Лекции по функциональному анализу Смолянова О.Г., страница 7

Îòñþäà ñëåäóåò, ÷òî (fb)0 (z) + z fb(z) = 0, òàê êàê\c0 (z) = i · iz fb(z) = −z fb(z).(fb)0 (z) = (−ixf(x))(z) = ifÎáùåå ðåøåíèå óðàâíåíèÿ fb0 (z) + z fb(z) = 0 èìååò âèä2zfb(z) = Ce− 2 ;ïîñòîÿííàÿ C âû÷èñëÿåòñÿ òàê:1fb(z) = √2π1⇒ fb(0) = C = √2πZ∞x2e− 2 e−ixz dx ⇒−∞Z∞−∞x21 √e− 2 dx = √2π = 1 ⇒2π2zfb(z) = e− 2 .4811 Ëåêöèÿ11.1 Ñâîéñòâà ïðåîáðàçîâàíèÿ Ôóðüå â S .Ïðîñòðàíñòâî S ïðåîáðàçóåòñÿ ïðåîáðàçîâàíèåì Ôóðüå â S è áîëåå òîãî,Φ : S → S íåïðåðûâíî (ýòî áóäåò äîêàçàíî íèæå).Ïðåäïîëîæèì, ÷òî ϕ ∈ S è ìû çíàåì, ÷òî ϕb ∈ S è ïî ϕb õîòèì âîññòàíîâèòü ϕ.Ïóñòü ψ, ψb ∈ S; òîãäàZ∞ϕ³z ´aZ∞b dz =ψ(z)−∞Z∞ba ϕ(x) ψ(ax)dx =−∞ϕ(x) ψb³x´aZ∞dx =−∞ϕ(x)b ψ³x´adx−∞(ìû âîñïîëüçîâàëèñü çàìåíîé z = ax, çàòåì ñâîéñòâîì 5 è ðàâåíñòâîì Ïàðñåâàëÿ).Ïåðåõîäÿ ê ïðåäåëó ïðè a → ∞, ïîëó÷èìZ∞ϕ(0)d dz = ψ(0) ·ψ(z)−∞Ïóñòü ϕ(x) =2x√1 e− 22π1√2πZ∞Z∞ϕ(z)b dz.−∞z2⇒ ϕ(z)b = e− 2 .

ÒîãäàZ∞√1b dz = ψ(0) 2π =⇒ ψ(0) =b dz.ψ(z)ψ(z)2π−∞−∞Äàëåå íóæíî âîñïîëüçîâàòüñÿ ôîðìóëîé (ñâîéñòâî 6):\f (x+ a)(z) = e−iza fb(z).Ïóñòü ψ1 (x) = ψ(x + a);òîãäà ψ1 (0) =à òàê êàê, ïî ñâîéñòâó 6,12πR∞−∞ψb1 (z) dz ,\bψb1 (z) = ψ(x+ a)(z) = eiza ψ(z),òî1ψ(a) =2πZ∞b dz.eiza ψ(z)−∞Òàêèì îáðàçîì, ìû ïîëó÷èëè1ψ(x) =2πZ∞b dz;eixz ψ(z)−∞ïîýòîìó, îïðåäåëèâ îáðàòíîå ïðåîáðàçîâàíèå Ôóðüå ôóíêöèè f ðàâåíñòâîì1fˇ(x) =2πZ∞eixz f (z) dz,−∞4912 Ëåêöèÿb̌ïîëó÷èì ψ = ψ.Èç îïðåäåëåíèÿ îáðàòíîãî ïðåîáðàçîâàíèÿ Ôóðüå ñëåäóåò, ÷òî åãî ñâîéñòâà àíàëîãè÷íûñâîéñòâàì (ïðÿìîãî) ïðåîáðàçîâàíèÿ Ôóðüå.  ÷àñòíîñòè, îáðàòíîå ïðåîáðàçîâàíèå Ôóðüåòîæå ïåðåâîäèò S â S .Äîêàæåì, ÷òî∀ g ∈ S ∃ ϕ ∈ S : g = ϕ.bÏðèìåíèì ê g îáðàòíîå ïðåîáðàçîâàíèå, ïîëó÷èì ôóíêöèþ ϕ èç S , ê íåé ïðèìåíèìïðÿìîå ïðåîáðàçîâàíèå Ôóðüå, ïîëó÷èì ôóíêöèþ g : ϕ = ǧ è g = ϕb = b̌g.Èíà÷å ãîâîðÿ, ∀g ∈ SΦΦ−1 g = g(Φ - ïðåîáðàçîâàíèå Ôóðüå).Àíàëîãè÷íî ìîæíî äîêàçàòü è ðàâåíñòâî∀ϕ ∈ SΦ−1 Φ ϕ = ϕ.Ñëåäîâàòåëüíî, Φ - âçàèìíî îäíîçíà÷íîå îòîáðàæåíèå S íà S .Îïðåäåëåíèå 26.

Ïðåîáðàçîâàíèå Ôóðüå îáîáùåííîé ôóíêöèè F ∈ S ∗ îïðåäåëÿåòñÿ ñëåäóþùèì îáðàçîì:(Fb, ϕ) = (F, ϕ).bÏóñòü g ∈ L1 . Òîãäà óòâåðæäàåì, ÷òî Fbg = Fgb (ñîãëàñîâàííîñòü îïðåäåëåíèé ïðåîáðàçîâàíèÿ Ôóðüå â S ∗ è â L1 ).Ïðîâåðèì ýòî:Z∞(Fbg , ϕ) = (Fg , ϕ)b = (g, ϕ)b =Z∞g(x)ϕ(x)b dx =−∞gb(x)ϕ(x) dx = (bg , ϕ) = (Fgb, ϕ).−∞Ïðîâåðèì, ÷òî òàê îïðåäåëåííûé ëèíåéíûé ôóíêöèîíàë Fb íà S íåïðåðûâåí, òî åñòüäåéñòâèòåëüíî ÿâëÿåòñÿ ýëåìåíòîì ïðîñòðàíñòâà S ∗ . Òàê êàê ôóíêöèîíàë Fb ëèíååí, äîñòàòî÷íî äîêàçàòü åãî íåïðåðûâíîñòü â íóëå.Èòàê, ïóñòü ϕn → 0 â S.

Òîãäà(Fb, ϕn ) = (F, ϕbn ) −→ 0,òàê êàê, â ñèëó íåïðåðûâíîñòè ïðåîáðàçîâàíèÿ Ôóðüå â S (îíà áóäåò äîêàçàíà ïîçæå),ϕbn → 0 â S.12 ËåêöèÿÏðåîáðàçîâàíèå Ôóðüå â S ∗ áóäåì îáîçíà÷àòü òåì æå ñèìâîëîì Φ; òàêèì îáðàçîì,Φ : S ∗ → S ∗.5012 Ëåêöèÿ12.1 Ñâîéñòâà ïðåîáðàçîâàíèÿ Ôóðüå â S ∗ .(1) Íàäåëèì S ∗ ñëàáîé òîïîëîãèåé (S ∗ , σ(S ∗ , S)). Òîãäà ñõîäèìîñòü â ñëàáîé òîïîëîãèèîïðåäåëÿåòñÿ òàê:Fn → F ⇐⇒ ∀ ϕ ∈ S (Fn , ϕ) → (F, ϕ).Ïîêàæåì, ÷òî Φ íåïðåðûâíî â (S ∗ , σ(S ∗ , S)).

Òàêèì îáðàçîì, ìû äîëæíû ïðîâåðèòü,÷òîåñëè Fn → F, òî Fbn → Fb ⇐⇒ ∀ ψ ∈ S (Fbn , ψ) → (Fb, ψ).b → (F, ψ)b = (Fb, ψ).Âîò ýòà ïðîâåðêà: (Fbn , ψ) = (Fn , ψ)Äîêàæåì àíàëîãèè ñâîéñòâ 1, 2 äëÿ ïðåîáðàçîâàíèÿ Ôóðüå â S ∗ .(2) Áûëî ïîêàçàíî, ÷òî ∀ ϕ ∈ S,ϕb0 (z) = iz ϕ(z)b . Ïóñòü f ∈ S ∗ , ∀ϕ ∈ S :d = −(fb, −ixϕ) = (ixfb, ϕ) ⇒(fb0 , ϕ) = (f 0 , ϕ)b = −(f, ϕb0 ) = −(f, −ixϕ)⇒ fb0 = ixfb.(3) Áûëî ïîêàçàíî, ÷òî ∀ ϕ ∈ S,³´\ (z). Ïóñòü F ∈ S ∗ , ∀ϕ ∈ S :(ϕ)b 0 (z) = −ixϕ(x)c0 , ϕ) = −(Fb, ϕ0 ) = −(F, (ϕ)d , ϕ) ⇒(Fb 0 ) = −(F, ixϕ(x))b= −(ixF, ϕ)b = −(ixFd.⇒ (Fb)0 = −ixFÀíàëîãè îñòàëüíûõ ñâîéñòâ ïðåäëàãàåòñÿ äîêàçàòü â êà÷åñòâå óïðàæíåíèÿ.ÓÒÂÅÐÆÄÅÍÈÅ 5. Îáðàç ëþáîé ôóíêöèè èç S ïðè ïðåîáðàçîâàíèè Ôóðüå ëåæèò â S .Äðóãèìè ñëîâàìè, Φ(S) ⊂ S (èëè ϕ ∈ S ⇒ ϕb ∈ S ).Äîêàçàòåëüñòâî.

Íóæíî ïðîâåðèòü, ÷òî∀n, k|(1 + x2 )n ϕb(k) (x)| → 0, ïðè |x| → ∞.Ñäåëàåì ýòî:¯¯ ¯¯ (1 + x2 )n¯¯¯ ¯¯¯ ¯2 n(k)2n+2\k\k¯(1 + x2 )n · ϕ¯b (x) = ¯(1 + x ) · (−ix) ϕ¯ = ¯ 2n+2 · (ix)· (−ix) ϕ¯¯ =x¯¯¯ (1 + x2 )n ¡ \¢2n+2 ¯¯ → 0 ïðè |x| → ∞,= ¯¯ 2n+2 (−ix)k ϕ¯xòàê êàê(1 + x2 )n→ 0 ïðè |x| → ∞,x2n+2à ôóíêöèÿ ïîä çíàêîì ïðåîáðàçîâàíèÿ Ôóðüå ñïðàâà ñîäåðæèòñÿ â S è òåì áîëåå â L1 è,çíà÷èò, åå ïðåîáðàçîâàíèå Ôóðüå îãðàíè÷åíî.¤5112 ËåêöèÿÒ Å Î Ð Å Ì À 14.

Ïðåîáðàçîâàíèå Ôóðüå Φ : S → S - íåïðåðûâíî.Äîêàçàòåëüñòâî.Ïóñòü ϕn → 0 â S ⇐⇒ ∀ k, m xk ϕ(m)n (x) → 0 ðàâíîìåðíî ïî x ïðè n → ∞.(m)Íàì íàäî äîêàçàòü, ÷òî ϕbn → 0 â S, òî åñòü, ÷òî ∀k, m z k ϕbn (x) → 0 ðàâíîìåðíî ïî xïðè n → ∞, òî åñòü, ÷òî:m ϕ(x))(k) n→∞(−i)k ((−ix)\→ 0ðàâíîìåðíî ïî x. Ïîñëåäîâàòåëüíîñòü ôóíêöèé ïîä çíàêîì ïðåîáðàçîâàíèé Ôóðüå ñõîäèòñÿ ê íóëþ â S . Ìû ïîêàæåì, ÷òî îíà ñõîäèòñÿ ê íóëþ â L1 ; òîãäà ïîñëåäîâàòåëüíîñòü ååïðåîáðàçîâàíèé Ôóðüå áóäåò ñõîäèòüñÿ ê íóëþ ðàâíîìåðíî, ÷òî è òðåáóåòñÿ.Ïóñòü ∀ m, kÄîêàæåì, ÷òîgnmk = ((−ix)m ϕn (x))kϕn → 0 â S =⇒ ∀ m, kgnmk → 0 â L1 .Åñëè ϕn → 0 â S, òîn→∞ϕn (x) → 0 è (1 + x2 ) ϕn (x) → 0 ðàâíîìåðíî ïî x (è äàæå â S).Êðîìå òîãî,¯Z ¯¯ (1 + x2 ) ϕn (x) ¯¯¯|ϕn (x)| dx =¯ (1 + x2 ) ¯ dx =Zkϕn kL1 =RZ−A=R|(1 + x2 ) ϕn (x)|dx +1 + x2−∞Z∞|(1 + x2 ) ϕn (x)|dx +1 + x2AZA|ϕn (x)| dx−AÎöåíèâàåì ïîëó÷èâøèåñÿ èíòåãðàëû:Z−A|(1 + x2 ) ϕn (x)|dx 6 max |(1 + x2 ) ϕn (x)| ·21+x−∞Z∞Z−A1dx;1 + x2−∞|(1 + x2 ) ϕn (x)|dx 6 max |(1 + x2 ) ϕn (x)| ·21+xAZ∞1dx;1 + x2AZ∞∀ ε ∃ A > 0:1dx < ε è1 + x2Z−A1dx < ε.1 + x2−∞AÒåïåðü âûáåðåì n0 òàêîå, ÷òî:ZA|ϕn (x)| dx < ε;∀ n > n0−A5212 Ëåêöèÿïîñëåäíåå âîçìîæíî, òàê êàê ϕn (x) → 0 ðàâíîìåðíî ïî x.

Ñëåäîâàòåëüíî, ìû äîêàçàëè, ÷òîkϕn k L1 → 0, à, ñëåäîâàòåëüíî, èõ ïðåîáðàçîâàíèÿ Ôóðüå ñòðåìÿòñÿ ê íóëþ ðàâíîìåðíî.¤Îïðåäåëèì ïðåîáðàçîâàíèå Ôóðüå â L2 (R). Äîêàæåì ñíà÷àëà, ÷òî åñëè ϕ, ψ ∈ S, òîbL2π(ϕ, ψ)L2 = (ϕ,b ψ)2(∗).Ðàâåíñòâî îçíà÷àåò, ÷òî ïðåîáðàçîâàíèå Ôóðüå ñîõðàíÿåò íîðìó L2 (ñ òî÷íîñòüþ äî óìíîæåíèÿ íà 2π ), à òàê êàê S - ïëîòíî â L2 , òî òîãäà ìîæíî ïðîäîëæèòü Φ íà L2 .Åñëè g ∈ L2 , òî (òàê êàê S ïëîòíî â L2 ):∃ ϕn ∈ S :ϕn → g â L2 .Çíà÷èò, ïîñëåäîâàòåëüíîñòü ϕn - ôóíäàìåíòàëüíà â L2 , òî åñòükϕn − ϕn+k kL2 → 0 ïðè n → ∞ ðàâíîìåðíî ïî k.Òàê êàê íîðìû ñîõðàíÿþòñÿ (ñ òî÷íîñòüþ äî óìíîæåíèÿ íà 2π ), òîkϕbn − ϕbn+k kL2 → 0.Èç óñëîâèÿ ïîëíîòû L2 ïîëó÷èì, ÷òî ∃ ôóíêöèÿ gb â L2 , òàêàÿ ÷òî ϕb → gb â L2 .Ýòó ôóíêöèþ gb ìû íàçûâàåì ïðåîáðàçîâàíèåì Ôóðüå ôóíêöèè g ∈ L2 .Äîêàçàòåëüñòâî.

(*) Èç ôîðìóë ïðÿìîãî è îáðàòíîãî ïðåîáðàçîâàíèÿ Ôóðüå:ZZ1−ixzϕb= eϕ(x) dx è ϕ̌ =eixz ϕ(z)b dz.(2π)RRb = (2π)ϕ̌. Èòàê,ñëåäóåò, ÷òî ϕZ∞(ϕ, ψ)L2 =b = 2π(ϕ̌, ψ̌) = 2π(ϕ̌, ψ̌) .b̌ ψ) = (ϕ̌, ψ)ϕ(x)ψ(x) dx = (ϕ, ψ) = (ϕ,L2−∞Ìû äîêàçàëè, ÷òîåñëè ϕ, ψ ∈ S, òî (ϕ, ψ)L2 = 2π(ϕ̌, ψ̌)L2 .Ïóñòü ϕ = ϕb1 , ψ = ψb1 , òîãäà(ϕb1 , ψb1 )L2 = 2π(ϕ1 , ψ1 )L2 .¤5313 Ëåêöèÿ13 ËåêöèÿÓïðàæíåíèå 16. Ïóñòü g ∈ L2 (R1 ); òîãäà∀ ϕ ∈ S g · ϕ ∈ L1 (R1 )Rè, áîëåå òîãî, ôóíêöèîíàë Fg (ϕ) = g · ϕ dx íåïðåðûâåí íà S, òî åñòü ÿâëÿåòñÿ ýëåìåíR1cg , ïðî êîòîðîå ìûòîì ïðîñòðàíñòâà S . Ïîýòîìó îïðåäåëåíî åãî ïðåîáðàçîâàíèå Ôóðüå Fãîâîðèì, ÷òî ýòî ïðåîáðàçîâàíèå Ôóðüå ôóíêöèè g â ñìûñëå òåîðèè îáîáùåííûõ ôóíêöèéíàä S.

Ñ äðóãîé ñòîðîíû, ìû òîëüêî ÷òî îïðåäåëèëè ïðåîáðàçîâàíèå Ôóðüå äëÿ ôóíêöèèèç L2 íåïîñðåäñòâåííî. Ïîêàæåì, ÷òî îíè ñîâïàäàþò , òî åñòü ÷òî∗Fbg = Fgb.Äîêàçàòåëüñòâî. Çàìåòèì, ÷òî â ñèëó (íåïîñðåäñòâåííîãî) îïðåäåëåíèÿ ïðåîáðàçîâàíèÿÔóðüå ôóíêöèè g ∈ L2 ∀ ϕ ∈ S(⊂ L2 ) :(bg , ϕ)b L2 = 2π(g, ϕ)L2 .Ïîýòîìó ∀ ϕ ∈ SZcg , ϕ) = (Fg , ϕ)(Fb =b L2 =g(t) ϕ(t)b dt = (g, ϕ)R1=1bb(bg , ϕ)g , ϕ)L2 = (Fgb, ϕ).L2 = (b2πÑëåäîâàòåëüíî Fbg = Fgb.

×òî è òðåáîâàëîñü äîêàçàòü.Çàìå÷àíèå. Èçâåñòíî, ÷òî ∀ t > 01√2πtZx2e− 2t e−ixz dx = e−tz 22.RÝòî çíà÷èò, ÷òî−tz 21\− x2e 2t (z) = e 2 .2πt£¤Ïóñòü òåïåðü äëÿ êàæäîãî γ = ρ eiα : ρ > 0, α ∈ − π2 , π2 ,√gγ (x) =1 − x2γ2e2πγèψγ (z) = e−γz 22.Òîãäà ôóíêöèè γ 7→ Fgγ ∈ S ∗ è γ 7→ Fψγ ∈ S ∗ àíàëèòè÷íû íà îáëàñòè³ π π´{γ = ρ eiα : ρ > 0, α ∈ − ,}2 2è íåïðåðûâíû íà ìíîæåñòâåh π πi{γ = ρ eiα : ρ > 0, α ∈ − ,}2 25413 ËåêöèÿÝòî çíà÷èò, ÷òî ∀ ϕ ∈ S, íà ýòèõ îáëàñòÿõ àíàëèòè÷íû (ñîîòâåòñòâåííî, íåïðåðûâíû)ôóíêöèèγ 7→ (Fψγ , ϕ), γ 7→ (Fgγ , ϕ).Ïðè ýòîì äëÿ γ ∈ (0, ∞) Fcgγ = Fψγ .

Ïîýòîìó, â ñèëó òåîðåìû åäèíñòâåííîñòè äëÿ àíàëèòè÷åñêèõ ôóíêöèé, ýòî ðàâåíñòâî ñïðàâåäëèâî äëÿ âñåõ äîïóñòèìûõ γ.πÄëÿ γ = ei 2 îíî îçíà÷àåò, ÷òî\iπ ix2iz 2e− 4 e 2√(z) = e− 2 .2πÒàêèì îáðàçîì, ñ ïîìîùüþ òåîðèè îáîáùåííûõ ôóíêöèé ìû ñîñ÷èòàëè ïðåîáðàçîâàíèåÔóðüå íåèíòåãðèðóåìîé ôóíêöèè íà ïðÿìîé.Çàìå÷àíèå.

Ïóñòü F ∈ S ∗ è g ∈ E(R1 ). Äëÿ íåêîòîðûõ ôóíêöèé ìîæíî âûáðàòü åñòå-ñòâåííîå ïðîäîëæåíèå ôóíêöèîíàëà F, ïåðâîíà÷àëüíî îïðåäåëåííîãî íà S, íà áîëåå øèðîêóþ îáëàñòü, ñîäåðæàùóþ g .Ïóñòü ψ ∈ D,1. ψ(t) = 1, åñëè t ∈ [−1, 1].2. ψ(t) = 0, åñëè t ∈/ [−1 − ε, 1 + ε].a→0Ïóñòü åùå äëÿ a > 0 ψa (t) = ψ(a t). ßñíî, ÷òî ∀ t ψa (t) → 1 (êðîìå òîãî, ψα → 1è â S ∗ ).Áîëåå òîãî, ∀ ϕ ∈ S,åñëè (ψa · ϕ)(t) = ψa (t) · ϕ(t),a→0òîψa · ϕ −→ ϕ â Sèψa (t) · ϕ(t) −→ ϕ(t) ∀ t(∗)Óïðàæíåíèå 17.

Äîêàçàòü, ÷òî âûïîëíÿåòñÿ (∗).Äîêàçàòåëüñòâî. Èç (∗) ïîëó÷àåì, ÷òîa→0ψa · F −→ F â S ∗(∗∗)×òîáû ïðîâåðèòü (∗∗), íóæíî ïîñìîòðåòü, âûïîëíÿåòñÿ ëè:∀ϕ ∈ Sa→0(ψa · F, ϕ) −→ (F, ϕ).5513 ËåêöèÿÏðîâåðèì ýòî:a→0(ψa · F, ϕ) = (F, ψa · ϕ) −→ (F, ϕ)Ýòèì ìû äîêàçàëè, ÷òî (∗) ⇒ (∗∗).Òåïåðü ïóñòü g ∈ E, g ∈/ S è F ∈ S ∗ îáëàäàþò ñëåäóþùèì ñâîéñòâîì:∃ lim (F, ψa · g).a→0Òîãäà ïî îïðåäåëåíèþ ïîëàãàåì, ÷òî:def(F, g) = lim (F, ψa · g).Òèïè÷íûé ïðèìåð òàêîãî ðàñøèðåíèÿ îáëàñòè îïðåäåëåíèÿ ôóíêöèîíàëà (îáîáùåííîé ôóíêöèè) F .Ïóñòü F1 = δ, F2 = δ 0 , òîãäà∀ g ∈ E (δ, g) = g(0) è (δ 0 , g) = −g 0 (0).Ýòî è åñòü ïðîäîëæåíèå.Ïóñòü ϕ ∈ S(R2 ) è F ∈ S ∗ (R1 ).

ßñíî, ÷òî∀ ϕ ∈ R1[z 7→ ϕ(x, z)] ∈ SÁîëåå òîãî, ìû ìîæåì ïðîèíòåãðèðîâàòü (ñåé÷àñ - â îáû÷íîì ñìûñëå) è ïîëó÷èì, ÷òîZz 7→ϕ(x, z) dz  ∈ SÌû õîòèì ïîñ÷èòàòü: (F,çíà÷åíèå äåéñòâèÿ:ïðè Φ ∈ S ∗RR1ϕ(x, z) dz). Äëÿ óäîáñòâà çàïèñè ââåäåì òðàäèöèîííîå îáî-R1Z(Φ, ψ) =Φ(t)ψ(t) dt −R- òî åñòü ìû ââîäèì ñèìâîë, êîòîðûé ïî îïðåäåëåíèþ ðàâåí ëåâîé ÷àñòè.Èòàê,ZZZ(F,ϕ (·, z) dz) =F (x) ϕ(x, z) dz =R1(1)ZRZ=RÄàëåå áåç èñïîëüçîâàíèÿ ñèìâîëàRRF (x)ϕ(x, z) dx dzRíàïèøåì òî æå ñàìîå:Z(F, ϕ (·, z)) dz=R56=13 ËåêöèÿÌû õîòèì äîêàçàòü(1).

Ñäåëàåì ýòî:RÝòîò èíòåãðàë ϕ (·, z) dz ìû áóäåì ïîíèìàòü, êàê èíòåãðàë Ðèìàíà, òàê ÷òîRRPϕ (x, z) dz − ýòî ïðåäåë ïîñëåäîâàòåëüíîñòè ðèìàíîâûõ ñóìì âèäàϕ(x, zj ) ∆zj , êàæjRäàÿ èç êîòîðûõ áóäåò çàâèñåòü îò x, è êàê ôóíêöèÿ x áóäåòR ñíîâà ýëåìåíòîì ïðîñòðàíñòâàS . Ïîëó÷åííàÿ ïîñëåäîâàòåëüíîñòü áóäåò ñõîäèòüñÿ ê ϕ (·, z) dz â ïðîñòðàíñòâå S.RÑëåäîâàòåëüíî, ïðèìåíåíèå F ê ýòîìó èíòåãðàëó ìîæíî ïîíèìàòü êàê ïðåäåë ðåçóëüòàòîâ ïðèìåíåíèé F ê ðèìàíîâûì ñóììàì:RP(F,ϕ (·, zj ) ∆zj ) −−−→ (F, ϕ (·, z) dz)j°°°°°°RP((F, ϕ (·, zj )) ∆zj ) −−−→(F, ϕ (·, z)) dzjR∗Ó÷èòûâàÿ, ÷òî F ∈ S , ϕ ∈ S, ïîëó÷àåì:Z(Fb, ϕ) = (F, ϕ)b = (F,RZ−ixz= lim (ψa · F,ϕ(x) · ea→0ϕ(x) · e−ixz dx) =RZψa (·) ϕ(x)e−ix· dx) =dx) = lim (F,a→0RZ¡¢ϕ(x) F, ψa (·) e−ix· dx= lima→0=RÏðåäïîëîæèì, ÷òî∃ lim (F, ψa (·) e−ix· ) = (F, e−ix· ). Òîãäàa→0Z¡¢=ϕ(x) lim (F, ψa (·) e−ix· ) dx = (F, e−ix· ), ϕa→0RÑëåäîâàòåëüíî, ìû ïîëó÷àåì, ÷òî: Fb(z) = (F, e−ixz ).Åñëè g ∈ L1 , òî ýëåìåíò Fg ∈ S ∗ ìîæíî ïðèìåíèòü ê ôóíêöèè x 7→ e−ixz (íå ÿâëÿþùåéñÿýëåìåíòîì S ):Z−ixzbFg (z) = (Fg , e) =g(x) e−ixz dx.RÒàêèì îáðàçîì, ýòî îïðåäåëåíèå ïðåîáðàçîâàíèÿ Ôóðüå äåéñòâèòåëüíî ñîãëàñóåòñÿ ñîáû÷íûì îïðåäåëåíèåì.

Ïðèìåðû:ZFb(z) = (F, e−ixz ) =b = (δ, e−ixz ) = 1,δ(z)δb0 (z) = (δ 0 , e−ixz ) = iz,F (x) e−ixz dz − ñèìâîë, êîòîðûé ìû ââåëè ðàíåå íà ñòð. 56RÀíàëîãè÷íî,F̌ (z) =1(F, eixz ).2π5714 Ëåêöèÿ14 Ëåêöèÿ14.1 Ïðÿìûå è îáðàòíûå îáðàçû îáîáùåííûõ ôóíêöèé.Îïðåäåëåíèå 27. Ïóñòü E1 = Rn , E2 = Rk è g - îòîáðàæåíèå E1 â E2 , ϕ - ôóíêöèÿ íàE2 è ν - ìåðà íà σ -àëãåáðå áîðåëåâñêèõ ïîäìíîæåñòâ E1 .Òîãäà ïðÿìîé îáðàç g∗ ν ìåðû ν (åñëè g èçìåðèìî) è îáðàòíûé îáðàç g ∗ ϕ ôóíêöèè ϕîïðåäåëÿþòñÿ òàê:g ∗ ϕ − ýòî ôóíêöèÿ íà E2 , îïðåäåëÿåìàÿ ðàâåíñòâîì:(g ∗ ϕ)(x) = ϕ(g(x)),à g∗ ν - ýòî ìåðà íà E2 (ìû åå îïðåäåëèëè â ðàçäåëå, ïîñâÿùåííîì òåîðèè ìåðû).Ïóñòü F1 ∈ S ∗ (E1 ) è F2 ∈ S ∗ (E2 ).