1611689564-c292adb9a0ffac330f3ca30cdb684bd6 (826864), страница 7
Текст из файла (страница 7)
Äîêàæèòå, ÷òî åñëè f : R → C áûñòðî óáûâàþùàÿ ôóíêöèÿ,òî ñóììà ðÿäà+∞X√2πf (x + 2πn)n=−∞ÿâëÿåòñÿ áåñêîíå÷íî äèôôåðåíöèðóåìîé ôóíêöèåé (à íå òîëüêî íåïðåðûâíî äèôôåðåíöèðóåìîé, êàê áûëî ïîêàçàíî â ïðèâåäåííîì âûøå äîêàçàòåëüñòâå ôîðìóëû Ïóàññîíà).80. Äîêàæèòå ñëåäóþùåå ñîîòíîøåíèå, íàçûâàåìîå θ-ôîðìóëîé èèãðàþùåå âàæíóþ ðîëü â òåîðèè ýëëèïòè÷åñêèõ ôóíêöèé è òåîðèè òåïëîïðîâîäíîñòè:r+∞+∞Xπ X − π 2 n2−tn2e=e t(t > 0).t n=−∞n=−∞81. Ñ ïîìîùüþ ôîðìóëû Ïóàññîíà âû÷èñëèòå ñóììó ðÿäà+∞Xn=−∞n21.+ a2Îáðàòèòå âíèìàíèå, ÷òî ó÷àñòâóþùàÿ â âû÷èñëåíèÿõ ôóíêöèÿ íå ÿâëÿåòñÿ áûñòðî óáûâàþùåé. Îáîñíóéòå äëÿ íåå çàêîííîñòü ïðèìåíåíèÿôîðìóëû Ïóàññîíà. 12. Òåîðåìà Êîòåëüíèêîâà ØåííîíàÒåîðåìà (Êîòåëüíèêîâà Øåííîíà). Ïóñòü f : R → C áûñòðîóáûâàþùàÿ ôóíêöèÿ, è ïóñòü ñóùåñòâóåò ïîëîæèòåëüíîå ÷èñëî aòàêîå, ÷òî fb(x) = 0 äëÿ âñåõ âåùåñòâåííûõ x òàêèõ, ÷òî |x| > a.Òîãäà äëÿ âñåõ x ∈ R ñïðàâåäëèâî ðàâåíñòâ µ¶¸+∞ µXπnπnf (x) =fsinc a x −,(30)aa−∞ãäå ôóíêöèÿ t 7→ sinc t îïðåäåëÿåòñÿ ðàâåíñòâîì sinc t = (sin t)/t èíàçûâàåòñÿ ôóíêöèåé îòñ÷åòîâ.45Äîêàçàòåëüñòâî.
Èñïîëüçóÿ ôîðìóëó îáðàùåíèÿ äëÿ ïðåîáðàçîâàíèÿ Ôóðüå è òîò ôàêò, ÷òî fb çàíóëÿåòñÿ âíå èíòåðâàëà [−a, a], ìîæåìíàïèñàòü1f (x) = √2π+∞ZZa1+ixybf (y)edy = √fb(y)eixy dy.2π−∞(31)−aÐàçëîæèì ôóíêöèþ fb â ðÿä Ôóðüå â êîìïëåêñíîé ôîðìå â èíòåðâàëå[−a, a]:fb(y) =+∞Xcn eiπna y,1cn =2aãäå−∞Zaiπnfb(y)e− a y dy.−aÏîäñòàâèì ýòîò ðÿä â ôîðìóëó (31) è ïðîèíòåãðèðóåì åãî ïî÷ëåííî,ïîëüçóÿñü òåì, ÷òî ðÿä Ôóðüå íåïðåðûâíî äèôôåðåíöèðóåìîé ôóíêöèèñõîäèòñÿ ðàâíîìåðíî:1f (x) = √2πZa µ X+∞−ai πna ycn e¶eixy dy =n=−∞¡¢ ¯Za ¡¢+∞+∞y=ai πnXXa +x y ¯πn11e¢ ¯¯=√cn ei a +x y dy = √cn ¡ πn= (32)2π n=−∞2π n=−∞ i a + x y=−a−a£¡¢ ¤· µ¶¸+∞+∞X2a sin πn+x a2a X1πna¢¡ πn=√=√cncn sinc a+x .aa a +x2π n=−∞2π n=−∞Òåïåðü îáðàòèìñÿ ê âû÷èñëåíèþ êîýôôèöèåíòîâ cn .
Ïðè ýòîì åùåðàç èñïîëüçóåì óñëîâèå òåîðåìû î òîì, ÷òî fb çàíóëÿåòñÿ âíå èíòåðâàëà[−a, a]:1cn =2aZafb(y)e−a−i πna y1dy =2a+∞Zπnfb(y)e−i a y dy =−∞+∞√√µ¶Z¡¢12π2ππni − πnybaf (y)edy ==·√f −.2a2aa2π(33)−∞Ïîñëåäíåå ðàâåíñòâî çäåñü íàïèñàíî íà îñíîâàíèè ôîðìóëû îáðàùåíèÿ.46Ïîäñòàâèâ âûðàæåíèå (33) â ôîðìóëó (32) è ïîìåíÿâ èíäåêñ ñóììèðîâàíèÿ n íà −n, ïîëó÷è쵶· µ¶¸+∞ √πnπn2a X2πsinc a x +=f (x) = √f −aa2π n=−∞ 2a=+∞Xn=−∞µf¶· µ¶¸πnπnsinc a x −,aa÷òî è òðåáîâàëîñü äîêàçàòü.Òåîðåìà Êîòåëüíèêîâà Øåííîíà ïðèìå÷àòåëüíà íå òîëüêî ýëåãàíòíîñòüþ êîìáèíèðîâàíèÿ ðÿäà è ïðåîáðàçîâàíèÿ Ôóðüå, íî è òåì, ÷òî îíàÿâëÿåòñÿ êðàåóãîëüíûì êàìíåì òåîðèè öèôðîâîé ïåðåäà÷è èíôîðìàöèè.
×òîáû ïîÿñíèòü ýòî, ïðåäïîëîæèì, ÷òî ìû íàìåðåíû ïåðåäàòü ïîöèôðîâîìó êàíàëó ñâÿçè íåïðåðûâíûé (òî÷íåå áûëî áû ñêàçàòü àíàëîãîâûé) ñèãíàë ϕ. Ìû íå ìîæåì ¾ïðîñòî¿ ïåðåäàâàòü çíà÷åíèå ôóíêöèèϕ â êàæäîé òî÷êå, ïîñêîëüêó âåùåñòâåííûõ ÷èñåë ñëèøêîì ìíîãî: êàêâû çíàåòå èç êóðñà ìàòåìàòè÷åñêîãî àíàëèçà, ìíîæåñòâî âåùåñòâåííûõ÷èñåë íå ñ÷åòíî. Ïîýòîìó ïðèõîäèòñÿ ïðèìåíÿòü èíòåëëåêò. Ñîãëàñíîôîðìóëå îáðàùåíèÿ, ïî áîëüøîìó ñ÷åòó íàì áåçðàçëè÷íî, ÷òî ïåðåäàâàòü: ñàì ñèãíàë ϕ èëè åãî ïðåîáðàçîâàíèå Ôóðüå, íàïðèìåð, åãî∨îáðàòíîå ïðåîáðàçîâàíèå Ôóðüå f =ϕ. Òåïåðü ïðèìåì âî âíèìàíèå, ÷òîè ÷åëîâå÷åñêèé ãëàç, è ÷åëîâå÷åñêîå óõî âîñïðèíèìàþò ñèãíàëû ëèøü âîãðàíè÷åííîé îáëàñòè ÷àñòîò (íàïðèìåð, óõî âîñïðèíèìàåò çâóêè òîëüêî â äèàïàçîíå îò 20 Ãö äî 20 êÃö).
Ïðåíåáðåãàÿ íåâîñïðèíèìàåìîé÷àñòüþ ñïåêòðà, ìû ìîæåì ñ÷èòàòü âûïîëíåííûì óñëîâèå òåîðåìû Êîòåëüíèêîâà Øåííîíà î òîì, ÷òî fb = ϕ çàíóëÿåòñÿ âíå íåêîòîðîãîêîíå÷íîãî èíòåðâàëà [−a, a].Òåïåðü ìû âèäèì, ÷òî ïåðåäà÷à íåïðåðûâíîãî ñèãíàëà ïî öèôðîâîìóêàíàëó ñâÿçè ìîæåò áûòü îðãàíèçîâàíà òàê: ïåðåäàò÷èê íàõîäèò îáðàò∨íîå ïðåîáðàçîâàíèå Ôóðüå f =ϕ èñõîäíîãî ñèãíàëà ϕ è ïåðåäàåò åãîçíà÷åíèÿ â òàê íàçûâàåìûõ òî÷êàõ îòñ÷åòà πn/a (−∞ < n < +∞). Ïîëó÷èâ çíà÷åíèÿ f (πn/a), ïðèåìíèê èñïîëüçóåò ôîðìóëó (30) äëÿ âîññòàíîâëåíèÿ çíà÷åíèÿ ôóíêöèè f â ïðîèçâîëüíîé òî÷êå è, ñîâåðøèâïðÿìîå ïðåîáðàçîâàíèå Ôóðüå, âûäàåò èñõîäíûé ñèãíàë ϕ = fb.Òåì ñàìûì, íèñêîëüêî íå óòðàòèâ êà÷åñòâà ïåðåäàâàåìîãî ñèãíàëà,ìû ñóùåñòâåííî óìåíüøèëè îáúåì ïåðåäàâàåìîé èíôîðìàöèè: òî÷åêîòñ÷åòà õîòÿ è áåñêîíå÷íî ìíîãî, íî ãîðàçäî ìåíüøå, ÷åì âåùåñòâåííûõ÷èñåë.47Ñëåäóþùèé øàã, åñòåñòâåííî, ñîñòîèò â ïðèìåíåíèè èçâåñòíîãî íàì∨ñâîéñòâà: f (πn/a) =ϕ (πn/a) → 0 ïðè n → ±∞.
Èñïîëüçóÿ åãî, è äîïóñêàÿ êîíòðîëèðóåìóþ ïîãðåøíîñòü, ìîæíî îñòàâèòü â ôîðìóëå (30)ëèøü êîíå÷íîå ÷èñëî ñëàãàåìûõ è, ñîîòâåòñòâåííî, ïåðåäàâàòü òîëüêîêîíå÷íîå êîëè÷åñòâî çíà÷åíèé f (èìåííî â ýòèõ òî÷êàõ). Âîïðîñó î òîì,êàêèå èìåííî ñëàãàåìûå îñòàâèòü, ÷òîáû, ìèíèìèçèðóÿ îáúåì ïåðåäàâàåìîé èíôîðìàöèè, íå âûõîäèòü çà ðàìêè äîïóñòèìûõ ïîãðåøíîñòåé,ïîñâÿùåíî áîëüøîå êîëè÷åñòâî ðàáîò, îòíîñÿùèõñÿ ñîáñòâåííî ê òåîðèèöèôðîâîé ïåðåäà÷è èíôîðìàöèè. 13.
Ïðèìåíåíèå ïðåîáðàçîâàíèÿ Ôóðüå ê ðåøåíèþóðàâíåíèÿ òåïëîïðîâîäíîñòèÄîïóñòèì, ÷òî n-ìåðíîå åâêëèäîâî ïðîñòðàíñòâî Rn çàïîëíåíî ñðåäîé ñ ïîñòîÿííûì êîýôôèöèåíòîì òåïëîïðîâîäíîñòè. Ïðåäïîëîæèì,÷òî â ïðîñòðàíñòâå îòñóòñòâóþò èñòî÷íèêè è ñòîêè òåïëà è îáîçíà÷èì÷åðåç u(t, x) òåìïåðàòóðó, êîòîðóþ èìååò òî÷êà x = (x1 , . . . , xn ) ∈ Rn âìîìåíò âðåìåíè t ≥ 0. Êàê èçâåñòíî, ïðè òàêèõ ïðåäïîëîæåíèÿõ òåìïåðàòóðà ïåðåðàñïðåäåëÿåòñÿ ñî âðåìåíåì òàê, ÷òî ñîáëþäàåòñÿ óðàâíåíè嶵 2∂2u∂u2 ∂ u+ ··· +,(34)=a∂t∂x21∂x2níàçûâàåìîå óðàâíåíèåì òåïëîïðîâîäíîñòè.
Ôèãóðèðóþùàÿ çäåñü ïîñòîÿííàÿ a > 0 âûðàæàåòñÿ íåêîòîðûì îáðàçîì ÷åðåç êîýôôèöèåíòòåïëîïðîâîäíîñòè ñðåäû. Àêêóðàòíûé âûâîä óðàâíåíèÿ òåïëîïðîâîäíîñòè èç ôèçè÷åñêèõ ïðåäïîñûëîê ìîæíî íàéòè, íàïðèìåð, â ó÷åáíèêåÑ. Ê. Ãîäóíîâà ¾Óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçèêè¿.Åñòåñòâåííî îæèäàòü, ÷òî, çíàÿ ðàñïðåäåëåíèå òåìïåðàòóðû â íåêîòîðûé ìîìåíò âðåìåíè, íàïðèìåð ïðè t = 0, ìîæíî âîññòàíîâèòü ðàñïðåäåëåíèå òåìïåðàòóðû â ëþáîé ïîñëåäóþùèé ìîìåíò âðåìåíè t > 0.×òîáû ñäåëàòü ýòî, íàäî ðåøèòü óðàâíåíèå (34) ïðè íà÷àëüíûõ óñëîâèÿõ u(0, x) = ϕ(x), ãäå ϕ : Rn → R íåêîòîðàÿ çàäàííàÿ ôóíêöèÿ.Ïîñêîëüêó ìû ñîáèðàåìñÿ èñïîëüçîâàòü ïðåîáðàçîâàíèå Ôóðüå, áóäåìïðåäïîëàãàòü, ÷òî è ôóíêöèÿ ϕ, è ôóíêöèÿ u (âçÿòàÿ ïðè ôèêñèðîâàííîì çíà÷åíèè t) ÿâëÿþòñÿ áûñòðî óáûâàþùèìè â Rn .Ôèêñèðîâàâ t ≥ 0, ââåäåì â ðàññìîòðåíèå íîâóþ ôóíêöèþ y 7→ v(t, y),êîòîðàÿ ÿâëÿåòñÿ (ïðÿìûì) ïðåîáðàçîâàíèåì Ôóðüå ôóíêöèè u ïî ïåðåìåííîé x:Z\v(t, y) = u(t,x)(y) = (2π)−n/2 u(t, x)e−i(x,y) dx.Rn48Âûÿñíèì, êàêîìó äèôôåðåíöèàëüíîìó óðàâíåíèþ óäîâëåòâîðÿåò ôóíêöèÿ v .
Äëÿ ýòîãî ïðîäåëàåì ñëåäóþùèå âû÷èñëåíèÿ:Z∂v−n/2 ∂= (2π)u(t, x)e−i(x,y) dx =∂t∂tRn=(äèôôåðåíöèðóåì ïîä çíàêîì èíòåãðàëà, êàê îáû÷íî, îïóñêàÿ îáîñíîâàíèÿ, ïîñêîëüêó ðå÷ü èäåò î áûñòðî óáûâàþùèõ ôóíêöèÿõ)=Z∂u= (2π)−n/2(t, x)e−i(x,y) dx =∂tRn=(èñïîëüçóåì (34))=µd¶¶Z µ 222u∂ u∂ 2 u −i(x,y)∂d2 ∂ u= a2 (2π)−n/2+···+edx=a+···+=∂x21∂x2n∂x21∂x2nRn=(ê êàæäîìó ñëàãàåìîìó ïðèìåíÿåì ñâîéñòâî 3) ïðåîáðàçîâàíèÿ Ôóðüå: F± [Dα f ] = (±iy)α F± [f ])=¡¢= a2 (iy1 )2 ub + · · · + (iyn )2 ub = −a2 (y12 + · · · + yn2 )bu = −a2 |y|2 v(t, y).Òàêèì îáðàçîì, ôóíêöèÿ v óäîâëåòâîðÿåò äèôôåðåíöèàëüíîìó óðàâíåíèþ∂v= −a2 |y|2 v,(35)∂tâ êîòîðîì y èãðàåò ðîëü ïàðàìåòðà. Ïðè êàæäîì ôèêñèðîâàííîì y ðå22øåíèå ýòîãî óðàâíåíèÿ, î÷åâèäíî, èìååò âèä v(t, y)=Ce−a |y| t , ãäå C íåêîòîðàÿ ïîñòîÿííàÿ, íå çàâèñÿùàÿ îò t.
Íî äëÿ äðóãîãî çíà÷åíèÿ yïîñòîÿííàÿ C ìîæåò ïðèíèìàòü äðóãîå çíà÷åíèå. Ïîýòîìó ìû çàïèøåìðåøåíèå óðàâíåíèÿ (35) â âèäåv(t, y) = C(y)e−a2|y|2 t.Ïîäñòàâèâ ñþäà t = 0 è èñïîëüçîâàâ íà÷àëüíûå óñëîâèÿ u(0, x) = ϕ(x),ïîëó÷èì\[C(y) = v(0, y) = u(0,x)(y) = ϕ(x)(y). ðåçóëüòàòå ìû íàøëè íå ñàìî ðåøåíèå u óðàâíåíèÿ (34), óäîâëåòâîðÿþùåå íà÷àëüíûì óñëîâèÿì u(0, x) = ϕ(x), à ïðåîáðàçîâàíèå Ôóðüå îòíåãî:−a2 |y|2 tv(t, y) = ϕ(y)eb.×òîáû íàéòè îòñþäà u, åñòåñòâåííî èñïîëüçîâàòü ôîðìóëó îáðàùåíèÿ.Íàõîæäåíèå æå îáðàòíîãî ïðåîáðàçîâàíèÿ Ôóðüå îò ïðîèçâåäåíèÿ äâóõ49ôóíêöèé íàì îáëåã÷èò ñâîéñòâî 6 ñâåðòêè áûñòðî óáûâàþùèõ ôóíêöèé(F± [f · g] = (2π)−n/2 F± [f ] ∗ F± [g]):£¤−a2 |y|2 tu(t, x) = F− [v(t, y)](x) = F− ϕ(y)eb(x) =££22 ¤22 ¤= (2π)−n/2 F− [ϕ]b ∗ F− e−a |y| t = (2π)−n/2 ϕ ∗ F− e−a |y| t .Âòîðîé ñîìíîæèòåëü â ñâåðòêå âû÷èñëèì îòäåëüíî:Z£22 ¤222F− e−a |y| t (z) = (2π)−n/2 e−a (y1 +···+yn )t e+i(y1 z1 +···+yn zn ) dy1 .
. . dyn =Rn=(ïîëüçóÿñü òåì, ÷òî ïîäûíòåãðàëüíàÿ ôóíêöèÿ ÿâëÿåòñÿ áûñòðî óáûâàþùåé, ïðåâðàùàåì êðàòíûé èíòåãðàë â ïîâòîðíûé)=Z222= (2π)−(n−1)/2e−a (y1 +···+yn−1 )t ei(y1 z1 +···+yn−1 zn−1 ) ×Rn−1+∞·¸Z2 21× √e−a yn t eiyn zn dyn dy1 . . . dyn−1 =2π−∞=(ïîñêîëüêó îäíîìåðíûé èíòåãðàë, ñòîÿùèé â êâàäðàòíûõ ñêîáêàõ, íåçàâèñèò îò ïåðåìåííûõ y1 , . . . , yn , òî ìîæåì âûíåñòè åãî êàê ïîñòîÿííóþèç-ïîä çíàêà (n − 1)-ìåðíîãî èíòåãðàëà)=·¸Z222= (2π)−(n−1)/2e−a (y1 +···+yn−1 )t ei(y1 z1 +···+yn−1 zn−1 ) dy1 . .
. dyn−1 ×Rn−1+∞·¸Z2 21× √e−a yn t eiyn zn dyn =2π−∞=(êàê âèäèì, íàì óäàëîñü ¾îòùåïèòü¿ îäíîìåðíûé èíòåãðàë îò êðàòíîãî; ïîâòîðèì ýòîò ïðîöåññ ìíîãîêðàòíî)=µ=1√2π+∞+∞¶ µ¶ZZ21−a2 y12 t iy1 z1−a2 ynt iyn zneedy1 . . . √eedyn .2π−∞−∞Êàæäûé èç ôèãóðèðóþùèõ â ïîñëåäíåé ôîðìóëå îäíîìåðíûõ èíòåãðàëîâ ïðåäñòàâëÿåò ñîáîé îäíîìåðíîå îáðàòíîå ïðåîáðàçîâàíèå Ôóðüå2îò ôóíêöèè âèäà y 7→ e−by , b > 0. Åãî ìû óæå ñ÷èòàëè â 6 è çíàåì,÷òî∨221−by 2 (z) =e−by (z) = √ e−z /4b .e[2b50Èñïîëüçóÿ ýòî ñîîáðàæåíèå, ìû ìîæåì çàêîí÷èòü âû÷èñëåíèå ìíîãîìåðíîãî ïðåîáðàçîâàíèÿ Ôóðüå ñëåäóþùèì îáðàçîì:¶ µ¶µz2z2£ −a2 |y|2 t ¤|z|2111− 4a12 t− 4an2 te... √e=F− e(z) = √e− 4a2 t ,2n/222(2a t)2a t2a t÷òî ïîçâîëÿåò íàì çàïèñàòü ðåøåíèå óðàâíåíèÿ òåïëîïðîâîäíîñòè (34)ñ íà÷àëüíûìè óñëîâèÿìè u(0, x) = ϕ(x) â âèä嵶Z|z|2|z|211− 4a2 tu(t, x) =ϕ(z)∗e(x)=ϕ(x − z)e− 4a2 t dz.2n/22n/2(4πa t)(4πa t)RnÏîñëåäíèé èíòåãðàë íàçûâàåòñÿ èíòåãðàëîì Ïóàññîíà äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè.Èçëîæåííûé â ýòîì ïàðàãðàôå ìåòîä ðåøåíèÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè îñíîâàí íà òîì, ÷òî ïðåîáðàçîâàíèå Ôóðüå çàìåíÿåò îïåðàöèþäèôôåðåíöèðîâàíèÿ îïåðàöèåé óìíîæåíèÿ íà íåçàâèñèìóþ ïåðåìåííóþ.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
