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Çíà÷èò,C|f (x)e∓i(x,y) | = |f (x)| ≤,1 + |x|pà, êàê èçâåñòíî èç êóðñà ìàòåìàòè÷åñêîãî àíàëèçà, ïîñëåäíÿÿ ôóíêöèÿèíòåãðèðóåìà ïî âñåìó ïðîñòðàíñòâó Rn , åñëè òîëüêî p > n.Îòìåòèì òàêæå, ÷òî ïðè n = 1 îïðåäåëåíèå ïðåîáðàçîâàíèÿ Ôóðüå,äàííîå â íàñòîÿùåì ïàðàãðàôå, ñîâïàäàåò ñ îïðåäåëåíèåì, äàííûì ðàíåå â 5.Ðàññìîòðèì íàèáîëåå óïîòðåáèòåëüíûå ñâîéñòâà ïðåîáðàçîâàíèÿ Ôóðüå áûñòðî óáûâàþùèõ ôóíêöèé.1) Ïðåîáðàçîâàíèå Ôóðüå ëèíåéíî, òî åñòü äëÿ ëþáûõ a, b ∈ C èëþáûõ f, g ∈ S(Rn ) ñïðàâåäëèâû ðàâåíñòâà F± [af +bg] = aF± [f ]+bF± [g].Äîêàçàòåëüñòâî íåìåäëåííî ñëåäóåò èç ëèíåéíîñòè èíòåãðàëà:ZF± [af + bg](y) = (2π)−n/2 [af (x) + bg(x)]e∓i(x,y) dx =Z= a(2π)−n/2RnZf (x)e∓i(x,y) dx + b(2π)−n/2g(x)e∓i(x,y) dx =RnRn= aF± [f ](y) + bF± [g](y).2) Äëÿ ëþáîãî ìóëüòèèíäåêñà α è ëþáîé áûñòðî óáûâàþùåé ôóíêöèè f ñïðàâåäëèâû ðàâåíñòâàF± [xα f (x)] = (±i)|α| Dα (F± [f ]).Äîêàçàòåëüñòâî âûòåêàåò èç ñëåäóþùåãî âû÷èñëåíèÿ:µ¶·¸ZDα F± [f ](y) = Dα (2π)−n/2 f (x)e∓i(x,y) dx =Rn31= (äèôôåðåíöèðóåì ïîä çíàêîì èíòåãðàëà) =Z= (2π)−n/2f (x1 , .
. . , xn )Rn∂ |α| e∓i(x1 y1 +···+xn yn )dx1 . . . dxn = (2π)−n/2 ×∂y1α1 . . . ∂ynαnZ(∓ix1 )α1 . . . (∓ixn )αn f (x1 , . . . , xn )e∓i(x1 y1 +···+xn yn ) dx1 . . . dxn =×RnZ|α|= (∓i)(2π)−n/2xα f (x)e∓i(x,y) dx = (∓i)|α| F± [xα f (x)](y).RnÄëÿ çàâåðøåíèÿ äîêàçàòåëüñòâà íàì îñòàëîñü îáîñíîâàòü çàêîííîñòüäèôôåðåíöèðîâàíèÿ ïîä çíàêîì èíòåãðàëà. Äëÿ ýòîãî, êàê èçâåñòíî,äîñòàòî÷íî óáåäèòüñÿ â íàëè÷èè èíòåãðèðóåìîé ìàæîðàíòû äëÿ ïðîèçâîäíîé x 7→ xα f (x)e∓i(x,y) îò åãî ïîäûíòåãðàëüíîé ôóíêöèè. Òàêàÿìàæîðàíòà äåéñòâèòåëüíî áåç òðóäà ìîæåò áûòü óêàçàíà, íàäî ëèøüïðèíÿòü âî âíèìàíèå, ÷òî ôóíêöèÿ x 7→ xα f (x) ÿâëÿåòñÿ áûñòðî óáûâàþùåé, à ìîäóëü ýêñïîíåíòû ñ ÷èñòî ìíèìûì ïîêàçàòåëåì ðàâåí åäèíèöå:|xα f (x)e∓i(x,y) | = |xα f (x)| ≤C,1 + |x|pãäå â êà÷åñòâå p ñëåäóåò âçÿòü ëþáîå ÷èñëî, áîëüøåå n. äàëüíåéøåì ìû íå áóäåì äåòàëüíî îáîñíîâûâàòü çàêîííîñòü îïåðàöèé, êîòîðûå íàì ïðèäåòñÿ ïðîèçâîäèòü ïîä çíàêîì èíòåãðàëà, îñòàâëÿÿ ñîîòâåòñòâóþùèå âîïðîñû ÷èòàòåëþ.
Ïðè÷èíà êðîåòñÿ â åäèíîîáðàçèè ïîäîáíûõ ðàññóæäåíèé: êàæäûé ðàç ðåøàþùóþ ðîëü èãðàåò íàëè÷èå èíòåãðèðóåìîé ìàæîðàíòû ó íåêîòîðîãî âûðàæåíèÿ; ñàìà æå ìàæîðàíòà êàæäûé ðàç ñòðîèòñÿ áåç ïðîáëåì, ïîñêîëüêó ìû ðàáîòàåì ñáûñòðî óáûâàþùèìè ôóíêöèÿìè.3) Äëÿ ëþáîãî ìóëüòèèíäåêñà α è ëþáîé áûñòðî óáûâàþùåé ôóíêöèè f ñïðàâåäëèâû ðàâåíñòâàF± [Dα f (x)](y) = (±iy)α (F± f )(y).Äîêàçàòåëüñòâî âûòåêàåò èç ñëåäóþùèõ âû÷èñëåíèé:ZF± [Dα f (x)](y) = (2π)−n/2Rn32[Dα f (x)]e∓i(x,y) dx ==(ïðåâðàùàåì êðàòíûé èíòåãðàë â ïîâòîðíûé, âûäåëÿÿ â êà÷åñòâå âíóòðåííåãî îäíîìåðíûé èíòåãðàë ïî ïåðåìåííîé x1 )=+∞Z ½Z−n/2= (2π)Rn−1−∞·¸¾∂ |α|−1 f∂∓ix1 y1edx1 ×αn2∂x1 ∂x1α1 −1 ∂xα2 .
. . ∂xn×e∓i(x2 y2 +···+xn yn ) dx2 . . . dxn .(25)Ïðèìåíÿÿ ê âíóòðåííåìó èíòåãðàëó ôîðìóëó èíòåãðèðîâàíèÿ ïî ÷àñòÿì, ïîëó÷àåì äëÿ íåãî ñëåäóþùåå âûðàæåíèå:+∞¯+∞ Z¯∂ |α|−1 f∂ |α| f∓ix1 y1∓ix1 y1 ¯−(∓iy1 ) α1 −1 α2dx1 .α1α2αn eαn e¯∂x1 ∂x2 . . . ∂xn∂x∂x...∂xn−∞12−∞Ïîñêîëüêó ôóíêöèÿ f áûñòðî óáûâàþùàÿ, òî âíåèíòåãðàëüíûå ñëàãàåìûå çàíóëÿþòñÿ. Ïîýòîìó ìû ìîæåì ïðîäîëæèòü ðàâåíñòâî (25) ñëåäóþùèì îáðàçîì:Z∂ |α|−1 f∓i(x,y)F± [Dα f (x)](y) = (±iy1 )(2π)−n/2dx.α1 −1αn e2∂x1∂xα...∂xn2RnÒàêèì îáðàçîì, íàì óäàëîñü ïîíèçèòü ïîðÿäîê äèôôåðåíöèðîâàíèÿïî ïåðåìåííîé x1 íà åäèíèöó, íî ïðè ýòîì èç-ïîä çíàêà èíòåãðàëà ¾âûñêî÷èë¿ äîïîëíèòåëüíûé ìíîæèòåëü ±iy1 .
Ïðèìåíÿÿ ïîäîáíûå ðàññóæäåíèÿ ìíîãîêðàòíî ê êàæäîé èç ïåðåìåííûõ x1 , x2 , . . . , xn , ìû ìîæåìïîëíîñòüþ èçáàâèòüñÿ îò ïðîèçâîäíûõ ôóíêöèè f . Ïðè ýòîì ïîëó÷èìZαα1αn−n/2f (x)e∓i(x,y) dx =F± [D f (x)](y) = (±iy1 ) . . . (±iyn ) (2π)Rnα= (±iy) F± [f ](y).Ñâîéñòâà 2 è 3 èíîãäà âûðàæàþò ñëîâàìè, ãîâîðÿ, ÷òî ïðåîáðàçîâàíèå Ôóðüå ïåðåâîäèò (ñ òî÷íîñòüþ äî ïîñòîÿííîãî ìíîæèòåëÿ) îïåðàöèþ óìíîæåíèÿ íà íåçàâèñèìóþ ïåðåìåííóþ â îïåðàöèþ äèôôåðåíöèðîâàíèÿ è íàîáîðîò.4) Ïóñòü A íåâûðîæäåííàÿ n × n-ìàòðèöà, b n-ìåðíûé âåêòîðè f : Rn → C áûñòðî óáûâàþùàÿ ôóíêöèÿ. ÒîãäàF± [f (Ax + b)](y) = | det A|−1 e±i(y,A−1b)F± [f (x)]((A−1 )∗ y).Çäåñü A−1 îáîçíà÷àåò ìàòðèöó, îáðàòíóþ ê A, à (A−1 )∗ îáîçíà÷àåò ìàòðèöó, ñîïðÿæåííóþ ê A−1 , ò.
å. òàêóþ (åäèíñòâåííûì îáðàçîì îïðåäåëåííóþ) ìàòðèöó, ÷òî äëÿ ëþáûõ âåêòîðîâ u, v ∈ Rn ñïðàâåäëèâîðàâåíñòâî (A−1 u, v) = (u, (A−1 )∗ v).33Äîêàçàòåëüñòâî íåïîñðåäñòâåííî ñëåäóåò èç ôîðìóëû çàìåíû ïåðåìåííîé â êðàòíîì èíòåãðàëå:ZF± [f (Ax + b)](y) = (2π)−n/2 f (Ax + b)e∓i(x,y) dx =Rn=(äåëàåì çàìåíó ïåðåìåííûõ Ax + b = z )=Z−1−1−n/2= (2π)f (z)e∓i(A z−A b,y) | det A|−1 dz =Rn= | det A|−1 ±i(y,A−1 b)eZ(2π)−n/2f (z)e∓i(z,(A−1 ∗) y)dz =Rn= | det A|−1 e±i(y,A−1b)F± [f (x)]((A−1 )∗ y).Ñâîéñòâî 4 ïîêàçûâàåò, êàê ìåíÿåòñÿ ïðåîáðàçîâàíèå Ôóðüå, êîãäà âèñõîäíîé ôóíêöèè äåëàåòñÿ ëèíåéíàÿ íåâûðîæäåííàÿ çàìåíà ïåðåìåííûõ. Íèæå ìû ïðèâîäèì äâà íàèáîëåå ÷àñòî èñïîëüçóåìûõ ñëåäñòâèÿñâîéñòâà 4.5) Åñëè f : Rn → C áûñòðî óáûâàþùàÿ ôóíêöèÿ, à x0 ∈ Rn , òîF± [f (x − x0 )](y) = e∓i(y,x0 ) F± [f ](y).Äîêàçàòåëüñòâî ìîæåò áûòü ïîëó÷åíî íåïîñðåäñòâåííûì ïðèìåíåíèåì ñâîéñòâà 4 äëÿ ñëó÷àÿ, êîãäà A åäèíè÷íàÿ ìàòðèöà, à b = −x0 .Îòìåòèì òîëüêî, ÷òî ïðè ýòîì A−1 = (A−1 )∗ òàêæå ÿâëÿåòñÿ åäèíè÷íîéìàòðèöåé.Ñâîéñòâî 5 èçâåñòíî â ëèòåðàòóðå êàê òåîðåìà î ñäâèãå.
Ñëîâàìè ååôîðìóëèðóþò òàê: ïðåîáðàçîâàíèå Ôóðüå ïåðåâîäèò ñäâèã ïî àðãóìåíòóâ ñäâèã ïî ôàçå.6) Åñëè f : Rn → C áûñòðî óáûâàþùàÿ ôóíêöèÿ, à a îòëè÷íîåîò íóëÿ âåùåñòâåííîå ÷èñëî, òîµ ¶1yF± [f (ax)](y) = n F± [f (x)].|a|aÄîêàçàòåëüñòâî îïÿòü ìîæåò áûòü ïîëó÷åíî íåïîñðåäñòâåííûì ïðèìåíåíèåì ñâîéñòâà 4. Íà ýòîò ðàç âåêòîð b íàäî ñ÷èòàòü ðàâíûì íóëþ,à ìàòðèöó A äèàãîíàëüíîé, ó êîòîðîé íà ãëàâíîé äèàãîíàëè ñòîèò÷èñëî a:a0 ...0 0a ...0 A= ... ... ... ... .00 ... a34Ïðè ýòîì det A = an èA−1a−10= (A−1 )∗ = ...00a−1...0............00 ,... a−1à çíà÷èò, (A−1 )∗ y = y/a.Ñâîéñòâî 6 îáû÷íî íàçûâàþò ïðàâèëîì èçìåíåíèÿ ìàñøòàáà.7) Êàê ïðÿìîå, òàê è îáðàòíîå ïðåîáðàçîâàíèå Ôóðüå ïåðåâîäèò ïðîñòðàíñòâî áûñòðî óáûâàþùèõ ôóíêöèé â ñåáÿ.
Äðóãèìè ñëîâàìè, êàêîâà áû íè áûëà ôóíêöèÿ f ∈ S(Rn ), îáå ôóíêöèè F± [f ] ïðèíàäëåæàòS(Rn ).Äîêàçàòåëüñòâî. Ñóùåñòâîâàíèå âñåõ ïðîèçâîäíûõ ó ôóíêöèé F± [f ]óæå óñòàíîâëåíî â ñâîéñòâå 2. Ïîýòîìó äîñòàòî÷íî ïðîâåðèòü ëèøü òî,÷òî äëÿ ëþáûõ ìóëüòèèíäåêñîâ α è β ôóíêöèÿ y 7→ |y α Dβ F± [f (x)](y)|îãðàíè÷åíà â Rn . Äëÿ ýòîãî, â ñâîþ î÷åðåäü, äîñòàòî÷íî óáåäèòüñÿ, ÷òîýòà ôóíêöèÿ ñòðåìèòñÿ ê íóëþ ïðè y , ñòðåìÿùåìñÿ ê áåñêîíå÷íîñòè. Íîåñëè âåêòîð y = (y1 , . . . , yn ) ñòðåìèòñÿ ê áåñêîíå÷íîñòè, òî õîòÿ áû îäíàèç åãî êîìïîíåíò ñòðåìèòñÿ ê áåñêîíå÷íîñòè.
Íå îãðàíè÷èâàÿ îáùíîñòè,ìîæåì ñ÷èòàòü, ÷òî èìåííî yn → +∞.Ñîãëàñíî ñâîéñòâàì 2 è 3 áûñòðî óáûâàþùèõ ôóíêöèé, ôóíêöèÿ x 7→Dα (xβ f (x)) ÿâëÿåòñÿ áûñòðî óáûâàþùåé. Îáîçíà÷èì åå ÷åðåç g . Òîãäàìîæåì íàïèñàòü¯Z¯¯¯α β−n/2 ¯∓i(x,y)|y D F± [f (x)](y)| = |F± [g(x)](y)| = (2π)dx¯¯ =¯ g(x)e¯ Z¯−n/2 ¯= (2π)¯Rne∓i(x1 y1 +···+xn−1 yn−1 ) ×(26)Rn−1+∞¯·Z¸¯∓ixn yn×g(x1 , . . . , xn )edxn dx1 .
. . dxn−1 ¯¯.−∞Ïðè ôèêñèðîâàííûõ çíà÷åíèÿõ ïåðåìåííûõ x1 , . . . , xn−1 ôóíêöèÿ xn 7→g(x1 , . . . , xn ) îäíîãî âåùåñòâåííîãî ïåðåìåííîãî xn ÿâëÿåòñÿ áûñòðî óáûâàþùåé.  ÷àñòíîñòè, îíà èíòåãðèðóåìà íà âñåé ÷èñëîâîé ïðÿìîé. Ñëåäîâàòåëüíî, óñëîâèÿ ëåììû Ðèìàíà Ëåáåãà äëÿ áåñêîíå÷íîãî ïðîìåæóòêà âûïîëíåíû äëÿ èíòåãðàëà, ñòîÿùåãî â ôîðìóëå (26) â êâàäðàòíûõ ñêîáêàõ. Çíà÷èò, ñàì ýòîò èíòåãðàë ñòðåìèòñÿ ê íóëþ ïðè yn →+∞.
Ïåðåõîäÿ â ôîðìóëå (26) ê ïðåäåëó ïðè yn → +∞ ïîä çíàêîì(n − 1)-ìåðíîãî èíòåãðàëà, âèäèì, ÷òî âñå âûðàæåíèå (26) ñòðåìèòñÿ êíóëþ ïðè yn → +∞.358) Äëÿ ëþáîé áûñòðî óáûâàþùåé ôóíêöèè f : Rn → C ñïðàâåäëèâûðàâåíñòâà F+ [F− [f ]] = f è F− [F+ [f ]] = f . Äðóãèìè ñëîâàìè, ïîñëåäîâàòåëüíîå ïðèìåíåíèå ïðÿìîãî è îáðàòíîãî ïðåîáðàçîâàíèé Ôóðüåíå èçìåíÿåò ôóíêöèè. Ñâîéñòâî 8 íàçûâàþò Ôîðìóëîé îáðàùåíèÿ äëÿïðåîáðàçîâàíèÿ Ôóðüå.Äîêàçàòåëüñòâî. Äëÿ n = 1 ôîðìóëà îáðàùåíèÿ óæå óñòàíîâëåíàíàìè â 5.Äîêàæåì ôîðìóëó F+ [F− [f ]] = f äëÿ n = 2:ZZ1F− [f ](y1 , y2 )e−i(x1 y1 +x2 y2 ) dy1 dy2 =F+ [F− [f ]](x) =2πR2=1(2π)2Z Z ·Z ZR2¸f (t1 , t2 )e+i(t1 y1 +t2 y2 ) dt1 dt2 e−i(x1 y1 +x2 y2 ) dy1 dy2 .R2Ïîëüçóÿñü òåì, ÷òî f áûñòðî óáûâàþùàÿ ôóíêöèÿ, èçìåíèì ïîðÿäîêèíòåãðèðîâàíèÿ:ZZ1F+ [F− [f ]](x) =e+i(t2 y2 −x2 y2 ) ×2πR2·1× √2π+∞½Z−∞1√2π+∞¾¸Zit1 y1−ix1 y1f (t1 , t2 )edt1 edy1 dt2 dy2 .−∞Âûðàæåíèå, ñòîÿùåå â ôèãóðíûõ ñêîáêàõ, ïðåäñòàâëÿåò ñîáîé îäíîìåðíîå (ïîñêîëüêó t2 íå ó÷àñòâóåò â èíòåãðèðîâàíèè è âûïîëíÿåò ðîëüïàðàìåòðà) îáðàòíîå ïðåîáðàçîâàíèå Ôóðüå ôóíêöèè t1 7→ f (t1 , t2 ) îäíîãî ïåðåìåííîãî t1 .
Êðîìå òîãî, âûðàæåíèå, ñòîÿùåå â êâàäðàòíûõñêîáêàõ, ïðåäñòàâëÿåò ñîáîé îäíîìåðíîå ïðÿìîå ïðåîáðàçîâàíèå Ôóðüåîò ôóíêöèè, çàïèñàííîé â ôèãóðíûõ ñêîáêàõ. Ïîýòîìó, íà îñíîâàíèèîäíîìåðíîé ôîðìóëû îáðàùåíèÿ, çàêëþ÷àåì, ÷òî âûðàæåíèå â êâàäðàòíûõ ñêîáêàõ ðàâíî f (x1 , t2 ). Ñëåäîâàòåëüíî,ZZ1F+ [F− [f ]](x) =e+i(t2 y2 −x2 y2 ) f (x1 , t2 ) dt2 dy2 .2πR2Ïîñëåäíèé èíòåãðàë ïðåîáðàçóåì ïîäîáíî òîìó, êàê ìû ïîñòóïàëè ðàíüøå, ò. å. ïðåâðàòèì êðàòíûé èíòåãðàë â ïîâòîðíûé, ðàçãëÿäèì òàì îäíîìåðíûå ïðÿìîå è îáðàòíîå ïðåîáðàçîâàíèÿ Ôóðüå è âîñïîëüçóåìñÿ36îäíîìåðíîé ôîðìóëîé îáðàùåíèÿ:+∞½+∞¾ZZ11√F+ [F− [f ]](x) = √f (x1 , t2 )e+it2 y2 dt2 e−ix2 y2 dy2 = f (x).2π2π−∞−∞Òåì ñàìûì ðàâåíñòâî F+ [F− [f ]] = f äîêàçàíî ïðè n = 2.
Åãî äîêàçàòåëüñòâî â îáùåì ñëó÷àå, ðàâíî êàê è äîêàçàòåëüñòâî âòîðîãî ðàâåíñòâà F− [F+ [f ]] = f , ïðîâîäèòñÿ ñîâåðøåííî àíàëîãè÷íî. Ïîýòîìó ìû èõîïóñêàåì.Çàäà÷à62. Äîêàæèòå, ÷òî ïðåîáðàçîâàíèå Ôóðüå ôóíêöèè x 7→ ea|x| (a > 0)áåñêîíå÷íî äèôôåðåíöèðóåìî íà âñåé ÷èñëîâîé ïðÿìîé, íî íå ÿâëÿåòñÿáûñòðî óáûâàþùåé ôóíêöèåé. 9. Ðàâåíñòâî ÏàðñåâàëÿÒåîðåìà (ðàâåíñòâî Ïàðñåâàëÿ).
Äëÿ ëþáûõ áûñòðî óáûâàþùèõ ôóíêöèé f, g : Rn → C ñïðàâåäëèâî ðàâåíñòâîZZZ ∨∨bf (x)g(x) dx = f (x)bg (x) dx =f (x)g (x) dx,RnRnRnãäå ÷åðòà, êàê îáû÷íî, îçíà÷àåò êîìïëåêñíîå ñîïðÿæåíèå.Äîêàçàòåëüñòâî ðàçîáüåì íà òðè ýòàïà.Íà ïåðâîì ýòàïå óñòàíîâèì ðàâåíñòâàZ ∨ZZZ∨g (x) dx èf (x)g(x) dx = f (x) g (x) dx,fb(x)g(x) dx = f (x)bRnRnRnRnòî åñòü ïîêàæåì, ÷òî ñèìâîëû ïðÿìîãî è îáðàòíîãî ïðåîáðàçîâàíèÿ Ôóðüå ìîæíî ïåðåíîñèòü ñ îäíîãî ñîìíîæèòåëÿ íà äðóãîé ïîä çíàêîì èíòåãðàëà.×òîáû óáåäèòüñÿ â ýòîì, äîñòàòî÷íî, ïîëüçóÿñü òåì, ÷òî ôóíêöèè fè g áûñòðî óáûâàþùèå, èçìåíèòü ïîðÿäîê èíòåãðèðîâàíèÿ:¸Z ·ZZ−i(x,y)−n/2bf (y)edy g(x) dx =f (x)g(x) dx =(2π)RnRnRn¸Z ·ZZ−n/2−i(x,y)=(2π)g(x)edx f (y) dy = f (y)bg (y) dy.RnRnRnÂòîðîå ðàâåíñòâî, ñîñòàâëÿþùåå ïåðâûé ýòàï, äîêàçûâàåòñÿ àíàëîãè÷íî.37Íà âòîðîì ýòàïå óñòàíîâèì ðàâåíñòâà∨fb =f∨f = fb,èòî åñòü ïîêàæåì, ÷òî ïðè ¾îïóñêàíèè¿ êîìïëåêñíîãî ñîïðÿæåíèÿ ïðÿìîå è îáðàòíîå ïðåîáðàçîâàíèÿ Ôóðüå ìåíÿþòñÿ ðîëÿìè. ñàìîì äåëå, ïîëüçóÿñü òåì, ÷òî êîìïëåêñíîå ñîïðÿæåíèå îò ïðîèçâåäåíèÿ êîìïëåêñíûõ ÷èñåë ðàâíî ïðîèçâåäåíèþ èõ êîìïëåêñíûõ ñîïðÿæåíèé, èìååìZZ∨−n/2−n/2−i(x,y)bf (x)edx = (2π)f (y) = (2π)f (x)e+i(x,y) dx =f (y).RnRnÂòîðîå ðàâåíñòâî, ñîñòàâëÿþùåå âòîðîé ýòàï, äîêàçûâàåòñÿ àíàëîãè÷íî.Íàêîíåö, íà òðåòüåì ýòàïå äîêàæåì ñîáñòâåííî ðàâåíñòâî Ïàðñåâàëÿ.
Äëÿ ýòîãî ïîñëåäîâàòåëüíî âîñïîëüçóåìñÿ ôîðìóëîé îáðàùåíèÿ èðåçóëüòàòàìè ïåðâûõ äâóõ ýòàïîâ äîêàçàòåëüñòâà:ZZ ∨ZZ∨f (x)g(x) dx =fb (x)g(x) dx = fb(x) g (x) dx = fb(x)bg (x) dx.RnRnRnRnÂòîðîå ðàâåíñòâî Ïàðñåâàëÿ äîêàçûâàåòñÿ àíàëîãè÷íî.Çàäà÷à63. Ïóñòü ϕ(x) è ψ(p) áûñòðî óáûâàþùèå ôóíêöèè âåùåñòâåííûõïåðåìåííûõ x è p ñîîòâåòñòâåííî, ïðè÷åì ïóñòü ψ ÿâëÿåòñÿ ïðåîáðàçî+∞+∞RRâàíèåì Ôóðüå îò ϕ (ò. å. ψ = ϕb) è|ϕ(x)|2 dx =|ψ(p)|2 dp = 1.−∞−∞ òàêîì ñëó÷àå ôóíêöèè |ϕ|2 è |ψ|2 ìîæíî ðàññìàòðèâàòü êàê ïëîòíîñòè ðàñïðåäåëåíèÿ âåðîÿòíîñòåé ñëó÷àéíûõ âåëè÷èí x è p.  êâàíòîâîéìåõàíèêå ïîêàçûâàåòñÿ, ÷òî ñîîòíîøåíèå ψ = ϕb ïîçâîëÿåò èíòåðïðåòèðîâàòü ñëó÷àéíûå âåëè÷èíû x è p êàê êîîðäèíàòó è èìïóëüñ êâàíòîâîé÷àñòèöû.(à) Ïîêàæèòå, ÷òî ñäâèãîì ïî àðãóìåíòó (ò. å.
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