Лекции по функциональному анализу Смолянова О.Г. (1134957), страница 8
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Ïðÿìîé îáðàç g∗ F îáîáùåííîé ôóíêöèè F ïðèîòîáðàæåíèè g îïðåäåëÿåòñÿ òàê:g∗ F ∈ S ∗ (E2 ),(g∗ F, ϕ) = (F, g ∗ ϕ),g ∗ ϕ ∈ S(E1 ),∀ ϕ ∈ S(E2 ).Åñëè ϕ ∈ S(E2 ), òî íåîáÿçàòåëüíî, ÷òî g ∗ ϕ ∈ S(E1 ), òî åñòü ïðÿìîé îáðàç îïðåäåëåíëèøü äëÿ òåõ F, êîòîðûå ìîæíî ïðîäîëæèòü íà îáðàòíûå îáðàçû ôóíêöèé g ∗ ϕ,ϕ ∈ S(E1 ).Îáðàòíûé îáðàç îáîáùåííîé ôóíêöèè F ∈ S ∗ (E2 ) ïðè îòîáðàæåíèè g îïðåäåëÿåòñÿòàê:(g ∗ F, ϕ) = (F, g∗ ϕ)∀ ϕ ∈ S(E1 )(ïðè îïðåäåëåíèè ïðÿìîãî îáðàçà g∗ ϕ ôóíêöèè ϕ ∈ S(E1 ); îíà ðàññìàòðèâàåòñÿ êàêýëåìåíò ïðîñòðàíñòâà S ∗ (E1 ))Åñëè ∀ ϕ ∈ S g∗ ϕ ∈ S, òî îáðàòíûé îáðàç îïðåäåëåí äëÿ âñåõ îáîáùåííûõ ôóíêöèé.Îáîçíà÷åíèå: (g ∗ F )(x) = F (g(x)).Çàìå÷àíèå.
Åñëè νϕ - ýòî ìåðà ñ ïëîòíîñòüþ ϕ è ψ - ïëîòíîñòü ìåðû g∗ νϕ , òîãäàg∗ ϕ = ψ.Óïðàæíåíèå 18. Äîêàçàòü, ÷òî åñëè F - îáû÷íàÿ ôóíêöèÿ, à g - âçàèìíî îäíîçíà÷íîåîòîáðàæåíèå, òî(g∗ F )(x) = F (g −1 (x)).Ï Ð È Ì Å Ð. Ïóñòü g : R → R,g(x) = ax + b è F = δ.Òîãäà, ÷òî òàêîå îáðàòíûé îáðàç δ -ôóíêöèè g ∗ δ ?Íàéäåì g ∗ δ (= δ(ax + b)) ñëåäóþùèì îáðàçîì:µ¶ZZz − b dz∗δ(z) ϕ=(g δ, ϕ) ≡ δ (ax + b) ϕ(x) dx =| {z }aaRzR5814 Ëåêöèÿ1bϕ(− ) = (δ(ax + b), ϕ) =aa ÷àñòíîñòè,RZδ(ax + b) ϕ(x) dx;R1δ(x + b) ϕ(x) dx = ϕ(−b).Óïðàæíåíèå 19. Ïóñòü åñòü àôôèííîå îòîáðàæåíèå g : R2 → R1(x1 , x2 ) 7→ ax1 + bx2 + cÍàéòè(g ∗ δ)(x) = δ(ax1 + bx2 + c).Óïðàæíåíèå 20.
Äîêàçàòü, ÷òî ñëåäóþùåå âûðàæåíèå âåðíî è íàéòè C :ixz = C · e−ixz ( ýêñïîíåíòû ñ÷èòàþòñÿ îáîáùåííûìè ôóíêöèÿìè äâóõ àðãóìåíòîâ).edÄîêàçàòåëüñòâî. Íàéäåì ïðåîáðàçîâàíèå Ôóðüå ôóíêöèè ei(x,z) :Z Z[i(x,z)e(x1 , z1 ) =ei(x,z) e−i(x,x1 ) ei(z,z1 ) dx dz =RZ−izz1=eRZRixzee−ixx1Zdx dz =R2π · δ(z − z1 ) · e−izz1 dz = 2π · e−ix1 z1 .R¤14.2 Ñâåðòêà îáîáùåííûõ ôóíêöèé.Îïðåäåëåíèå 28. Ïóñòü F1 , F2 ∈ S ∗ (R1 ). Òîãäà òåíçîðíîå ïðîèçâåäåíèå îáîáùåííûõôóíêöèé F1 ⊗ F2 ∈ S ∗ (R2 ) îïðåäåëÿåòñÿ ñëåäóþùèì îáðàçîì:(F1 ⊗ F2 , ϕ1 (x) · ϕ2 (z)) = (F1 , ϕ1 ) · (F2 , ϕ2 )Ïî ëèíåéíîñòè è íåïðåðûâíîñòè åãî ìîæíî ïðîäîëæèòü íà âñå S. Òåíçîðíîå ïðîèçâåäåíèå ìîæíî îïðåäåëèòü è äðóãèì îáðàçîì, íàïðèìåð:(F1 ⊗ F2 , ϕ(x1 , x2 )) = (F1 , (F2 , ϕ(x1 , ·))).Ï Ð È Ì Å Ð.
Ïóñòü, íàïðèìåð:F1 = g1 (·) ∈ L1 ;F2 = g2 (·) ∈ L1 .Òîãäà òåíçîðíûì ïðîèçâåäåíèåì ýòèõ ôóíêöèé áóäåò èõ ïðîñòîå ïðîèçâåäåíèå:(g1 ⊗ g2 )(x1 , x2 ) = g1 (x1 ) · g2 (x2 )5914 ËåêöèÿÎïðåäåëåíèå 29. Îïðåäåëèì ñâåðòêó äâóõ îáîáùåííûõ ôóíêöèé. ÏóñòüR2 → R1 .Φ : (x1 , x2 ) 7→ x1 + x2 ,Òîãäà ñâåðòêà îáîáùåííûõ ôóíêöèé F1 , F2 îïðåäåëÿåòñÿ òàê:(F1 ∗ F2 ) = Φ∗ (F1 ⊗ F2 )(êîíå÷íî, ýòî çíà÷èò, ÷òî ñâåðòêà îáîáùåííûõ ôóíêöèé F1 è F2 îïðåäåëåíà íå âñåãäà).Ïóñòü f1 è f2 ∈ L1 (R1 ), òîãäà(f1 ∗ f2 , ϕ) = (g∗ (f1 · f2 ), ϕ) = (f1 · f2 , g ∗ ϕ) = (f1 · f2 , ϕ(x1 + x2 )) =ZVz }| {f1 (x1 ) · f2 (x2 ) ϕ (x1 + x2 ) dx1 dx2 ==R2Z=ZR2R1Z=f1 (x1 ) f2 (V − x1 ) ϕ(V ) dV dx1 =f1 (x1 ) f2 (V − x1 ) dx1 ϕ(V ) dV = R1Zf1 (x1 ) f2 (V − x1 ) dx1 , ϕR1Ñëåäîâàòåëüíî,Z(f1 ∗ f2 )(V ) =f1 (x) f2 (V − x) dx1R1(îòìåòèì åùå ðàç, ÷òî [(x1 , x2 ) 7→ ϕ(x1 + x2 )] ∈/ S(R2 ), äàæå åñëè ϕ ∈ S(R1 )).×òîáû áûëî âñå êîððåêòíî, íåîáõîäèìî ïðîâåðèòü, ÷òî f1 (x1 ) · f2 (x2 ) ∈ L1 (R2 ).Ïîëüçóÿñü òåîðåìîé Ôóáèíè, ïîëó÷èì:Z Z|f1 (x1 ) · f2 (x2 )| dx1 dx2 =R2Z|f1 (x1 )| =R1Z|f2 (x2 )| dx2 dx1 < ∞R1Ñëåäîâàòåëüíî, ìû äîêàçàëè, ÷òî f1 (x1 ) · f2 (x2 ) ∈ L1 (R2 ).Òåïåðü ïðîâåðèì, ÷òîf1 (x1 ) · f2 (V − x1 ) ∈ L1 (R2 ),ïîëüçóÿñü òåîðåìîé Ôóáèíè:Z Z|f1 (x1 ) · f2 (V − x1 )| dx1 dV =R26014 ËåêöèÿZZ|f1 (x1 )| =R1|f2 (V − x1 )| dV dx1 < ∞.R1Çàìåòèì, ÷òî R∀ x1 ôóíêöèÿ [V 7→ |f2 (V − x1 )|] ∈ L1 , òàê êàê f2 (x1 ) ∈ L1 (R1 ), ñëåäîâàòåëüíî, èíòåãðàë |f2 (V − x1 )| dV êîíå÷åí.R1Ïîýòîìó èíòåãðàëRf1 (x1 ) · f2 (V − x1 )dx1 îïðåäåëåí äëÿ ïî÷òè âñåõ V è ôóíêöèÿf1 ∗ f2 ∈ L1 .Òî, ÷òî ìû òîëüêî ÷òî ïðîâåðèëè, ìîæíî çàïèñàòü åùå â òàêîé ôîðìå:Ff1 ∗f2 = Ff1 ∗ Ff2 .Ñäåëàåì çàìåíó ïåðåìåííûõ: ïóñòü V − x1 = z, òîãäàZZf1 ∗ f2 = f1 (x1 ) · f2 (V − x1 ) dx1 =f1 (V − z) · f2 (z) dzR1R1Ñëåäîâàòåëüíî, ìû ìîæåì ñäåëàòü âûâîä, ÷òî ñâåðòêà ÿâëÿåòñÿ êîììóòàòèâíîé îïåðàöèåé.Îïðåäåëèì ïðåîáðàçîâàíèå Ôóðüå ñâåðòêè ñíà÷àëà äëÿ ôóíêöèé èç ïðîñòðàíñòâà L1 .Z Z f1 (x1 ) · f2 (V − x1 ) dx1 e−iV z dV =f\1 ∗ f2 (z) =R1R1Îïÿòü çäåñü ïðèìåíèìà òåîðåìà Ôóáèíè.
ÏóñòüV − x1 = x2 è V = x1 + x2 ,òîãäàZ Zf1 (x1 ) · f2 (x2 ) e−iz (x1 +x2 ) dx1 dx2 ==Z=f1 (x1 ) eR1Ïóñòü F ∈ S ∗ (R1 )R1 R1−izx1Zf2 (x2 ) e−izx2 dx2 = fb1 (z) · fb2 (z)dx1 ·R1èZ(Fb, ϕ) = ((F, e−ixz ), ϕ) =ZR1Òîãäà:F (x) e−ixz dx ϕ(z) dz.R1−iz·(F\) =1 ∗ F2 , e6114 ËåêöèÿZ Zïî îáîáùåííîé ò.Ôóáèíè c c=F1 (x1 ) · F2 (x2 ) e−iz (x1 +x2 ) dx1 dx2=F1 · F2R1 R1Ñâåðòêà îáîáùåííûõ ôóíêöèé îïðåäåëåíà íå âñåãäà.bδ(z)= (δ, e−ixz ) = 1.Åñëè ϕ ∈ S, òî\ϕ(x+ a)(z) = eiaz ϕ(z),bòîãäà\b = eiazδ(x+ a)(z) = eiaz δ(z)ˇˇ = δ(x + a)\δ(x+ a) = eiazèψa (x) = ψ(ax)a→0ψ(a) → 1 â S ∗ .Ìû ìîæåì íàïèñàòü äëÿ îáðàòíîãî ïðåîáðàçîâàíèÿ Ôóðüå ðàâåíñòâîZ1ψ̌(a)(x) =ψa (z) eixz dz2πR1Z11̌(x) = lima→0 2πixzψa (z) eR111̌(x) =2πeixz dzR1ZeZ1dz =2πixz1dx = limn→∞ 2πRˇ (x) = δ(x + a) =eiaz12πZZneixz dx−neiz(x+a) dxR1ÓÒÂÅÐÆÄÅÍÈÅ 6.
Îïåðàöèÿ äèôôåðåíöèðîâàíèÿ ñâåðòêè âûãëÿäèò òàê:(F1 ∗ F2 )0 = (F10 ∗ F2 ) = (F1 ∗ F20 )6214 ËåêöèÿÄîêàçàòåëüñòâî.=¡0cc(F\1 ∗ F2 ) (z) = (iz)F1 (z) · F2 (z) =¢0c0 (z) · Fc1 (z) · Fc2 (z) = Fc2 (z) = F\iz · F11 ∗ F2¤Ïóñòü D - äèôôåðåíöèàëüíûé îïåðàòîð â ïðîñòðàíñòâå îáîáùåííûé ôóíêöèé:nX(Dϕ)(x) =aj ϕ(j) (x)j=0Ôóíäàìåíòàëüíàÿ ôóíêöèÿ F äèôôåðåíöèàëüíîãî îïåðàòîðà îïðåäåëÿåòñÿ ðàâåíñòâîì:DF = δ.Ò Å Î Ð Å Ì À 15. Ïóñòü F - ôóíäàìåíòàëüíàÿ ôóíêöèÿ äèôôåðåíöèàëüíîãî îïåðàòîðàD, òîãäà ðåøåíèåì óðàâíåíèÿ DΦ = G ∈ S ∗ ÿâëÿåòñÿ:Φ = F ∗ G.Äîêàçàòåëüñòâî.DΦ = D(F ∗ G) = (DF ∗ G) = δ ∗ G.Äîêàæåì, ÷òî δ ∗ G = G :b = 1·Gb = Gb ⇒ δ∗G = Gδ[∗ G = δb · G¤Îïðåäåëåíèå 30.
Ðàññìîòðèì çàäà÷ó Êîøè: ∂F = D(F (t))∂tt→0F (t) → G ∈ S ∗Ãäå F : (0, a) → S ∗ , ∀ t F (t) ∈ S ∗ , G ∈ S ∗ .Òîãäà ôóíêöèÿ F íàçûâàåòñÿ ðåøåíèåì çàäà÷è Êîøè. Ôóíäàìåíòàëüíûì ðåøåíèåìçàäà÷è Êîøè íàçûâàåòñÿ òàêîå F, ÷òît→0F (t) −→ δ.Ò Å Î Ð Å Ì À 16.
Ðàññìîòðèì òàêæå çàäà÷ó Êîøè: ∂Φ = D(Φ(t))∂tt→0Φ(t) → G ∈ S ∗Òîãäà åå ðåøåíèåì ÿâëÿåòñÿ: Φ(t) = (F (t) ∗ G).Äîêàçàòåëüñòâî.∂Φ∂(F (·) ∗ G)=(t) =∂t∂tµ¶∂F∗ G = (DF (t) ∗ G) = D(F (t) ∗ G) = D(Φ(t)).∂tt→0Òàê êàê F (t) → δ, òî:t→0Φ(t) → (δ ∗ G) = G.¤6315 Ëåêöèÿ15 ËåêöèÿÇàìå÷àíèå. Áûëî S ⊃ D è S ∗ ⊃ D∗ . Ïðåîáðàçîâàíèå Ôóðüå íå ïåðåâîäèò D â ñåáÿ. Ïóñòüb - ìíîæåñòâî ïðåîáðàçîâàíèé Ôóðüå è Ď - ìíîæåñòâî îáðàòíûõ ïðåîáðàçîâàíèé ÔóðüåDôóíêöèé èç D ñ òîïîëîãèÿìè, çàèìñòâîâàííûìè èç D.b ≡ Z ⊂ S,Òîãäà Ď = Díî òîïîëîãèÿ â Z - íå òà, êîòîðàÿ ïîðîæäàåòñÿ òîïîëîãèåé ïðîñòðàíñòâà S (à ñèëüíåå).Ïðåîáðàçîâàíèå Ôóðüå (îáðàòíîå ïðåîáðàçîâàíèå Ôóðüå) ýëåìåíòà F ∈ D∗ - ýòî ôóíêöèîíàë íà Z, îïðåäåëÿåìûé òàê:(Fb, ϕ) = (F, ϕ);b∀ ϕ ∈ Z, F ∈ D∗ , ϕ ∈ D :Ïðåäïîëîæèì, ÷òî:f (x) = ebx ,(F, ϕ̌).S ∗ 63 Ff ∈ D∗ ;\ðàíåå áûëî ïîêàçàíî, ÷òî δ(z+ a)(x) = e−iax (z).Àíàëîãè÷íî,\δ(z+ ib)(x) = ebx ;(δ(z − ib), ψ) = ψ(ib),b∀ψ ∈ Z = D15.1 Òåîðèÿ ëèíåéíûõ îïåðàòîðîâ â ãèëüáåðòîâîì ïðîñòðàíñòâå. ýòîì ðàçäåëå ðàññìàòðèâàþòñÿ ãèëüáåðòîâû ïðîñòðàíñòâà íàä ïîëåì êîìïëåêñíûõ ÷èñåë.
Ñêàëÿðíîå ïðîèçâåäåíèå (·, ·) â êîìïëåêñíîì ãèëüáåðòîâîì ïðîñòðàíñòâå H îáëàäàåòñâîéñòâàìè:(1) ∀ x ∈ H (x, x) > 0,(x, x) = 0 ⇔ x = 0;(2) ∀ x1 , x2 , z ∈ H, ∀ α, β ∈ C(αx1 + βx2 , z) = α(x1 , z) + β(x2 , z);(3) ∀ x1 , x2 ∈ H(x1 , x2 ) = (x1 , x2 ).Èç (2) è (3) ñëåäóåò, ÷òî ∀ x1 , x2 , z ∈ H,∀ α, β ∈ C :(z, αx1 + βx2 ) = α · (z, x1 ) + β · (z, x2 ).Äåéñòâèòåëüíî,(z, αx1 + βx2 ) = (αx1 + βx2 , z) = α · (x1 , z) + β · (x2 , z) = α · (z, x1 ) + β · (z, x2 ).Ïðèìåðîì êîìïëåêñíîãî ãèëüáåðòîâà ïðîñòðàíñòâà ÿâëÿåòñÿ êîìïëåêñíîå ïðîñòðàíñòâî L2 (R1 ), ñîñòîÿùåå èç âñåõ (êëàññîâ ýêâèâàëåíòíîñòè) - êîìïëåêñíîçíà÷íûõ ôóíêöèéf, äëÿ êîòîðûõ:Z|f |2 dx < ∞,R1ñî ñêàëÿðíûì ïðîèçâåäåíèåì (·, ·), îïðåäåëÿåìûì òàê:Z(f1 , f2 ) =f1 (x) f2 (x) dx.R16415 ËåêöèÿÊîððåêòíîñòü îïðåäåëåíèÿ ïðîâåðÿåòñÿ àíàëîãè÷íî òîìó, êàê ýòî áûëî ñäåëàíî äëÿ âåùåñòâåííîãî ñëó÷àÿ.
Àíàëîãè÷íî îïðåäåëÿåòñÿ äëÿ êàæäîãî ïðîñòðàíñòâà ñ ìåðîé (Ω, A, ν)êîìïëåêñíîå ïðîñòðàíñòâî L2 (Ω, A, ν) è, â ÷àñòíîñòè, êîìïëåêñíîå l2 .Íàïîìíèì îïðåäåëåíèå ñîïðÿæåííîãî îïåðàòîðà:(A∗ x, z) = (x, Az)∀ x, z ∈ HÎïðåäåëåíèå 31. Ïóñòü H - ãèëüáåðòîâî ïðîñòðàíñòâî. Îïåðàòîð A ∈ L(H) íàçûâà-åòñÿ íîðìàëüíûì, åñëè âûïîëíÿåòñÿ ñëåäóþùåå óñëîâèå:A∗ · A = A · A∗Îïðåäåëåíèå 32. Îïåðàòîð V íàçûâàåòñÿ óíèòàðíûì, åñëè:V∗·V = V ·V∗ =I∀x I · x = xÎïðåäåëåíèå 33.
Îïåðàòîð A íàçûâàåòñÿ ñàìîñîïðÿæåííûì, åñëè:A∗ = A.Óíèòàðíûå è ñàìîñîïðÿæåííûå îïåðàòîðû ÿâëÿþòñÿ íîðìàëüíûìè.Óïðàæíåíèå 21. Ïðèâåñòè ïðèìåð íå åäèíè÷íîãî îïåðàòîðà, ÿâëÿþùåãîñÿ îäíîâðåìåííîñàìîñîïðÿæåííûì è óíèòàðíûì.15.2 Ñïåêòð îïåðàòîðîâ â ãèëüáåðòîâîì ïðîñòðàíñòâå.³´Ïóñòü A ∈ L(H). Ãîâîðÿò, ÷òî λ ∈ C íå ïðèíàäëåæèò ñïåêòðó A λ ∈/ spec A , åñëèîïåðàòîð Aλ = A − λI îáëàäàåò îáðàòíûì, êîòîðûé âñþäó îïðåäåëåí, òî åñòü òîãäà èòîëüêî òîãäà, êîãäà∃ A−1λ (∈ L(H)).Ï Ð È Ì Å Ð.
Ïóñòü H = L2 (R1 ) è(Af )(x) = gA (x) · f (x),ãäå gA ÿâëÿåòñÿ íåïðåðûâíîé êîìïëåêñíîçíà÷íîé ôóíêöèåé. Äëÿ òîãî, ÷òîáû îïåðàòîð áûëëèíåéíûì è íåïðåðûâíûì íåîáõîäèìî è äîñòàòî÷íî, ÷òîáû ôóíêöèÿ g áûëà îãðàíè÷åííîé.Óïðàæíåíèå 22. Ïîêàçàòü, ÷òîkAk = sup |gA (x)|.xÓïðàæíåíèå 23. Äîêàçàòü, ÷òî:(A∗ f )(x) = g A (x) · f (x).Ñëåäîâàòåëüíî, A∗ · A = A · A∗ , à, çíà÷èò, êàæäûé òàêîé îïåðàòîð íîðìàëåí.6515 ËåêöèÿÓïðàæíåíèå 24. Îïåðàòîð A ÿâëÿåòñÿ óíèòàðíûì ⇐⇒ |gA (x)| = 1 ∀ x.Óïðàæíåíèå 25.
Îïåðàòîð A ÿâëÿåòñÿ ñàìîñîïðÿæåííûì ⇐⇒ gA (x) ∈ R1 ∀ x.Àíàëîãè÷íî îïðåäåëÿåòñÿ â L2 (Ω, B, ν) îïåðàòîð óìíîæåíèÿ íà (êîìïëåêñíóþ) ñóùåñòâåííî îãðàíè÷åííóþ ôóíêöèþ.Óïðàæíåíèå 26. Äîêàçàòü, ÷òî:kAk = esssup |f (ω)| = (òî åñòü ñóùåcòâåííàÿ âåðõíÿÿ ãðàíü) =ω∈Ω= inf sup{|f (ω)| : ω ∈ Ω\N }.N ∈BνN =0Ò Å Î Ð Å Ì À 17. Åñëè A - íîðìàëüíûé ëèíåéíûé íåïðåðûâíûé îïåðàòîð â ãèëüáåðòîâîì ïðîñòðàíñòâå H, òî ñóùåñòâóþò ïðîñòðàíñòâî (Ω, A, ν) ñî ñ÷åòíî-àääèòèâíîé σ êîíå÷íîé ìåðîé ν è òàêîé èçîìîðôèçì èç H â L2 (Ω, A, ν), ÷òî ïðè íåì îïåðàòîð Aïåðåõîäèò â(Af )(x) = gA (x) · f (x),òî åñòü â îïåðàòîð óìíîæåíèÿ íà êàêóþ-òî êîìïëåêñíîçíà÷íóþ ôóíêöèþ gA (x).15.3 Êëàññèôèêàöèÿ òî÷åê ñïåêòðà.Ïóñòü λ ∈ spec A:1.Åñëè Ker Aλ 6= {0}, òî ãîâîðÿò, ÷òî ýëåìåíò λ ïðèíàäëåæèò òî÷å÷íîìó ñïåêòðó;òàêèì îáðàçîì, λ ïðèíàäëåæèò òî÷å÷íîìó ñïåêòðó òîãäà è òîëüêî òîãäà, êîãäà∃ x ∈ H,x 6= 0 :Aλ x = 0 (x ∈ Ker Aλ ), òàê ÷òîâñå âåêòîðû èç Aλ ÿâëÿþòñÿ ñîáñòâåííûìè âåêòîðàìè.2.
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