Добавления к лекциям 4 семестр - Балашов

ÌÈÍÈÑÒÅÐÑÒÂÎ ÎÁÐÀÇÎÂÀÍÈß È ÍÀÓÊÈ ÐÎÑÑÈÉÑÊÎÉ ÔÅÄÅÐÀÖÈÈÌÎÑÊÎÂÑÊÈÉ ÔÈÇÈÊÎ-ÒÅÕÍÈ×ÅÑÊÈÉ ÈÍÑÒÈÒÓÒ(ÃÎÑÓÄÀÐÑÒÂÅÍÍÛÉ ÓÍÈÂÅÐÑÈÒÅÒ)Êàôåäðà âûñøåé ìàòåìàòèêèÌ. Â. ÁàëàøîâÄÎÁÀÂËÅÍÈÅ Ê ËÅÊÖÈßÌ ÏÎÌÀÒÅÌÀÒÈ×ÅÑÊÎÌÓ ÀÍÀËÈÇÓ4 ÑÅÌÅÑÒÐÓ÷åáíî-ìåòîäè÷åñêîå ïîñîáèåïî êóðñó Ìàòåìàòè÷åñêèé àíàëèçÌÎÑÊÂÀÌÔÒÈ2016ÓÄÊ 517ÐåöåíçåíòÄîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðîôåññîð Â. Æ.

ÑàêáàåâÁàëàøîâ, Ì. Â.Äîáàâëåíèå ê ëåêöèÿì ïî ìàòåìàòè÷åñêîìó àíàëèçó. 4 ñåìåñòð: ó÷åáíî-ìåòîäè÷åñêîå ïîñîáèå ïî êóðñó Ìàòåìàòè÷åñêèé àíàëèç/ Ì. Â. Áàëàøîâ. Ì. : ÌÔÒÈ, 2016. 28 ñ.Ïðåäíàçíà÷åíî äëÿ ñòóäåíòîâ 2 êóðñà ÌÔÒÈ, à òàêæå äëÿ ïðåïîäàâàòåëåé.Ó÷åáíîå èçäàíèåÁàëàøîâ Ìàêñèì Âèêòîðîâè÷Äîáàâëåíèå ê ëåêöèÿì ïî ìàòåìàòè÷åñêîìó àíàëèçó. 4 ñåìåñòðÓ÷åáíî-ìåòîäè÷åñêîå ïîñîáèå ïî êóðñó Ìàòåìàòè÷åñêèé àíàëèçÐåäàêòîð Î. Ï. Êîòîâà. Êîððåêòîð Ë. Â. Ñåáîâà.Êîìïüþòåðíàÿ âåðñòêà Ì. Â. Áàëàøîâ.Ïîäïèñàíî â ïå÷àòü: 17.10.2016. Ôîðìàò 60 × 841 /16 . Óñë. ïå÷. ë. 1,7.Ó÷.-èçä.

ë. 1,6. Òèðàæ 250 ýêç. Çàêàç ¹ 422.Ôåäåðàëüíîå ãîñóäàðñòâåííîå àâòîíîìíîå îáðàçîâàòåëüíîåó÷ðåæäåíèå âûñøåãî îáðàçîâàíèÿ ¾Ìîñêîâñêèéôèçèêî-òåõíè÷åñêèé èíñòèòóò (ãîñóäàðñòâåííûé óíèâåðñèòåò)¿141700, Ìîñêîâñêàÿ îáë., ã. Äîëãîïðóäíûé, Èíñòèòóòñêèé ïåð., 9Òåë. (495) 408-58-22, e-mail: rio@mipt.ruÎòäåë îïåðàòèâíîé ïîëèãðàôèè ¾Ôèçòåõ-ïîëèãðàô¿141700, Ìîñêîâñêàÿ îáë., ã. Äîëãîïðóäíûé, Èíñòèòóòñêèé ïåð., 9Òåë. (495) 408-84-30, e-mail: polygraph@mipt.ruc Ôåäåðàëüíîå ãîñóäàðñòâåííîå àâòîíîìíîå°îáðàçîâàòåëüíîå ó÷ðåæäåíèåâûñøåãî îáðàçîâàíèÿ¾Ìîñêîâñêèé ôèçèêî-òåõíè÷åñêèé èíñòèòóò¿(ãîñóäàðñòâåííûé óíèâåðñèòåò)c Áàëàøîâ Ì. Â., 2016°ÑîäåðæàíèåÂâåäåíèå .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41. Ðàñõîäèìîñòü ðÿäà Ôóðüå â òî÷êå äëÿ íåïðåðûâíîé ôóíêöèè . . . . 52. Òðèãîíîìåòðè÷åñêèå ðÿäû è ðÿäû Ôóðüå . . . . . . . . . . . . . . . . . . . . . . . . . 73.

Ðàâíîìåðíàÿ ñõîäèìîñòü ðÿäà Ôóðüå. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124. Ðàâåíñòâî ïàðàëëåëîãðàììà . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145. Òåîðåìà Ôåéåðà â RL1 . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166. Òåîðåìà Ìþíöà . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187. Ðàâåíñòâî Ïàðñåâàëÿ â S è òåîðåìà Êîòåëüíèêîâà . . . . . . . . . . . . . . . 24Ëèòåðàòóðà . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283Ââåäåíèå ìåòîäè÷åñêîå ïîñîáèå âîøëè íåêîòîðûå òåìû, ïðî÷èòàííûå àâòîðîìñòóäåíòàì 2 êóðñà ÔÓÏÌ â 4-ì ñåìåñòðå (2015 2016 ó÷åáíûé ãîä), àòàêæå íåêîòîðûå òåìû, êîòîðûå àâòîð ïëàíèðóåò ÷èòàòü â äàëüíåéøåìâ ðàìêàõ êóðñà ¾Ãàðìîíè÷åñêèé àíàëèç¿ â ÌÔÒÈ.Àâòîð áëàãîäàðåí äîöåíòó êàôåäðû âûñøåé ìàòåìàòèêè Â. Â. Ðåäêîçóáîâó çà ïðèìåð ôóíêöèè èç ïåðâîãî ðàçäåëà è ïîëåçíûå îáñóæäåíèÿ.Ì. Â.

ÁàëàøîâÑåíòÿáðü, 2016Äîëãîïðóäíûé41. Ðàñõîäèìîñòü ðÿäà Ôóðüå â òî÷êåäëÿ íåïðåðûâíîé ôóíêöèè ëþáûõ äîñòàòî÷íûõ óñëîâèÿõ ñõîäèìîñòè ðÿäà Ôóðüå ôóíêöèè â òî÷êå ê çíà÷åíèþ ôóíêöèè (ïî ñòàíäàðòíîé òðèãîíîìåòðè÷åñêîé ñèñòåìå)ôèãóðèðóåò íåêîòîðîå óñëîâèå, áîëåå ñèëüíîå, ÷åì íåïðåðûâíîñòü (íàïðèìåð, ñõîäèìîñòü íåêîòîðîãî èíòåãðàëà â ïðèçíàêå Äèíè).

Âîçíèêàåò âîïðîñ: âåðíî ëè, ÷òî äëÿ íåïðåðûâíîé ôóíêöèè ðÿä Ôóðüå ìîæåòâ íåêîòîðîé òî÷êå íå ñõîäèòüñÿ ê çíà÷åíèþ ýòîé ôóíêöèè â äàííîéòî÷êå èëè æå âîîáùå ðàñõîäèòüñÿ? Îêàçûâàåòñÿ, ÷òî îòâåò íà ýòîò âîïðîñ óòâåðäèòåëüíûé. Ìû ïðèâåäåì î÷åíü ïðîñòîé ïðèìåð, êîòîðûéïðèíàäëåæèò Ñ.À. Òåëÿêîâñêîìó è ñîîáùåí ìíå Â.Â.

Ðåäêîçóáîâûì.Ðàññìîòðèì ïðè x ∈ [0, π] ôóíêöèþf (x) =∞Xbk sin kx,k=1ãäå bk ≥ 0 è ðÿä∞Pk=1bk ñõîäèòñÿ. Áîëåå òî÷íî ÷èñëà bk îïðåäåëèìïîçæå. Ïî ïðèçíàêó Âåéåðøòðàññà ôóíêöèÿ f íåïðåðûâíà íà îòðåçêå [0, π]. Ïðîäîëæèì f ÷åòíî íà îòðåçîê [−π, 0]. Ïîñòðîåííàÿ òàêèìîáðàçîì ôóíêöèÿ f (x), x ∈ [−π, π], íåïðåðûâíàÿ, ÷åòíàÿ è ìîæåò áûòü2π -ïåðèîäè÷íî ïðîäîëæåíà íà R. ñèëó ôîðìóëû äëÿ n-é ÷àñòè÷íîé ñóììû ðÿäà Ôóðüå ÷åðåç ÿäðîÄèðèõëå¡¢sin n + 12 t1,Dn (t) = + cos t + · · · + cos nt =22 sin 2tà èìåííî:1Sn (f, x) =πZπ(f (x + t) + f (x − t)) Dn (t) dt,0ïîëó÷àåì, ÷òî äëÿ ïîñòðîåííîé íàìè ôóíêöèè fSn (f, 0) =2πZπf (t)Dn (t) dt.05Ðàññìîòðèì ïîäûíòåãðàëüíóþ ôóíêöèþf (t)Dn (t) =∞Xbk sin ktDn (t).k=1Ïîëîæèìβk,n2=πZπbk sin ktDn (t) dt.0Èç ðàâåíñòâà2 sin ktDn (t) =nP= sin kt +2 sin kt cos mt == sin kt +m=1nP(sin(k + m)t + sin(k − m)t)m=1â ñëó÷àå k > n âûòåêàåò, ÷òî 2 sin ktDn (t) åñòü ñóììà ñèíóñîâ âèäàsin mt ïðè íàòóðàëüíûõ m.Ïóñòü k ≤ n.

ÒîãäànPsin kt +(sin(k + m)t + sin(k − m)t) =m=1n+kP= sin kt ++==kPm=1n+kPsin mt+m=k+1nPsin(k − m)t +sin mt +m=kn+kPk−1Psin(m − k)t =m=k+1sin mt −m=1n−kP(1)sin mt =m=1sin mt,m=n−k+1à çíà÷èò, 2 sin ktDn (t) òàêæå åñòü ñóììà ñèíóñîâ âèäà sin mt ïðè íàòóðàëüíûõ m.Çàìåòèì, ÷òî äëÿ âñåõ m ≥ 1Zπ0¯π1 − (−1)mcos mt ¯¯≥ 0,=sin mt dt = −¯m 0mïîýòîìóSn (f, 0) ≥ βn,n2= bnπ6Zπsin ntDn (t) dt,0à â ñèëó (1)Rπ0=≥2 sin ntDn (t) dt =2nPRπµ01−(−1)mmm=1Rn 1x dx1nP=m=1¶sin mt dt =2nPm=122m−1=nPm=11m− 12≥= ln n.Ñëåäîâàòåëüíî,Sn (f, 0) ≥bnln n.π3Óòî÷íèì òåïåðü âûáîð ïîñëåäîâàòåëüíîñòè bn : ïðè n = 2m îïðåäåëèì3bn = m12 , à ïðè n 6= 2m ïîëîæèì bn = 0.3Òîãäà ïðè n = 2m èìååìSn (f, 0) ≥31 1ln 2m.ln 2m =2πmπÒàêèì îáðàçîì,S2m3 (f, 0) → +∞,m → ∞.Ðÿä Ôóðüå íåïðåðûâíîé 2π -ïåðèîäè÷åñêîé ôóíêöèè f íå ñõîäèòñÿ âòî÷êå x = 0.2. Òðèãîíîìåòðè÷åñêèå ðÿäû è ðÿäû ÔóðüåÏðèìåð 1. Ðàññìîòðèì ðÿä∞Xbk sin kx.(2)k=1Ïóñòü ïîñëåäîâàòåëüíîñòü bk ìîíîòîííî óáûâàåò ê íóëþ (è òåì ñàìûì∞Pbkbk ≥ 0 ïðè âñåõ k ), à ðÿäk ñõîäèòñÿ.

Ïîêàæåì, ÷òî ðÿä (2) åñòük=1ðÿä Ôóðüå ñâîåé ñóììû S(x).Ïðåæäå âñåãî çàìåòèì, ÷òî íà ëþáîì îòðåçêå [δ, π], δ ∈ (0, π), ðÿä(2) ñõîäèòñÿ ðàâíîìåðíî ïî ïðèçíàêó Äèðèõëå, à çíà÷èò, S(x) íåïðåðûâíà íà îòðåçêå [δ, π] ïðè ëþáîì ñêîëü óãîäíî ìàëîì δ > 0.7Ïóñòü Am , Bm êîýôôèöèåíòû Ôóðüå ôóíêöèè S(x), êîòîðàÿíå÷åòíà.¶µ∞Rπ PBm = π2bk sin kx sin mx dx =k=10∞Rπ P2bk sin kx sin mx dx =πδ→+0δ k=1ïπ∞P¯lim π1bk sin(k−m)x¯ + bm (πk−mδ→+0δk=1, k6=m= lim=Ã=lim 1δ→+0 π−∞Pk=1, k6=mbk sin(k−m)δk−m− δ) +íî (èç ñõîäèìîñòè ðÿäÿ∞Pk=1bkk ),k=1+ bm (π − δ) +∞Pk=1Ïîñêîëüêó â ñèëó óñëîâèÿ ïðèìåðà 1 ðÿäû¯π∞P∞Pk=1¯bk sin(k+m)x¯k+mbk sin(k+m)δk+m!δ!=.(3)bkk±mñõîäÿòñÿ àáñîëþò-òî ðÿä â ïðàâîé ÷àñòè ôîðìóëû (3)ñõîäèòñÿ ðàâíîìåðíî, è ïðåäåë ïðè δ → +0 åñòü çíà÷åíèå ýòîãî ðÿäàâ òî÷êå δ = 0, ò.å.

Bm = bm .Äëÿ íàõîæäåíèÿ Am ïðîâåäèòå àíàëîãè÷íûå âû÷èñëåíèÿ ñàìîñòîÿòåëüíî, âîñïîëüçîâàâøèñü ðàâåíñòâîì −δ π  Ã!ZZ∞XAm = lim  + bk sin kx cos mx dx  ,δ→+0−πk=1δè ïîêàæèòå, ÷òî âûðàæåíèå â ïðàâîé ÷àñòè èìååò ïðåäåë 0.Îòìåòèì, ÷òî, íàïðèìåð, ðÿä∞Xsin kxk=2ln2 k(4)ÿâëÿåòñÿ ðÿäîì Ôóðüå ñâîåé ñóììû.Ïðèìåð 2. Ðàññìîòðèì ðÿä∞Xsin kxk=2ln k.(5)Ïîêàæåì, ÷òî äëÿ ëþáîé êóñî÷íî-íåïðåðûâíîé ôóíêöèè ðÿä (5) íåÿâëÿåòñÿ åå ðÿäîì Ôóðüå.Îò ïðîòèâíîãî, äîïóñòèì, f êóñî÷íî-íåïðåðûâíàÿ ôóíêöèÿ íàîòðåçêå [−π, π], ðÿä Ôóðüå êîòîðîé åñòü ðÿä (5). Òîãäà ïî òåîðåìå îá8èíòåãðèðîâàíèè ðÿäà Ôóðüå [1, òåîðåìà 2, Ÿ 4, ñ. 117] èìååìZx∞f (t) dt =A0 X 1−cos kx,2k ln kk=20ïðè÷åì ðÿä â ïðàâîé ÷àñòè ñõîäèòñÿ ïðè âñåõ x. Çäåñü A0 =ãäå Φ(x) =Rx01πRπ−πΦ(x) dx,f (t) dt.

Ïðè x = 0 ïîëó÷àåì∞0=A0 X 1−,2k ln kk=2÷òî ïðîòèâîðå÷èò ðàñõîäèìîñòè ðÿäà∞Pk=21k ln k(ïî èíòåãðàëüíîìó ïðèç-íàêó). Ïðîòèâîðå÷èå ïîêàçûâàåò, ÷òî äîïóùåíèå ñóùåñòâîâàíèÿ òàêîéêóñî÷íî-íåïðåðûâíîé ôóíêöèè íåâåðíî.Ìîæíî äîêàçàòü, ÷òî òðèãîíîìåòðè÷åñêèé ðÿä (5) íå ÿâëÿåòñÿ ðÿäîìÔóðüå íèêàêîé àáñîëþòíî èíòåãðèðóåìîé ôóíêöèè.Óïðàæíåíèå 1.

Äîêàæèòå, ÷òî ðÿä (5) íå ÿâëÿåòñÿ ðÿäîì Ôóðüåñâîåé ñóììû.Óòâåðæäåíèå óïðàæíåíèÿ 1 îñîáåííî èíòåðåñíî ñîïîñòàâèòü ñ ïðèìåðîì (4) òðèãîíîìåòðè÷åñêîãî ðÿäà, êîòîðûé ÿâëÿåòñÿ ðÿäîì Ôóðüåñâîåé ñóììû.Ïðèìåð 3. Ðàññìîòðèì ðÿä∞Xsin kx2k=1k3.Ýòîò ðÿä ñõîäèòñÿ ïðè âñåõ x, îáîçíà÷èì åãî ñóììó S(x). Äîêàæåì,÷òî S(x) ∈ RL2 [−π, π].Êàê è â ïðåäûäóùèõ ïðèìåðàõ, ëåãêî âèäåòü, ÷òî ôóíêöèÿ S(x)íåïðåðûâíà íà ëþáîì îòðåçêå âèäà [δ, π], δ ∈ (0, π). Íàì îñòàåòñÿ ïîêàçàòü, ÷òî íåñîáñòâåííûé èíòåãðàëZπS 2 (x) dx0ñõîäèòñÿ.9ÈìååìZπZπ2S 2 (x) dx.S (x) dx = limδ→+00δ ñèëó ðàâíîìåðíîé ñõîäèìîñòè íàøåãî òðèãîíîìåòðè÷åñêîãî ðÿäàíà îòðåçêå [δ, π] ïðè δ > 0, èíòåãðàëZπS 2 (x) dxδåñòü ïðåäåë ïî N → ∞ èíòåãðàëîâZπ ÃXNsin kx!2dx2δk=1(îòêóäà ýòî âûòåêàåò?).ÈìååìÃN!2X sin kxNXsin2 kx+4kk3k=1NX2 sin kx sin mx+=22k3 m3k=1 k<mNX1 − cos 2kx=+42k 3k=1N XNXcos(m − k)x − cos(m + k)x.+22k3 m3k=1 k<mk=1NXÑëàãàåìîå23=k3NX1 − cos 2kx4k=12k 3åñòü ÷àñòè÷íàÿ ñóììà àáñîëþòíî è ðàâíîìåðíî ñõîäÿùåãîñÿ (ïî x)ðÿäà è ïðè èíòåãðèðîâàíèè ïî îòðåçêó [δ, π] äàåò àáñîëþòíî è ðàâíîìåðíî ñõîäÿùèéñÿ ðÿä (ïî ïàðàìåòðó δ , ïðè âñåõ çíà÷åíèÿõ ïàðàìåòðà).Ðàññìîòðèì èíòåãðàë îò ñóììûN XNXcos(m − k)x2k=1 k<m2k3 m310,ò.å.N XNXsin(m − k)δÐÿä.2222(m − k)k 3 m 3k=1 k<m∞ X∞X1k=1 k<m(m − k)k 3 m 3ñõîäèòñÿ àáñîëþòíî, ò.ê.

åãî ñõîäèìîñòü â ñèëó èíòåãðàëüíîãî ïðèçíàêà îäíîâðåìåííà ñî ñõîäèìîñòüþ∞ Z∞Xk=1k+1123(x − k)k xZ∞∞X1dx =23k=1k23dx2k+1(x − k)x 3(ïî÷åìó?). Ðàçîáüåì èíòåãðàë íà ÷àñòèZ∞dx=2k+1(x − k)x 3Z2k=Z∞dx+2k+1Èìååì îöåíêè(x − k)x 3Z2k22kdx2k+1(x − k)x 3Zk=1Z∞dtt(t + k)dx=(x − k)x 3Z∞dtkÈòàê,Z∞k+1t(t + k)23=≤ln k2k3,=22kdx(x − k)x 3Z∞≤23(x − k)xtkdx23≤11dt53ln kk23=+3 1.2 k 23322k 3..Èç àáñîëþòíîé ñõîäèìîñòè ñëåäóåò ðàâíîìåðíàÿ ñõîäèìîñòü ïî δ(íà R) ðÿäà∞ X∞Xsin(m − k)δ22 .(m− k)k 3 m 3k=1 k<mÀíàëîãè÷íî äîêàçûâàåòñÿ, ÷òî ðÿä∞ X∞Xsin(m + k)δ2k=1 k<m2(m + k)k 3 m 3òàêæå ñõîäèòñÿ ðàâíîìåðíî ïî δ ∈ R.Òàêèì îáðàçîì, èíòåãðàëZπS 2 (x) dxδåñòü íåïðåðûâíàÿ ïî δ ôóíêöèÿ (êàê ñóììà ðàâíîìåðíî ñõîäÿùèõñÿðÿäîâ) è, çíà÷èò,ZπZπ2S 2 (x) dx ∈ R.S (x) dx = limδ→+00δ3.