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SIN − è COS-ïðåîáðàçîâàíèÿ Ôóðüå140Òîãäà èíòåãðàëû J1 (R), J2 (R) ñòðåìÿòñÿ ê íóëþ ïðè R → ∞ â ñèëó ëåììûÐèìàíà-Ëåáåãà. Èíòåãðàë J3 (R) ïîñëå çàìåíû Rx = s ïðèâîäèòñÿ ê âèäóZJ3 (R) = −ϕ(0))sin sds.s|x|>RδÏîýòîìó J3 (R) → 0 ïðè R → ∞ ââèäó óñëîâíîé ñõîäèìîñòè èíòåãðàëà+∞R−∞sin ssds.Çàäà÷à 7.7. Íàéòè ïðåîáðàçîâàíèå Ôóðüå îáîáùåííîé ôóíêöèè f (t) = tn .Çàäà÷à 7.8. Äîêàçàòü, ÷òî äëÿ îáîáùåííîé ôóíêöèè f êëàññà S 0 (R) ñïðàâåäëèâû ôîðìóëûFf (n) = (ix)n Ff,F(tn f ) = in (Ff )(n) .7.4 sin − è cos-ïðåîáðàçîâàíèÿ ÔóðüåÏðåäâàðèòåëüíî îòìåòèì ñëåäóþùèå ñâîéñòâà èíâàðèàíòíîñòè ïðåîáðàçîâàíèÿ Ôóðüå.Òåîðåìà 9.
Ñïðàâåäëèâû ñëåäóþùèå óòâåðæäåíèÿ:1. ïðåîáðàçîâàíèå Ôóðüå fb(x) ÷åòíî òîãäà è òîëüêî òîãäà, êîãäà ôóíêöèÿ f (t) ÷åòíà;2. ïðåîáðàçîâàíèå Ôóðüå fb(x) íå÷åòíî òîãäà è òîëüêî òîãäà, êîãäàôóíêöèÿ f (t) íå÷åòíà;3. ïðåîáðàçîâàíèå Ôóðüå fb(x) óäîâëåòâîðÿåò ñîîòíîøåíèþ fb(x) =fb(−x) òîãäà è òîëüêî òîãäà, êîãäà ôóíêöèÿ f (t) âåùåñòâåííà.Äîêàçàòåëüñòâî. Ïóñòü f (t) ÷åòíà, òîãäà1fb(−x) = √2πZ+∞eixt f (t) dt.−∞7.4.
SIN − è COS-ïðåîáðàçîâàíèÿ Ôóðüå141Ñîâåðøàÿ çàìåíó ïåðåìåííîé t → −t, ïîëó÷àåì, ÷òî1fb(−x) = √2πZ+∞e−ixt f (t) dt = fb(x).−∞Çàäà÷à 7.9. Äîêàçàòü óòâåðæäåíèÿ 2 è 3.Èñïîëüçóÿ ñâîéñòâà èíâàðèàíòíîñòè ïðåîáðàçîâàíèÿ Ôóðüå ìîæíî ââåñòè íà ïîëîæèòåëüíîé ïîëóîñè sin − è cos-ïðåîáðàçîâàíèÿ Ôóðüå.Ïóñòü, íàïðèìåð, f (t) ÷åòíàÿ ôóíêöèÿ. òîãäà åå ïðåîáðàçîâàíèå Ôóðüå òàêæå ÷åòíàÿ ôóíêöèÿ. Ñâåðíåì ôîðìóëû ïðÿìîãî è îáðàòíîãî ïðåîáðàçîâàíèÿ Ôóðüå íà ïîëóîñü. Òîãäà ïîëó÷àþòñÿ ôîðìóëû ïðÿìîãî è îáðàòíîãî cos-ïðåîáðàçîâàíèÿ Ôóðüår Z∞2fb(x) =cos xt f (t) dt,π0r Z∞2f (t) =cos xt fb(t) dt.π0Óñëîâèÿ ïðèìåíèìîñòè ýòèõ ôîðìóë ñîâïàäàþò ñ óñëîâèÿìè ïðèìåíèìîñòèäëÿ ïðåîáðàçîâàíèÿ Ôóðüå.Çàäà÷à 7.10.Óñòàíîâèòü ôîðìóëû ïðÿìîãî è îáðàòíîãî sin-ïðåîáðàçîâàíèÿ Ôóðüår Z∞2fb(x) =sin xt f (t) dt,π0r Z∞2sin xt fb(t) dt.f (t) =π0Ïðèìåð 7.7. Íàéòè cos-ïðåîáðàçîâàíèå Ôóðüå âòîðîé ïðîèçâîäíîé ôóíêöèè f (t).7.4.
SIN − è COS-ïðåîáðàçîâàíèÿ Ôóðüå142Ðåøåíèå. Ïðåäïîëàãàåì, ÷òî f (t) äâàæäû íåïðåðûâíî äèôôåðåíöèðóåìà íà [0, +∞] è àáñîëþòíî èíòåãðèðóåìà âìåñòå ñî ñâîèìè ïðîèçâîäíûìèf 0 (t) è f 00 (t). Ïóñòü F (x) cos-ïðåîáðàçîâàíèå Ôóðüå ôóíêöèè f (t). Ðàññìîòðèì cos-ïðåîáðàçîâàíèå Ôóðüå äëÿ f 00 (t):r Z∞2cos xt f 00 (t) dt.fc00 (x) =π0Îñóùåñòâëÿÿ äâàæäû èíòåãðèðîâàíèå ïî ÷àñòÿì, ïðèõîäèì ê ôîðìóëår(Fc f )(x) = −2 0f (0) − x2 (Fc f )(x).πÇäåñü è äàëåå äëÿ òîãî, ÷òîáû ðàçëè÷àòü sin- è cos-ïðåîáðàçîâàíèÿ Ôóðüåèñïîëüçóåì îáîçíà÷åíèÿ Fs è Fc .Çàäà÷à 7.11. Íàéòè ôîðìóëó äëÿ sin-ïðåîáðàçîâàíèÿ Ôóðüå âòîðîé ïðîèçâîäíîé ôóíêöèè f (t).Îòâåò.r(Fs f )(x) =2xf (0) − x2 (Fs f )(x).πÇàäà÷à 7.12.
Íàéòè ôîðìóëû äëÿ cos- è sin- ïðåîáðàçîâàíèé Ôóðüå ïðîèçâîäíûõ ÷åòíîãî ïîðÿäêà.Ïðèìåð 7.8. Íàéòè cos-ïðåîáðàçîâàíèÿ Ôóðüå ñëåäóþùèõ ôóíêöèé221). e−αx , 2). e−αx cos βx (α > 0).Ðåøåíèå.1. Èñïîëüçóÿ ñîîáðàæåíèÿ ÷åòíîñòè è ðåçóëüòàòû ïðèìåðà 7.3, èìååìFc (e−αx2r Z∞22e−αx cos xy dx =)(y) =π0r= Re 12πZ∞e−αx−∞2y2e− 4αeixy dx = √ .2α7.4. SIN − è COS-ïðåîáðàçîâàíèÿ Ôóðüå1432. Èñïîëüçóÿ ôîðìóëócos βx cos xy = 1/2(cos(y + β)x + cos(y − β)x),ñâîäèì âû÷èñëåíèå ê óæå ðàññìîòðåííîìó ñëó÷àþ.Îòâåò.2(y+β)√1 (e− 4α2 2α+e2− (y−β)4α).Çàäà÷à 7.13.
Íàéòè sin-ïðåîáðàçîâàíèå Ôóðüå ñëåäóþùèõ ôóíêöèé221). xe−αx ,Îòâåò:1).y2e−√4α,y 2α2α2).1√4α 2α2). xe−αx cos βx (α > 0).((y + β)e2− (y+β)4α+ (y − β)e2− (y−β)4α).Ïðèìåð 7.9. Íàéòè ôîðìóëó îáðàùåíèÿ äëÿ èíòåãðàëüíîãî ïðåîáðàçîâàíèÿr Z∞2U (λ) = (Gu)(λ) =(λ cos λx + β sin λx)u(x) dx.π0Ðåøåíèå. Ñ÷èòàåì, ÷òî u(x) ãëàäêàÿ ôóíêöèÿ, ðàâíàÿ íóëþ âíå íåêîòîðîãî êîíå÷íîãî ïðîìåæóòêà. Òàêèå ôóíêöèè íàçûâàþòñÿ ôèíèòíûìè.Èñïîëüçóÿ ñîîòíîøåíèå λ cos λx =ëó÷àåìddx (sin λx)è èíòåãðèðóÿ ïî ÷àñòÿì, ïî-r Z∞2U (λ) =sin λx(βu(x) − u0 (x)) dx.π0Ñîãëàñíî ôîðìóëå îáðàùåíèÿ äëÿ sin-ïðåîáðàçîâàíèÿ Ôóðüå, èìååìr Z∞2u0 (x) − βu(x) = −sin λxU (λ) dλ.π0Ïîñëå ýòîãî u(x) ìîæíî îïðåäåëèòü, ðåøàÿ ëèíåéíîå äèôôåðåíöèàëüíîåóðàâíåíèå.
Ïðèìåíÿÿ ìåòîä èíòåãðèðóþùåãî ìíîæèòåëÿ, óñòàíàâëèâàåìðàâåíñòâîr Z∞d2(exp{−βx}u(x)) = −exp{−βx} sin λx U (λ) dλ.dxπ07.5. Ìíîãîìåðíîå ïðåîáðàçîâàíèå Ôóðüå144Ñ÷èòàÿ, ÷òî β > 0, ïðîèíòåãðèðóåì îò x äî ∞. Ó÷èòûâàÿ, ÷òîZ∞exp{−βx} sin λx dx = Im xZ∞exp{−βx}eiλx dx =xe−βxe−βx+iλx= 2= Im(β sin λx + λ cos λx),β − iλβ + λ2óñòàíàâëèâàåì ñëåäóþùóþ ôîðìóëó îáðàùåíèÿr Z∞2λ cos λx + β sin λxu(x) =U (λ) dλ.πβ 2 + λ2(7.3)0Çàìåòèì, ÷òî ïîêà îíà óñòàíîâëåíà ëèøü ïðè β > 0.Çàäà÷à 7.14. Óñòàíîâèòü íåïîñðåäñòâåííîé ïðîâåðêîé ñïðàâåäëèâîñòüôîðìóëû 7.3 ïðè β < 0.Çàäà÷à 7.15. Íàéòè (Gu00 )(λ) äëÿ ïðåîáðàçîâàíèÿ G, îïðåäåëåííîãî âïðåäûäóùåì ïðèìåðå.q00Îòâåò.
(Gu )(λ) =20π λ(−u (0)+ βu(0)) − λ2 (Gu)(λ).7.5 Ìíîãîìåðíîå ïðåîáðàçîâàíèå ÔóðüåÀíàëîãè÷íî îäíîìåðíîìó ñëó÷àþ ìíîãîìåðíîå ïðåîáðàçîâàíèå Ôóðüå äëÿôóíêöèé íà Rn îïðåäåëÿåòñÿ ôîðìóëîé1(Ff )(y) =(2π)n/2nãäå x, y ∈ R , (x, y) =nPj=0Zf (x)e−i(x,y) dx,Rnxj yj . Êàê è â îäíîìåðíîì ñëó÷àå ïðîñòðàí-ñòâî áûñòðî óáûâàþùèõ áåñêîíå÷íî äèôôåðåíöèðóåìûõ ôóíêöèé S(Rn )îïðåäåëÿåòñÿ êàê ìíîæåñòâî âñåõ áåñêîíå÷íî äèôôåðåíöèðóåìûõ ôóíêöèé f (x) íà Rn , äëÿ êîòîðûõ âûïîëíÿþòñÿ óñëîâèÿsup |xα ∂ β f (x)| < ∞x7.5.
Ìíîãîìåðíîå ïðåîáðàçîâàíèå Ôóðüå145äëÿ ëþáûõ ìóëüòèèíäåêñîâ α = (α1 , α2 , ..αn ), β = (β1 , ...βn ), ãäå xα =xα1 1 , ...xαnn .Äàëüíåéøèå ñâîéñòâà ìíîãîìåðíîãî ïðåîáðàçîâàíèÿ Ôóðüå ñôîðìóëèðóåì â ôîðìå óïðàæíåíèé.Çàäà÷à 7.16. Åñëè f (x) ïðèíàäëåæèò ïðîñòðàíñòâó S(Rn ), òî äëÿ ïðåîáðàçîâàíèÿ Ôóðüå ñïðàâåäëèâû îïåðàöèîííûå ñîîòíîøåíèÿ(F∂ α f )(y) = (iy)α (Ff )(y), (∂ α (Ff )(y) = (F(−ix)α f )(y).Çàäà÷à 7.17.
Äîêàçàòü, ÷òî îäíîìåðíîå ïðåîáðàçîâàíèå ôóðüå Fxj →yj âçàèìíî îäíîçíà÷íî è âçàèìíî íåïðåðûâíî îòîáðàæàåò S(Rn ) íà ñåáÿ, ïðè÷åìîáðàòíîå îòîáðàæåíèå îïðåäåëÿåòñÿ ôîðìóëîé1(F −1 g)(y1 , ...xj , ...yj ) = √2πZ+∞g(y)eixj yj dyj .−∞Çàäà÷à 7.18. Äîêàçàòü, ÷òî ìíîãîìåðíîå ïðåîáðàçîâàíèå Ôóðüå ïðåäñòàâëÿåòñÿ â âèäå ïðîèçâåäåíèÿ îäíîìåðíûõ ïðåîáðàçîâàíèé:Fx→y = Fx1 →y1 Fx2 →y2 ...Fxn →ynè, ñëåäîâàòåëüíî, âçàèìíî îäíîçíà÷íî è âçàèìíî íåïðåðûâíî îòîáðàæàåòS(Rn ) íà ñåáÿ.
Îáðàòíîå ïðåîáðàçîâàíèå Ôóðüå îïðåäåëÿåòñÿ ôîðìóëîéZ1(F −1 f )(y) =f (x)ei(x,y) dx,n/2(2π)RnÌåòîä êîìïîçèöèè ìîæíî èñïîëüçîâàòü äëÿ îïðåäåëåíèÿ íîâûõ èíòåãðàëüíûõ ïðåîáðàçîâàíèé â òîì ñëó÷àå, åñëè ÷àñòü ïåðåìåííûõ ìåíÿåòñÿ íàïîëîæèòåëüíîé ïîëóîñè, à îñòàâøàÿñÿ ÷àñòü ïåðåìåííûõ ìåíÿåòñÿ íà âñåéîñè. Îãðàíè÷èìñÿ, íàïðèìåð, ñëó÷àåì ÷åòâåðòè ïëîñêîñòè. Òîãäà ìîæíîðàññìîòðåòü ñëåäóþùèå ïðåîáðàçîâàíèÿ:Fs,x1 →y1 Fs,x2 →y2 , Fc,x1 →y1 Fs,x2 →y2 , Fc,x1 →y1 Fc,x2 →y2 .Çàäà÷à 7.19. Äëÿ ââåäåííûõ ïðåîáðàçîâàíèé âûâåñòè ôîðìóëû ïðåîáðàçîâàíèÿ ôóíêöèè 4f .7.6.
Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè1467.6 Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþ òåïëîïðîâîäíîñòèÓðàâíåíèå òåïëîïðîâîäíîñòè íà âñåé îñèÏðèìåð 7.10. Ïðèìåíÿÿ èíòåãðàëüíîå ïðåîáðàçîâàíèå Ôóðüå, ðåøèòüíà÷àëüíóþ çàäà÷ó(ut = a2 uxxx ∈ R, t > 0u|t=0 = ϕ(x).Ðåøåíèå.
Áóäåì ïðåäïîëàãàòü, ÷òî ϕ ∈ S(R). Áóäåì òàêæå ñ÷èòàòü, ÷òîçàäà÷à èìååò ðåøåíèå u(x, t), óäîâëåòâîðÿþùåå óñëîâèÿì u, ut , ux , uxx ∈S(R) ïðè t ≥ 0. Äàííûé ôàêò ïîçâîëÿåò èñïîëüçîâàòü ñëåäóþùèå îïåðàöèîííûå ñîîòíîøåíèÿ2ubt = (bu)t , ucbxx = Fx→y uxx = −y uè áóäåò îáîñíîâàí â ïðîöåññå ðåøåíèÿ çàäà÷è.Ïîñëå ïðèìåíåíèÿ ïðåîáðàçîâàíèÿ Ôóðüå óñòàíàâëèâàåì, ÷òî ub(y, t)ÿâëÿåòñÿ ðåøåíèåì çàäà÷è Êîøè(dbdt u= −a2 y 2 ubub|t=0 = ϕ(y).bÏðèìåíÿÿ ìåòîä èíòåãðèðóþùåãî ìíîæèòåëÿ, óñòàíàâëèâàåì, ÷òîd2 2(buea y t ) = 0.dt ðåçóëüòàòå èíòåãðèðîâàíèÿ ïîëó÷àåì äëÿ ub ñëåäóùóþ ôîðìóëó2 2−aub(y, t) = ϕ(y)eby t.Èç ýòîé ôîðìóëû â ñèëó ñäåëàííûõ ïðåäïîëîæåíèé óæå ñëåäóåò, ÷òîub(y, t) ∈ S(R) ðàâíîìåðíî îòíîñèòåëüíî t ≥ 0.
Òåì ñàìûì îáîñíîâàíûïðåäûäóùèå ðàññóæäåíèÿ.Ðåøåíèå èñõîäíîé çàäà÷è ïîëó÷àåòñÿ ïî ôîðìóëå îáðàùåíèÿ ïðåîáðàçîâàíèÿ Ôóðüå è èìååò âèä7.6. Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè1471u(x, t) = √2πZ+∞−a2 y 2 t iyxϕ(y)ebe dy.(7.4)−∞Äëÿ âûâîäà ôîðìóëû Ïóàññîíà ïðåäñòàâèì ϕ(y)bâ ïîñëåäíåé ôîðìóëå ââèäå èíòåãðàëà è èçìåíèì ïîðÿäîê èíòåãðèðîâàíèÿ1u(x, t) =2πZ+∞Z+∞2 2ϕ(s)e−a y t eiy(x−s) dy ds.−∞−∞Èñïîëüçóÿ ðåçóëüòàòû ïðèìåðà 7.3 âûâîäèì ôîðìóëó ÏóàññîíàZ+∞u(x, t) =−∞Ââåäåì ôóíêöèþ− (x−s)4a2 t2e√ ϕ(s) ds.2a πt(7.5)x2e− 4a2 t(7.6)G0 (x, t) = √ .2a πt ýòèõ îáîçíà÷åíèÿõ ôîðìóëà Ïóàññîíà 7.6 çàïèñûâàåòñÿ â âèäåZ+∞u(x, t) =G0 (x − s, t)ϕ(s) ds.(7.7)−∞Ôóíêöèÿ G0 (x − s, t) íàçûâàåòñÿ ÿäðîì Ïóàññîíà.Çàìå÷àíèå 7.1.
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