Домрина. Лекции (2009) v4.0 (1118430), страница 4
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Ïîäñòàâèì â (∗∗) :1πnRπ−π20bÔn (u)du = 1 (∗∗)[−π,π]Òåîðåìà 4 (Ôåéåðà). Ïóñòü f (x) ∈ C[−π, π], f (−π) = f (π). Òîãäà σn (f, x) ⇒ f (x)Äîê-âî: Ïðîäîëæèì ôóíêöèþ f ñ ïåðèîäîì 2π íà âñþ ïðÿìóþ R.Ò.ê. f ∈ C(R) è ïåðèîäè÷íà, òî f ðàâíîìåðíî ¯íåïðåðûâíà íà R¯,ò.å. ∀ε > 0 ∃δ(ε) > 0 : ∀u ∈ [−δ, δ], ∀ x ∈ R ¯f (x + u) − f (x)¯ < 2εÊðîìå òîãî, f (x) îãðàíè÷åíà íà R: ∃M > 0 : |f (x)| ≤ M ∀ x ∈ RRπRπ11b = πnσn (f, x) − f (x) = (∗∗)Ôn (u)f (u + x)du − f (x) πnÔn (u)du =−π=1πnRπ−π−π¡¢Ôn (u) f (u + x) − f (x) du =1πnRδ−δ¯ Rδ¯¯1¯• ¯ πn |f (x + u) − f (x)|Ôn (u)du¯ ≤−δ¯¯¯¯R¯1¯(.
. . )du¯ ≤• ¯ πn¯ δ≤|u|≤π¯1πnRε2·1πnRδ1πnR(. . . )duδ≤|u|≤πÔn (u)du ≤−δ2M |Ôn (u)|du ≤δ≤|u|≤π∀ε ∃N0 (ε) : |σn (f, x) − f (x)| ≤2MπnRπ−πε2·1πnRπ−πÔn (u)du =1du ≤sin2 2δε2ε2ïðè n> N0 (ε)[−π,π]ε ε+ ≤ ε ∀ x ∈ [−π, π] ⇒ σn (f, x) ⇒ f (x)2 2Òðèãîíîìåòðè÷åñêèì ìíîãî÷ëåíîì íàçîâåì ôóíêöèþ âèäà:Îïðåäåëåíèå.Tn (x) = c0 +(. . . )du +nPk=1ck cos kx + dk sin kxÍàïðèìåð, Sn (f, x), σn (f, x) - òðèãîíîìåòðè÷åñêèå ìíîãî÷ëåíûÒåîðåìà 5 (ïåðâàÿ òåîðåìà Âåéåðøòðàññà). Ïóñòü f ∈ C[−π, π], f (−π) = f (π).Òîãäà f ìîæíî íà [−π, π] ðàâíîìåðíî ïðèáëèçèòü òðèãîíîìåòðè÷åñêèì ìíîãî÷ëåíîì,ò.å.
∀ε > 0 ∃T (x) - òðèãîíîìåòðè÷åñêèé ìíîãî÷ëåí: max |T (x) − f (x)| < εx∈[−π,π]Äîê-âî: Ïî ò. Ôåéåðà σn (f, x) ⇒ f (x), ãäå σn (f, x) - òðèã. ìíîãî÷ëåí.[−π,π]Òåîðåìà 6 (âòîðàÿ òåîðåìà Âåéåðøòðàññà). Ïóñòü f ∈ C[a, b].Òîãäà f ìîæíî íà [a,b] ðàâíîìåðíî ïðèáëèçèòü ìíîãî÷ëåíîì (îáû÷íûì),ò.å. ∀ε > 0 ∃P (x) - ìíîãî÷ëåí: max |P (x) − f (x)| < εx∈[a,b]Äîê-âî:I.Äîêàæåì äëÿ ÷àñòíîãî ñëó÷àÿ : [a, b] = [−π, π], f (π) = f (−π).nPck cos kx + dk sin kx : max |T (x) − f (x)| < ε.Òîãäà ∀ε > 0 ∃ Tn (x) = c0 +x∈[−π,π]k=1T(x) - ëèíåéíàÿ êîìáèíàöèÿ sin kx, cos kx, 0 ≤ k ≤ n, êàæäàÿ èç ýòèõ ôóíêöèéïðåäñòàâèìà ñòåïåííûì ðÿäîì ñ ðàäèóñîì ñõîäèìîñòè R = ∞∞Pak xk , R = ∞⇒ T (x) =k=0Íà [−π, π] ñòåïåííîé ðÿä ñõîäèòñÿ ðàâíîìåðíî ïî òåîðåìå ÀáåëÿNNPPak xkak xk | < ε/2, x ∈ [−π, π] , îáîçíà÷èì P (x) =⇒ ∃N : |T (x) −k=0k=0⇒ |f (x) − P (x)| ≤ |f (x) − T (x)| + |T (x) − P (x)| < ε/2 + ε/2 = ε,21∀ x ∈ [−π, π]II.Òåïåðü ñëó÷àé : [a, b] = [−π, π], íî óñëîâèé íà çíà÷åíèÿ íåòÏîäáåðåì g(x) âèäà: g(x) = f (x) − αx, ñ óñëîâèåì g(−π) = g(π),(−π)ò.å.
f (−π) + απ = f (π) − απ ⇒ α = f (π)−f2π∀ ε > 0 ∃Q(x) - ìíîãî÷ëåí: max |g(x) − Q(x)| < ε (èç I øàãà)x∈[−π,π]|g(x) − Q(x)| = |f (x) − αx − Q(x)| = |f (x) − P (x)|,P (x) = αx + Q(x) - ìíîãî÷ëåí , max |f (x) − P (x)| < ε, ÷.ò.ä.x∈[−π,π]III.Îáîáùèì äëÿ ïðîèçâîëüíîãî îòðåçêà [a, b] ïðè ïîìîùèëèíåéíîéçàìåíû: x = αt + β , ïåðåâîäÿùåé x ∈ [a, b] â îòðåçîê t ∈ [−π, π](−πα + β = ax−βb−a⇒ β = a+b2 , α = 2π 6= 0, t = απα + β = bf (x) ∈ C[a, b]. Ïóñòü F (t) = f (αt + β).
Òîãäà F (t) ∈ C[−π, π]⇒ ∃ Q(t) - ìíîãî÷ëåí: max |F (t) − Q(t)| < ε ( ïî øàãó II )⇒ max |f (x) −x∈[a,b]t∈[−π,π]x−βQ( α )| < ε ⇒P (x) = Q( x−βα ) - èñêîìûé ìíîãî÷ëåíÇàìêíóòîñòü òðèãîíîìåòðè÷åñêîé ñèñòåìû√ x , sin√ x , . . . } çàìêíóòà â L2 [−π, π]Òåîðåìà 7. Òðèãîíîìåòðè÷åñêàÿ ñèñòåìà { √12π , cosRππÄîê-âî:Íóæíî ïîêàçàòü, ÷òî ∀ f ∈ L2R [−π, π] ∀ ε > 0 ∃ T (x) − òðèã. ìíîãî÷ëåí : kf − T (x)k < ε1.
∃g1 ∈ Ĉ[−π, π] : ||f − g1 || < ε/3f èíòåãðèðóåìà ïî Ðèìàíó ⇒a) f îãðàíè÷åíà íà [−π, π], ò.å. ∃M > 0 : |f (x)| ≤ M ∀x ∈ [−π, π]Rπε2á) ∃ τ - ðàçáèåíèå [−π, π] : −π = x0 < · · · < xl = π :f (x)dx − sτ (f ) <18M−πlPsτ (f ) =mk ∆xk , mk = inf f (x)[xk−1 ,xk ]k=1x ∈ (xk−1 , xk ), 1 ≤ k ≤ lmk ,mk +mk+1g1 (x) =, x = xk , k = 1, · · · , l − 1 ⇒ |g1 (x)| ≤ M , ò.ê. |mk | ≤ M2 m1 +mn,x = ±π2Rπg1 (x)dx = sτ (f )−πÒàêèì îáðàçîì,Rπ−πf (x)dx − sτ (f ) =Rπ−π(f (x) − g1 (x))dx =Rπε2|f (x) − g1 (x)|dx <= {f (x) − g1 (x) ≥ 0 êðîìå êîíå÷íîãî ÷èñëà òî÷åê} =18M−πssππRR(f (x) − g1 (x)) 2 dx ≤|f (x) − g1 (x)| (|f (x)| + |g1 (x)|) dx ≤||f − g1 || =−π−πsqRπε2≤ 2M|f (x) − g1 (x)|dx ≤ 2M 18M= ε/3−π222. . Ñòðîèì g2 ∈ C[−π, π], g2 (−π) = g2 (π), ||g2 − g1 || < ε/3 . g2 (x) δ -îêðåñòíîñòè òî÷åê ðàçðûâà g1 (x) çàìåíÿåì ëèíåéíûìè ôóíêöèÿìèg2 (x) = g1 (x) âíå [−π, −π + δ] ∪ [x1 − δ, x1 + δ] ∪ · · · ∪ [π − δ, π]g2 (x) ëèíåéíà íà êàæäîì ñåãìåíòå, ïðè÷åì g2 (x) ∈ C[−π, π],g2 (−π) = g2 (π), |g2 (x)| ≤ M, x ∈ [−π, π] ⇒ |g1 (x) − g2 (x)| ≤ 2MrpR||g1 − g2 || =(g1 (x) − g2 (x))2 dx ≤ (2M )2 2δl < ε/3 äëÿ[−π,−π+δ]∪[x1 −δ,x1 +δ]∪···∪[π−δ,π]äîñòàòî÷íî ìàëûõ δ3.
Ïîêàæåì, ÷òî ∃T (x) : ||g2 − T || < ε/3Äëÿ g2 (x) (î ÷óäî! ) âûïîëíåíû óñëîâèÿ ïåðâîé òåîðåìû Âåéåðøòðàññà⇒ ∃T (x) : òðèãîíîìåòðè÷åñêèé ìíîãî÷ëåí: max |g2 (x) − T (x)| ≤ 3√ε2πx∈[−π,π]ssrπRRπ ε2ε2 12||g2 (x) − T (x)|| =(g2 (x) − T (x)) dx ≤2π = ε/3 ⇒dx=9·2π9 2π−π−π||f (x) − T (x)|| ≤ ||f − g1 || + ||g1 − g2 || + ||g2 − T || ≤ ε/3 + ε/3 + ε/3 = ε⇒ ∀ε > 0 ∃T (x) − òðèãîíîìåòðè÷åñêèé ìíîãî÷ëåí : ||f (x) − T (x)|| < εÑëåäñòâèÿ:1.a202+∞Pk=1(a2k + b2k ) =1πRπ−πf 2 (x)dx, f ∈ L2R [−π, π] - ðàâåíñòâî Ïàðñåâàëÿ2.
lim ||f (x) − Sn (f, x)|| = 0 (ñì. ñëåäñòâèå èç òåîðåìû 2)n→∞3. {Sn (f, x)} ñõîäèòñÿ ê f(x) â ñðåäíåì íà [−π, π], ò.ê. ||f (x)−Sn (f, x)|| =4. ∀[a, b] ⊂ [−π, π]Rbaf (x)dx = limRbn→∞ aRπ−π(f (x) − Sn (f, x))2 dxSn (f, x)dxÄîê-âî: Cõîäèìîñòü â ñðåäíåì íà [−π, π] ⇒ ñõîäèìîñòüâ ñðåäíåì íà ∀[a, b] ⊂ [−π, π] ⇒ ïî÷ëåííàÿ èíòåãðèðóåìîñòü íà [a, b] ,÷.ò.ä.5. Ĉ[−π, π] ⊂ L2R [−π, π] ⇒ òðèãîíîìåòðè÷åñêàÿ ñèñòåìà çàìêíóòà â Ĉ[−π, π] è (+åâêëèäîâîñòüïðîñòðàíñòâà) ïîëíà â Ĉ[−π, π]6. f, g ∈ Ĉ[−π, π], f 6= g.
Òîãäà ðÿäû Ôóðüå äëÿ f è g íå ìîãóò ñîâïàäàòü( âñå êîýôôèöèåíòû Ôóðüå f 6= g íå ìîãóò ñîâïàäàòü - ñâîéñòâî ïîëíûõ ÎÍÑ )Çàäà÷è ê ýêçàìåíó√nx } â L2 [−π, π]?1. Áóäåò ëè çàìêíóòîé { sinRπ2. Áóäåò ëè çàìêíóòîé { sincnnx }, c2n =Rπ0sin2 nxdx â L2R [0, π] ?23Òåîðåìà 8 (Ëîêàëüíàÿ òåîðåìà Ôåéåðà).Ïóñòü f èíòåãðèðóåìà íà [−π, π], ïåðèîä f = 2π ; â òî÷êå x0Òîãäàn→∞σn (f, x0 ) −→ 21Äîê-âî: σn (f, x0 ) =1πn12Rπ0Ôn (t)dt =1πnR0−π∃ f (x0 + 0), f (x0 − 0).[f (x0 + 0) + f (x0 − 0)]1πnRπ−πˆ ):f (x0 + t) Ôn (t) dt.  ñèëó ÷åòíîñòè ÿäðà Ôåéåðà è (∗∗Ôn (t)dt =[f (x0 + 0) + f (x0 − 0)] =1πnRπ01 12 πnRπ−π12Ôn (t)dt =f (x0 + 0) Ôn (t) dt +1πnR0−πf (x0 − 0) Ôn (t) dtÒàêèì îáðàçîì, σn (f, x) − 12 [f (x0 + 0) + f (x0 − 0)] =RπR011= πn[f (x0 + t) − f (x0 + 0)] Ôn (t) dt + πn[f (x0 + t) − f (x0 − 0)] Ôn (t) dt = I1n + I2n−π0Ïîêàæåì, ÷òî lim I1n = 0 (àíàëîãè÷íî lim I2n = 0)n→∞n→∞• ∀ ε > 0 ∃ δ(ε) > 0 (ìîæíî ñ÷èòàòü δ < π) : |f (x0 + t) − f (x0 + 0)| < ε ∀t ∈ [0, δ)• f îãðàíè÷åíà íà R (ò.ê.
èíòåãðèðóåìà ïî Ðèìàíó è ïåðèîäè÷íà)⇒ ∃M > 0 : |f (x)| ≤ M ∀x ∈ RI1n =1πnRδ0[f (x0 + t) − f (x0 + 0)] Ôn (t) dt +1πn¯¯¯¯ 1 Rδ¯• ¯ πn [f (x0 + t) − f (x0 + 0)] Ôn (t) dt¯¯ <0Rπ[f (x0 + t) − f (x0 + 0)]δ1πnRδ01ε Ôn (t) dt < ε πnRπ0Ôn (t) dt =11• |f (x0 + t) − f (x0 + 0)| ≤ 2M sin2 nt2 ≤ 1, 2 sin2 t ≤ 2 sin2 δ22¯¯π2 nt¯ 1 Rπ¯Rsin1πM2M2¯¯¯ πn [f (x0 + t) − f (x0 + 0)] 2 sin2 2t dt¯ ≤ 2πn sin2 2δ dt ≤ πn sin2 2δ =δ⇒ |I1n | ≤ε2δ+ε2sin2 nt2dt2 sin2 2tCn≤ε2ε2∀n ∈ Nïðè n ≥ N0 (ε)= ε ïðè n ≥ N0 (ε) ⇒ lim I1n = 0n→∞Ñëåäñòâèå: Ïóñòü f ∈ L2R [−π, π] , ïåðèîäè÷íà ñ ïåðèîäîì 2π ,∃ f (x0 ± 0), ÒÐÔ ôóíêöèè f(x) ñõîäèòñÿ â òî÷êå x0 : ∃ lim Sn (f, x0 )n→∞òîãäà lim Sn (f, x0 ) =n→∞12[f (x0 + 0) + f (x0 − 0)]Äîê-âî: ∃ lim σn (f, x0 ) = lim Sn (f, x0 ) (ðåãóëÿðíûé ìåòîä ×åçàðî), ïðè÷åì ïî ëîêàëüíîén→∞n→∞òåîðåìå Ôåéåðà ýòîò ïðåäåë ðàâåí12[f (x0 + 0) + f (x0 − 0)]Ïðîñòåéøèå óñëîâèÿ ðàâíîìåðíîé ñõîäèìîñòè è ïî÷ëåííîéäèôôåðåíöèðóåìîñòè òðèãîíîìåòðè÷åñêîãî ðÿäà Ôóðüåf(x) èìååò íà [a,b] êóñî÷íî-íåïðåðûâíóþ ïðîèçâîäíóþ,lS{xj },åñëè ∃ a = x0 < · · · < xl = b, f äèôôåðåíöèðóåìà íà [a, b]\Îïðåäåëåíèå.∃f 0 (xj−1 + 0), f 0 (xj − 0), f 0 ∈ C(xj−1 , xj ) ïðè j = 1, .., l 66j=0îöåíèòå ðàçíèöó ñ ïîíÿòèåì: ïðîèçâîäíàÿ ôóíêöèè êóñî÷íî-íåïðåðûâíà24Ïðèìåð: f (x) = |x| íà [−π, π] èìååò êóñî÷íî-íåïðåðûâíóþ ïðîèçâîäíóþÒåîðåìà 9.
Ïóñòü f (x) ∈ C[−π, π], f (−π) = f (π),a02f (x) èìååò êóñî÷íî-íåïðåðûâíóþ ïðîèçâîäíóþ íà îòðåçêå [−π, π]∞P+(ak cos (kx) + bk sin (kx)) à f (x) ÒÐÔ ôóíêöèè f(x)k=1∞Pα0(αk2 +k=1cos (kx) + βk sin (kx)) à f 0 (x) ÒÐÔ ôóíêöèè f '(x), äîîïðåäåëåííîé ïðîèçâîëüíûìîáðàçîì â òî÷êàõ ðàçðûâàÒîãäà1. ÒÐÔ ôóíêöèè f '(x) ïîëó÷åí ïî÷ëåííûì äèôôåðåíöèðîâàíèåì TÐÔ f(x),ò.å. α0 = 0, αk = −kbk , βk = kak , k ∈ N,∞∞¡ ¢0 PPèíûìè ñëîâàìè, α20 +[ak cos0 (kx) + bk sin0 (kx)]7[αk cos (kx) + βk sin (kx)] = a20 +k=1k=12. ÒÐÔ ôóíêöèè f(x) ñõîäèòñÿ íà [−π, π] àáñîëþòíî è ðàâíîìåðíî ê f(x)Äîê-âî:1.
αk ==1π+ πk1πRπ−πlPj=1Rπ−πf 0 (x) cos kxdx ="¯xf (x) cos kx¯xjj−1+1πlRxjPj=1 xj−1Rxjf 0 (x) cos kxdx = {èíòåãðèðîâàíèå ïî ÷àñòÿì} =#f (x)k sin kxdx =xj−11π[ f(π )cosπ k-f(-π )cos(-π k) ]+f (x) sin kxdx = 0 + kbk ⇒ αk = kbk . Àíàëîãè÷íî βk = −kαk , k ∈ N2. Äîñòàòî÷íî ïîêàçàòü ñõîäèìîñòü ðÿäà∞Pk=1|ak | + |bk |.Òîãäà ÒÐÔ f(x) ñõîäèòñÿ àáñîëþòíî è ðàâíîìåðíî íà [−π, π] ïî ïð.
Âåéåðøòðàññà;(à ðÿä ñàìîé ôóíêöèè áóäåò ñõîäèòüñÿ ê íåé ñàìîé: ïåðèîäè÷åñêè ïðîäîëæàåì f (x) íàR è ïîëüçóåìñÿ ñëåäñòâèåì ëîêàëüíîé òåîðåìû Ôåéåðà ⇒ ÒÐÔ ôóíêöèè f(x) ñõîäèòñÿê f(x)).α2 +β 2|ak | + |bk | = |αkk | + |βkk | ≤ 12 (βk2 + k12 ) + 12 (αk2 + k12 ) = k 2 k + k12∞∞PP1(αk2 + βk2 ) ñõîäèòñÿ â ñèëó ðàâåíñòâà Ïàðñåâàëÿ,ñõîäèòñÿk2k=1⇒∞Pk=1k=1|ak | + |bk | ñõîäèòñÿÇàìå÷àíèå:  óñëîâèè ïðåäûäóùåé òåîðåìû lim k(|ak | + |bk |) = 0k→∞Äîê-âî:∞Pk=1(αk2 + βk2 ) ñõîäèòñÿ ⇒ lim |αk | = lim |βk | = 0.k→∞k→∞Íî |αk | + |βk | = k(|ak | + |bk |)Ïðèìåð: f (x) = sin x10Èññëåäîâàòü íà ðàâíîìåðíóþ ñõîäèìîñòü ÒÐÔ íà [−π, π]f (x) ∈ C[−π, π], f (−π) = f (π), f 0 (x) ∈ C[−π, π]⇒ ÒÐÔ ôóíêöèè f(x) ðàâíîìåðíî ñõîäèòñÿ ê f(x) íà [−π, π].7ïîäðàçóìåâàåòñÿ ïî÷ëåííîå ðàâåíñòâî ðÿäîâ, à íå ðàâåíñòâî ñóìì25Òåîðåìà 10.
Ïóñòü f (x), f 0 (x), · · · , f (m) (x) ∈ C[−π, π],f (m) (x) èìååò êóñî÷íî-íåïðåðûâíóþ ïðîèçâîäíóþ íà [−π, π],f (−π) = f (π), f 0 (−π) = f 0 (π), · · · , f (m) (−π) = f (m) (π).∞Pa0+(ak cos (kx) + bk sin (kx)) à f (x),2(j)α02Òîãäàk=1∞P+k=1(j)(j)(αk cos (kx) + βk sin (kx)) à f (j) (x), 1 ≤ j ≤ m + 1(j)1. α0 = 0(j)(j)αk cos kx + βk sin kx = ak (cos kx)(j) + bk (sin kx)(j)1≤j ≤m+12. ÒÐÔ ôóíêöèè f (j−1) (x) ðàâíîìåðíî ñõîäèòñÿ ê f (j−1) (x) íà [−π, π] 1 ≤ j ≤ m + 1Äîê-âî: Íàäî m ðàç ïðèìåíèòü ïðåäûäóùóþ òåîðåìóÇàìå÷àíèå:  óñëîâèè ïðåäûäóùåé òåîðåìû lim k m+1 (|ak | + |bk |) = 0k→∞Äîê-âî:(m+1)|αk|(m+1)|βk|(m+1)(m+1)= k m+1 (|ak | + |bk |), lim |αk| + |βk|=0k→∞∞P (m+1) 2(m+1) 2(αk) + (βk) < ∞.ò.ê. ñïðàâåäëèâî ðàâåíñòâî Ïàðñåâàëÿ+k=1Êîìïëåêñíàÿ çàïèñü òðèãîíîìåòðè÷åñêîãî ðÿäà Ôóðüåk ikxk −ikxak cos kx + bk sin kx = ak −ibe + ak +ibe= ck eikx + c−k e−ikx , k = 0, 1, · · ·22Åñëè f(x) âåùåñòâåííàÿ ⇒ c−k = ck , k = 0, 1, · · ·Rπa01c=f (x)dx ⇒ c0 ei0x = a20=022π−πRπRπ11f (x)(cos kx − i sin kx)dx = 2πf (x)e−ikx dx k ∈ Nck = 2π−π−πRπRπ11f (x)(cos kx + i sin kx)dx = 2πf (x)e ikx dx k ∈ Nc−k = 2π−π−πèòîã : f (x) Ã+∞Xck eikxk=−∞1ck =2πZπf (x)e−ikx dx, k ∈ Z−πÊàê è ñëåäîâàëî îæèäàòü, â êîìïëåêñíûõ ÷èñëàõ âñå ñòàëî áîëåå åñòåñòâåííî è ëîãè÷íî èñ÷åçëà íåñèììåòðè÷íîñòü (∃a0 , @b0 ), ïðîïàë êîýôôèöèåíò 1/2∞∞PPf (φ = x) Ãck eikφ = {z = eiφ } =ck z k - ðÿä Ëîðàíà íà åäèíè÷íîé îêðóæíîñòè8k=−∞k=−∞Ìû ïî çíà÷åíèþ ôóíêöèè íà åäèíè÷íîé îêðóæíîñòè âîññòàíîâèëè å¼ ðÿä Ëîðàíà(ïðè óñëîâèÿõ íà ôóíêöèþ, ñ òî÷íîñòüþ äî çíà÷åíèé â òî÷êàõ ðàçðûâà)Ïðè ýòîì êîýôôèöèåíòû ðÿäà íàõîäÿòñÿ ñîãëàñíî èçâåñòíîìó ïðàâèëó:RπR f1 (z)11ck = 2πf (x)e−ikx dx = {z = eix , f1 (z) = f (x)} = 2πidz, k ∈ Zk+1−π|z|=1 z óñëîâèè òåîðåìû 9: f 0 (x) Ã8+∞Pk=−∞ck ik eikxâîò è òàéíà óäîáñòâà ìíîæåñòâà [−π, π] - ýòî óãîë, à òàêæå òðåáîâàíèé: 2π -ïåðèîäè÷íîñòü,f (−π) = f (π) è ò.ï.
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