈ L (íåðàâåíñòâî òðåóãîëüíèêà)3) kxk ≥ 0 ∀ x ∈ L4) kxk = 0 ⇔ x = 0Îïðåäåëåíèå. Ïðîñòðàíñòâî ïî÷òè íîðìèðîâàííîå, åñëè ∀ x ∈ L îïðåäåëåíî kxk ∈R, óäîâëåòâîðÿþùåå 1),2),3).Çàìå÷àíèå: Åñëè L - íîðìèðîâàííîå, òî L - ïî÷òè íîðìèðîâàííîå.Óòâåðæäåíèå: Ïóñòü L - ïî÷òè åâêëèäîâî ïðîñòðàíñòâî.
Òîãäà ∀(x, y) ∈ L ñïðàâåäëèâîíåðàâåíñòâî Êîøè-Áóíÿêîâñêîãî:|(x, y)| ≤p(x, x) · (y, y)(∗∗)Äîê-âî: ∀ x, y ∈ L,∀λ ∈ R (x + λy, x + λy) ≥ 0 ⇒ (x, x) + 2λ(x, y) + λ2 (y, y) ≥ 0.Åñëè (y, y) = 0, òî (x, x) + 2λ(x, y) ≥ 0, ∀λ ∈ R ⇒ (x, y) = 0 ⇒ (∗∗)2Åñëè (y, y) 6= 0 ⇒ D4 = (x, y) − (x, x) · (y, y) ≤ 0 ⇒ (∗∗).¯Rb¯¯¯Â L2R [a, b] íåðàâåíñòâî Êîøè-Áóíÿêîâñêîãî èìååò âèä ¯ x(t)y(t)dt¯ ≤sRbax2 (t)dt ·a(÷àñòíûé ñëó÷àé íåðàâåíñòâà Ãåëüäåðà äëÿ èíòåãðàëîâ)Rby 2 (t)dtaÓòâåðæäåíèå: Ïóñòü L - ïî÷òè åâêëèäîâî (ñîîòâåòñòâåííî åâêëèäîâî).Òîãäà Lpìîæíî ñäåëàòü ïî÷òè íîðìèðîâàííûì (ñîîòâåòñòâåííî íîðìèðîâàííûì), ïîëîæèâkxk = (x, x) (íîðìà, ñîãëàñîâàííàÿ ñî ñêàëÿðíûì ïðîèçâåäåíèåì).Äîê-âî: Äîñòàòî÷íî ïðîâåðèòü, ÷òî kx + yk ≤ kxk + kyk:pkx + yk2 = (x + y, x + y) = (x, x) + 2(x, y) + (y, y) ≤ (x, x) + 2 (x, x) · (y, y) + (y, y) = (x, x) +pppp2 (x, x) · (y, y) + (y, y) = ( (x, x) + (y, y))2 = (kxk + kyk)2 ⇒ kx + yk ≤ kxk + kykÇàäà÷è ê ýêçàìåíó1.
L - íîðìèðîâàííî. Ïîêàçàòü, ÷òî åñëè kxk ñîãëàñîâàíà ñ íåêîòîðûì ñêàëÿðíûì ïðîèçâåäåíèåì,òî 2(kxk2 + kyk2 ) = (kx + yk2 + kx − yk2 ) (Êðèòåðèé åâêëèäîâîñòè íîðìû ïîìíèòå?)2. C[a, b]. Ïîëîæèì kxk = max |x(t)|. Ïîêàçàòü, ÷òî k · k - íîðìà, íå ñîãëàñîâàííàÿ íè ñt∈[a,b]êàêèì ñêàëÿðíûì ïðîèçâåäåíèåì.3.  ïðîñòðàíñòâå ôóíêöèé, èíòåãðèðóåìûõ ïî Ðèìàíó íà [a, b], kxk =¯Rb ¯¯x(t)¯dt. Ïîêàçàòü,a÷òî ïðîñòðàíñòâî ïî÷òè íîðìèðîâàíî, íî íîðìà íå ñîãëàñîâàíà íè ñ êàêèì ñêàëÿðíûìïðîèçâåäåíèåì.4. Ïîêàçàòü,÷òî (∗) íå ÿâëÿåòñÿ ñêàëÿðíûì ïðîèçâåäåíèåì äëÿ ïðîñòðàíñòâà C(a, b) .Îïðåäåëåíèå. {ψn } - ñèñòåìà ýëåìåíòîâ èç ïî÷òè åâêëèäîâà ëèíåéíîãî ïðîñòðàíñòâàíàçûâàåòñÿ îðòîíîðìèðîâàííîé ñèñòåìîé (ÎÍÑ), åñëè (ψi , ψj ) = δij ∀i, j ∈ NÎïðåäåëåíèå.Ðÿä∞Pk=1f ∈ L, {ψn } - ÎÍÑ â ïî÷òè åâêëèäîâîì ëèíåéíîì ïðîñòðàíñòâå L.fk · ψk , ãäå fk = (f, ψk ) íàçûâàåòñÿ ðÿäîì Ôóðüå ýëåìåíòîâ f ïî ñèñòåìå {ψk },fk - êîýôôèöèåíòû Ôóðüå.
55cÀíàëîãèÿ ñ ðàçëîæåíèåì ïî ÎÍÁ â êîíå÷íîìåðíîì ëèíåéíîì ïðîñòðàíñòâå äîëæíà áûòü î÷åâèäíà °17Çàäà÷à î íàèëó÷øåì ïðèáëèæåíèèLn =n©Pk=1ªck ψk , c1 , ..., ck ∈ R - ëèíåéíàÿ îáîëî÷êà ïåðâûõ n âåêòîðîâ ñèñòåìûÍåîáõîäèìî íàéòènPkf −k=1k=1c1 ,...,cn ∈RnPck ψk k2 = (f −nP= kf k2 − 2nPinfk=1ck fk +k=1nPk=1nPkf −k=1ck ψk , f −ck ψk knPk=1c2k = kf k2 −ck ψk ) = (f, f ) − 2nPk=1fk2 +nPk=1nPk=1nPfk2 − 2ck (f, ψk ) + (k=1ck fk +nPk=1nPk=1ck ψk ,nPk=1c2k = kf k2 −ck ψk ) =nPk=1fk2 +(fk − ck )2 , ãäå fk = (f, ψk ) - êîýôôèöèåíòû Ôóðüå.nP(fk − ck )2 = 0 äîñòèãàåòñÿ òî÷íàÿ íèæíÿÿ ãðàíü, ïîäâåäåì èòîã:k=1snnPPfk2 è äîñòèãàåòñÿ ⇔ ck = fk , 1 ≤ k ≤ nck ψk k = kf k2 −Òåîðåìà 1. inf kf −Ïðèc1 ,...,cn ∈Rk=1k=1Ñëåäñòâèå: ∀n ∈ N kf k2 −nPk=1fk2 ≥ 0 (âîîáùå-òî, ìû èì óæå âîñïîëüçîâàëèñü)Ñëåäñòâèå: ∀f ∈ L ñïðàâåäëèâî íåðàâåíñòâî Áåññåëÿ:Äîê-âî:nPk=1∞Pk=1fk2 ≤ kf k2fk2 ≤ kf k2 , ∀n ∈ N ⇒ âñå ÷ëåíû íåóáûâàþùåé ïîñëåäîâàòåëüíîñòè îãðàíè÷åíûêîíñòàíòîé kf k2 ⇒∞Pk=1fk2 ≤ kf k2Ïðèìåð: Òðèãîíîìåòðè÷åñêàÿ ñèñòåìà â L2R [−π, π] (â Ĉ[−π, π]):√1 , cos√ x , sin√ x , cos√ 2x , sin√ 2x , ...
- ÿâëÿåòñÿ ÎÍÑ (äîêàæèòå ýòî)ππππ2π³´∞P(1)(2) sin kx0√ kx + f√f à √f2π+fk cos- ðÿä Ôóðüå ôóíêöèè f ïî ýòîé ñèñòåìåkππk=1qRπ 1Rπ21√ f (x)dxf=a=f=f (x)dx000ππ2π−π−πRπ 1Rπ(1)(1)1√ f (x) cos kxdx√1 ffk =a==f (x) cos kxdx⇒Ïîëîæèìkπππ k−π−πRπ 1Rπ(2)(2)1√ f (x) sin kxdx√1 fb==f (x) sin kxdxfk =kπππ k−π−π∞ ¡¢Pak cos kx + bk sin kx - òðèãîíîìåòðè÷åñêèé ðÿä Ôóðüå (ÒÐÔ)Òîãäà f à a20 +k=1Íåðàâåíñòâî Áåññåëÿ äëÿ ÒÐÔ, çàïèñàííîå â ðàçíûõ îáîçíà÷åíèÿõ:∞ ³¡¡ (2) ¢2 ´Rπ 2P(1) ¢2f02 +fk+ fk≤f (x)dxπ 22 a0k=1nPπ(a2k+a20+2k=1∞X+b2k )≤(a2k + b2k ) ≤k=1Ïðèìåð:∞Pn=1(èíà÷å bn =sin√nxn√1 ,n−πRπf 2 (x)dx−πZπ1πf 2 (x)dx−πñõîäèòñÿ, íî íå ÿâëÿåòñÿ ðÿäîì Ôóðüå íè äëÿ êàêîé ôóíêöèè f ∈ L2R [−π, π]an = 0,∞Pn=1b2n ðàñõîäèòñÿ).18Çàìêíóòûå è ïîëíûå îðòîíîðìèðîâàííûå ñèñòåìûÎïðåäåëåíèå.Ïóñòü {ψn } - ÎÍÑ â ïî÷òè åâêëèäîâîì ïðîñòðàíñòâå L.nnPP{ψn } íàçûâàåòñÿ çàìêíóòîé, åñëè ∀f ∈ L, ∀ε ≥ 0 ∃ck ψk : kf −ck ψk k ≤ εk=1k=1Òåîðåìà 2 (Ðàâåíñòâî Ïàðñåâàëÿ).
Ïóñòü {ψn } çàìêíóòà â L. ÒîãäàÄîê-âî: {ψn } - çàìêíóòà ⇒ ∀ ε > 0 ∃nPk=1ck ψk : kf −⇒ {çàäà÷à î íàèëó÷øåì ïðèáëèæåíèè} ⇒ kf −Ò.å. ∀ ε > 0 ∃n :nPk=1fk2≥Ñëåäñòâèå: ∀f ∈ LÄîê-âî: kf −nPk=1kf k2nPk=1nPk=1lim kf −n→∞fk ψk k2 = kf k2 −k=1nPk=1nPk=1fk2n=1fk2 = kf k2√εfk ψk k2 = kf k2 −− ε. Íî ïî íåð-âó ÁåññåëÿnPck ψk k <∞PnPk=1fk2 < ε≤ kf k2 ⇒∞Pk=1fk2 = kf k2 .fk ψk k = 0, ãäå {ψn } - çàìêíóòàÿ ÎÍÑ.n→∞fk2 −−−→ 0Çàìå÷àíèå:s  L2R [a, b] ñõîäèìîñòü ïî íîðìå ðàâíîñèëüíà ñõîäèìîñòè â ñðåäíåì ò.ê.Rbkf − gk =(f − g)2 dxaÎïðåäåëåíèå.Ïóñòü {ψn } - ÎÍÑ â åâêëèäîâîì ïðîñòðàíñòâå L.{ψn } íàçûâàåòñÿ ïîëíîé â L, åñëè (f, ψn ) = 0 ∀n ∈ N ⇔ f = 0Óòâåðæäåíèå: Ïóñòü {ψn } ïîëíà â åâêëèäîâîì ïðîñòðàíñòâå L.Òîãäà ∀ f, g ∈ L, f 6= g ∃n : (f, ψn ) 6= (g, ψn )Äîê-âî: h = f − g, h 6= 0. Ò.ê. {ψn } ïîëíà, ∃n : (h, ψn ) 6= 0, ò.å.
(f, ψn ) 6= (g, ψn )Òåîðåìà 3. Ïóñòü L-åâêëèäîâî è {ψn } - çàìêíóòàÿ ÎÍÑ â L.Òîãäà {ψn } ïîëíà â L.Äîê-âî: Ïóñòü ∃f ∈ L : (f, ψn ) = fn = 0 ∀n ∈ N.Èç ðàâåíñòâà Ïàðñåâàëÿ kf k =∞Pn=0fn2 = 0 ⇒ f = 0 (ïî 4-ìó ñâîéñòâó íîðìû).Ëåììà (îá èíòåãðèðóåìîñòè ïî ïåðèîäó). Ïóñòü f èíòåãðèðóåìà íà îòðåçêå [−π, π] èèìååò ïåðèîä 2π .π−xRπRf (t)dtf (t)dt =Òîãäà ∀ x ∈ RÄîê-âî:π−xRπ−π−πR−π−xπ−xRf (t)dt =−π−xRπf (t)dt +−πf (t)dt =f (t)dt +−π−xRπ−ππ−xRf (t)dt +πRπf (t)dt = {τ = t + 2π} =Rπf (τ )dτ +π−x−πf (t)dt (äîêàæèòå, ÷òî ïðîìåæóòî÷íûå èíòåãðàëû îïðåäåëåíû)Çàäà÷à: Ïóñòü L - åâêëèäîâî, ñèñòåìà {ψn } : ∀f ∈ L, ∀ε ≥ 0 ∃íî íå ÎÍÑ. Ïî ýòîé ñèñòåìå ïîñòðîèòü çàìêíóòóþ ÎÍÑ â L.19nPk=1ck ψk : kf −nPk=1ck ψk k ≤ ε,Âûðàæåíèÿ äëÿ ÷àñòè÷íûõ ñóìì òðèãîíîìåòðè÷åñêîãî ðÿäàÔóðüå è ÷åçàðîâñêèõ ñóìì òðèãîíîìåòðè÷åñêîãî ðÿäà Ôóðüåf ∈ L2R [−π, π], ïåðèîäè÷åñêàÿ ñ ïåðèîäîì 2πSn (f, x) =n¢a0 X ¡+ak cos kx + bk sin kx − ÷àñòè÷íûå ñóììû ÒÐÔ2k=1S0 (f, x) + ...
+ Sn−1 (f, x)− ÷åçàðîâñêèå ñóììû ÒÐÔnn ¡ Rπ¢RπRπP111Sn (f, x) = 2πf (t)dt +f(t)cosktdt·coskx+f (t) sin kt dt · sin kx =ππ−π−π ·k=1¸¸· −π nnRπRπPP1111=π(cos kt cos kx + sin kt sin kx) dt = πcos k(t − x) dt =f (t) 2 +f (t) 2 +−π−πk=1k=1¸·nRπP{u = t − x, ïîäûíòåãðàëüíàÿ ôóíêöèÿ ïåðèîäè÷íà} = π1f (u + x) 12 +cos ku du;σn (f, x) =·¸12·−π¸k=1¸11uuu11cos ku =sin 2 +2 sin 2 cos ku =sin 2 +sin (k + 2 )u − sin (k − 2 )u =+2 sin u22 sin u2k=1k=1k=11 sin (n + 2 )u u 6= 0 äàëåå áóäåì ïîäðàçóìåâàòü èìåííî ýòó ôóíêöèþ,2 sin u2=,n + 1u = 0 ò.ê.
îíà − äðîáü ñâåðõó, äîîïðåäåëåííàÿ ïî íåïðåðûâíîñòè2Zπsin (n + 12 )u1f (u + x)du (∗)Sn (f, x) =π2 sin u2nPnP·nP−πDn (u) =sin (n + 12 )u2 sin u2Äëÿ f ≡ 1 : a0 =1πRπ−πÏîäñòàâèì â (∗): 1 =σn (f, x) =n−1Pk=0=- ÿäðî Äèðèõëå.dt = 2; an =Rπcos ntdt = bn =−ππ sin (n + 1 )uR21duuπ2sin−π2S0 (f,x)+...+Sn−1 (f,x)nsin (k + 21 )u =1π=1nn−1Pk=0"1πRπ−πsin ntdt = 0 ⇒ Sn (f, x) = 1, ∀n ∈ N(b∗)n−1P#1 RπDk (u)f (u + x)du =π −π1πnRπ−πsin (k + 12 )uk=0f (u + x)du2 sin u2n−1n−1¢11PP¡1u)usin=cos ku − cos(k + 1)u =·2sin(k+·uu222 sin 2 k=02 sin 2 k=01 − cos un 2 sin2 unsin2 un22==2 sin u22 sin u2sin u21σn (f, x) =πnÎáîçíà÷èì Ôn (u) =nu22 sin2 u2sin2Zπ−πnu2f (u2 sin2 u2sin2+ x)du (∗∗)- ÿäðî Ôåéåðà, òîæå äîîïðåäåëåíî ïðè u=0 ïî íåïðåðûâíîñòèÄëÿ f ≡ 1 : σn (f, x) ≡ 1.
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