Диссертация (Некоторые методы проекционного типа численного решения одного класса слабо сингулярных интегральных уравнений), страница 4

(1.4.24)31Äîêàçàòåëüñòâî. Ïóñòü f ∈ Lp (J), 1 ≤ p < ∞. Äëÿ τ ∈ (τi−1 , τi ) èìååìpZτ∗hZτii00 =(I − π h )Λf (τ )p = E(|τ − t|) − h−1E(|τ−t|)dτf(t)dtiτi−10Zτ∗= 0Zτ∗≤ Zτih−1ihp 00E(|τ − t|) − E(|τ − t|) dτ f (t) dt ≤τi−1Zτih−1i|E(|τ − t|) − E(|τ 0 − t|)|s dτ 0i1/sp|f (t)| dt ≤τi−10Zτ∗Zτiip−1−10s0≤ hi|E(|τ − t|) − E(|τ − t|)| dτ dt×0τZ∗×0hh−1iτi−1Zτi0s|E(|τ − t|) − E(|τ − t|)| dτ0ip/q|f (t)|p dt ≤τi−1s(p−1)≤ ωs (E; hmax )Zτ∗hh−1iZτi0s|E(|τ − t|) − E(|τ − t|)| dτ0ip/q|f (t)|p dt.τi−10Èñïîëüçóÿ îáîáùåííîå íåðàâåíñòâî Ìèíêîâñêîãî, èìååìk(I − π h )Λf kpLq (J) = k |(I − π h )Λf |p kLq/p (J) ≤p/qZτ∗XZτi Zτinsωs (E, hmax )s(p−1)h−1|E(|τ − t|) − E(|τ 0 − t|) dτ 0 dτ|f (t)|p dt ≤i0i=0τi−1 τi−1≤ ωs (E, hmax )s(p−1) ω s (E; J h )sp/q kf kpLp (J) .Èç âûâåäåííîãî íåðàâåíñòâà ïðè 1 ≤ p < ∞ ñëåäóåò îöåíêàk(I − π h )ΛkLp (J)→Lq (J) ≤ ωs (E, hmax )1−s/q ω s (E; J h )s/q .(1.4.25)Ïóñòü p = q = ∞.

Òîãäà äëÿ τ ∈ (τi−1 , τi ) ñïðàâåäëèâà îöåíêà(I − π h )Λf (τ ) ≤ h−1iZτi Zτ∗τi−1≤ ω1 (E, hmax )kf kL∞ (J) ,0E(|τ − t|) − E(|τ 0 − t|) dt dτ 0 kf kL (J) ≤∞32èç êîòîðîé ñëåäóåò, ÷òîk(I − π h )Λf kL∞ (J) ≤ ω1 (E, hmax )kf kL∞ (J) .Òàêèì îáðàçîì, íåðàâåíñòâî (1.4.25) âåðíî è ïðè p = ∞. Îöåíèâàÿ åãî ïðàâóþ÷àñòü ñ ïîìîùüþ íåðàâåíñòâ (1.4.6), (1.4.9), âûâîäèì îöåíêó (1.4.19).Çàìå÷àÿ, ÷òî îïåðàòîð Λ(I − π h ) ÿâëÿåòñÿ ñîïðÿæåííûì ê îïåðàòîðó(I − π h )Λ è ó÷èòûâàÿ, ÷òî 1 − 1/q 0 + 1/p0 = 1/s, 1 − s/p0 = s/q , èìååìkΛ(I − π h )kLp (J)→Lq (J) = k(I − π h )ΛkLq0 (J)→Lp0 (J) ≤≤ ωs (E, hmax )s/q ω s (E; J h )1−s/q ≤ 23/s−1/q kEkLs (0,hmax /2) .Ïóñòü µ òàêîâî, ÷òî 2/µ = 1/p + 1/q . Çàìåòèì, ÷òî p ≤ µ ≤ q è 1 − 1/p +1/µ = 1/r, 1 − 1/µ + 1/q = 1/r.Èñïîëüçóÿ îöåíêè (1.4.19), (1.4.20), è ó÷èòûâàÿ, ÷òî I − π h = (I − π h )2 ,èìååìkΛ(I − π h )ΛkLp (J)→Lq (J) = kΛ(I − π h )(I − π h )ΛkLp (J)→Lq (J) ≤≤ kΛ(I − π h )kLµ (J)→Lq (J) k(I − π h )ΛkLp (J)→Lµ (J) ≤≤ ωr (E, hmax )1−r/q ω r (E; J h )r/q ωr (E, hmax )r/µ ω r (E; J h )1−r/µ == ωr (E, hmax )r ω r (E; J h )2−r ≤ 22+3/s kEk2Lr (0,hmax /2) .îöåíêè (1.4.22)(1.4.24) ñëåäóþò èç îöåíîê (1.4.19)(1.4.21) â ñèëó ñâîéñòâà(1.4.25).33ÃËÀÂÀ 2.

Ìåòîäû ïðîåêöèîííîãî òèïàñ èñïîëüçîâàíèåì ïðîñòðàíñòâà êóñî÷íîëèíåéíûõ ôóíêöèé è îöåíêè èõïîãðåøíîñòåéÍàïîìíèì, ÷òî âñþäó â ðàáîòå ïðåäïîëàãàåòñÿ, ÷òî ÿäðî E èíòåãðàëüíîãîîïåðàòîðà Λ îáëàäàåò ñëåäóþùèìè ñâîéñòâàìè:A1 ) E ÿâëÿåòñÿ ïîëîæèòåëüíîé óáûâàþùåé ôóíêöèåé, çàäàííîé íà R+ èòàêîé, ÷òî E(0+ ) = +∞;A2 ) E ∈ Lr (R+ ) äëÿ âñåõ r ∈ [1, +∞); êðîìå òîãî, kEkL1 (R+ ) ≤ 1/2.A3 ) Äëÿ âñåõ r ≥ 1 âûïîëíåíî ñâîéñòâîZδE r (τ ) dτ ∼ δE r (δ) ïðè δ → 0+ .0Ââåäåì îïåðàòîðû σ h , πbh , `h , êîòîðûå äàëåå áóäóò èñïîëüçîâàíû â ðîëèîïåðàòîðà P h è óñòàíîâèì íåêîòîðûå èõ ñâîéñòâà.Ââåäåì ïðîñòðàíñòâî êóñî÷íî ëèíåéíûõ ôóíêöèé Sbh (J), ýëåìåíòàìèêîòîðîãî ÿâëÿþòñÿ ôóíêöèè âèäà f (τ ) =nPi=0f (τi )ehi (τ ), ãäå ehi ñòàíäàðòíûå34áàçèñíûå ôóíêöèè - "øàïî÷êè".(τ − τi−1 )/hi , τ ∈ [τi−1 , τi ],h0 < i < n,ei (τ ) = (τi+1 − τ )/hi+1 , τ ∈ [τi , τi+1 ],0, τ ∈/ [τi−1 , τi+1 ],(τ − τn−1 )/hn , τ ∈ [τn−1 , τn ],(τ1 − τ )/h1 , τ ∈ [τ0 , τ1 ],hhen (τ ) =e0 (τ ) =0, τ ∈/ [τn−1 , τn ].0, τ ∈/ [τ0 , τ1 ],Äëÿ äàëüíåéøåãî óäîáñòâà äîîïðåäåëèì ñåòêó J h óçëàìè τ−1 = τ0 , τn+1 =τn .

Ïîëîæèì Jih = (τi−1 , τi+1 ) è hi+1/2 = (hi + hi+1 )/2 äëÿ 0 ≤ i ≤ n, ñ÷èòàÿh0 = 0, hn+1 = 0. Çàìåòèì, ÷òî (1, ehi ) = hi+1/2 äëÿ âñåõ 0 ≤ i ≤ n.2.1Óñðåäíÿþùèé îïåðàòîðσhè íåêîòîðûå åãî ñâîéñòâàÎïðåäåëèì îïåðàòîð σ h : Lp (J) → Sbh (J), ãäå 1 ≤ p ≤ ∞, ôîðìóëîéh(σ f )(τ ) =nXσ h f (τi )ehi (τ ),τ ∈ J,i=0hhãäå σ h f (τi ) = h−1i+1/2 (f, ei ). Êàê íåòðóäíî âèäåòü, îïåðàòîð σ íå ÿâëÿåòñÿîïåðàòîðîì ïðîåêòèðîâàíèÿ, îäíàêî Im σ h = Sbh (J) è kσ h kLp (J)→Lp (J) = 1äëÿ âñåõ p ∈ [1, ∞].Äëÿ f ∈ Lp (J) ïîëîæèìωp (f ; J h ) =1/pnR RP−10p h 0 h0, 1 ≤ p < ∞, i=0 hi+1/2 h h |f (τ ) − f (τ )| ei (τ )ei (τ ) dτ dτJi Jimax ess sup |f (τ 0 ) − f (τ )|,1≤i≤nh0p = ∞.τ ,τ ∈JiËåììà 2.1.1.

Ïóñòü f ∈ Lp (J), 1 ≤ p ≤ ∞. Òîãäàk(I − σ h )f kLp (J) ≤ ωp (f ; J h ).(2.1.1).35Äîêàçàòåëüñòâî. Çàìåòèì, ÷òîh(I − σ )f (τ ) = f (τ ) −nXh−1i+1/2 (f, ei )ei (τ ) =i=0=nXh−1i+1/2i=0Z(f (τ ) − f (τ 0 ))ehi (τ 0 ) dτ 0 ehi (τ ). (2.1.2)JihÏîýòîìó ïðè 1 ≤ p < ∞ ñïðàâåäëèâî íåðàâåíñòâî|(I − σ h )f (τ )|p ≤p−1XXZZnnehi (τ 0 ) dτ 0 ehi (τ )|f (τ )−f (τ 0 )|p ehi (τ 0 ) dτ 0 ehi (τ )h−1.≤h−1i+1/2i+1/2i=0i=0JihJihÓ÷èòûâàÿ, ÷òînXh−1i+1/2i=0Zehi (τ 0 ) dτ 0 ehi (τ )Jih=nXehi (τ ) = 1(2.1.3)i=0è èíòåãðèðóÿ âûâåäåííîå íåðàâåíñòâî, ïðèõîäèì ê îöåíêå (2.1.1).Ïóñòü òåïåðü f ∈ L∞ (J). Òîãäàk(I − σ h )f kLp (J) ≤ ωp (f ; J h ) ≤1/pXn0p≤ess sup |f (τ ) − f (τ )| hi+1/2≤ ω∞ (f ; J h )τ∗1/p .i=0τ 0 ,τ ∈JihÏåðåõîäÿ ê ïðåäåëó ïðè p → ∞, ïðèõîäèì ê îöåíêå (2.1.1) ñ p = ∞.Ñëåäñòâèå 2.1.1.

Ïóñòü f ∈ Lp (J), 1 ≤ p < ∞. Òîãäàk(I − σ h )f kLp (J) → 0 ïðè hmax → 0.Äîêàçàòåëüñòâî. Ïîñêîëüêó ïðîñòðàíñòâî C(J) âñþäó ïëîòíî â Lp (J), òî äëÿâñÿêîãî ε > 0 ñóùåñòâóåò fε ∈ C(J) òàêîå, ÷òî kf − fε kLp (J) < ε. Ïîýòîìók(I−σ h )f kLp (J) ≤ k(I−σ h )(f −fε )kLp (J) +k(I−σ h )fε kLp (J) < 2ε+ω∞ (fε ; J h )τ∗1/p .36Òàêèì îáðàçîì,lim k(I − σ h )f kLp (J) ≤ 2ε.hmax →0Ñëåäñòâèå äîêàçàíî.Ïóñòü 1 ≤ r < ∞. Ïîëîæèìωbr (E; J h ) = supXnt∈JZ Zhi+1/2i=0|E(|τ 0 − t|) − E(|τ − t|)|r ehi (τ 0 )ehi (τ ) dτ 0 dτ1/r.Jih JihËåììà 2.1.2. Ñïðàâåäëèâà îöåíêàωbr (E; J h ) ≤ 24/r h1/rmax E(hmax /2)(1 + o(1)) ïðè hmax → 0.(2.1.4)Äîêàçàòåëüñòâî. ÏîëîæèìIih (t)=h−1i+1/2Z Z|E(|τ 0 − t|) − E(|τ − t|)|r ehi (τ 0 )ehi (τ ) dτ 0 dτJih Jihè ïðåäñòàâèì ωbr (E; J h ) â âèäåhωbr (E; J ) = suph X0Iih (t)tt∈JP0Çäåñü çíàêt+X00Iih (t)t+X000i1/rIih (t)t.îçíà÷àåò ñóììèðîâàíèå ïî òåì èíäåêñàì i, äëÿ êîòîðûõ|t − τi | ≤ 3hmax /2, çíàêP00îçíà÷àåò ñóììèðîâàíèå ïî òåì i, äëÿ êîòîðûõt > τi + 3hmax /2, à çíàêP000îçíà÷àåò ñóììèðîâàíèå ïî òåì i, äëÿ êîòîðûõttt < τi − 3hmax /2.Ïóñòü |t − τi | ≤ 3hmax /2.

ÒîãäàIih (t)≤h−1i+1/2Z Z[E(|τ 0 − t|)r + E(|τ − t|)r ] ehi (τ 0 )ehi (τ ) dτ 0 dτ =Jih JihZ0E(|τ −=Jiht|)r ehi (τ 0 ) dτ 0Z0E(|τ −+Jiht|)r ehi (τ ) dτZ=2JihE(|τ − t|)r ehi (τ ) dτ.37ÑëåäîâàòåëüíîX0tIih (t) ≤ 2Zτ∗E(|τ − t|)rX0ehi (τ ) dτ ≤0t+5hZ max /2E(|τ − t|)r dτ = 4≤25hZmax /2E(τ )s dτ.0t−5hmax /2Ïóñòü òåïåðü t > τi + 3hmax /2. Ïîëüçóÿñü äëÿ τ, τ 0 ∈ Jih îöåíêîé|E(τ − t) − E(τ 0 − t)|r ≤ E r (τi−1 − t) − E r (τi+1 − t),èìååìIih (t) ≤ hi+1/2 E r (τi−1 − t) − E r (τi+1 − t) ≤ hmax E r (τi−1 − t) − E r (τi+1 − t) .ÑëåäîâàòåëüíîP00ht Ii (t) ≤ hmaxP00 rrE(τ−t)−E(τ−t)≤ 2hmax E r (hmax /2).i−1i+1tÀíàëîãè÷íîP000tIih (t) ≤ 2hmax E r (hmax /2).Îáúåäèíÿÿ ïîëó÷åííûå îöåíêè, ïðèõîäèì ê íåðàâåíñòâóh5hZmax /21/rE (t) dt + 4hmax E (hmax /2).ωbr (E; J ) ≤ 4rr0Ïîëüçóÿñü òåì, ÷òî5hZmax /2E r (t) dt ∼ (5hmax /2)E(5hmax /2),0èìååìi1/rωbr (E; J ) ≤ 4(5hmax /2)E (5hmax /2)(1 + o(1)) + 4hmax E (hmax /2)≤hh1/rrr1/r≤ 141/r hmax E(hmax /2)(1 + o(1)) ≤ 24/r hmax E(hmax /2)(1 + o(1)).38Ëåììà 2.1.3.

Ïóñòü 1 ≤ p ≤ q ≤ ∞, 1/s = 1 − 1/p + 1/q > 0. Òîãäàñïðàâåäëèâû îöåíêèk(I − σ h )ΛkLp (J)→Lq (J) ≤ ωs (E, 2hmax )1−s/q ωbs (E; J h )s/q ,(2.1.5)kΛ(I − σ h )kLp (J)→Lq (J) ≤ ωs (E, 2hmax )s/q ωbs (E; J h )1−s/q ,(2.1.6)kΛ(I − σ h )ΛkLp (J)→Lq (J) ≤ Ms ω1 (E, 2hmax )1/(2s) ωb1 (E; J h )1/(2s) .(2.1.7)Äîêàçàòåëüñòâî. Ïóñòü f ∈ Lp (J), 1 ≤ p < ∞. Êàê íåòðóäíî âèäåòü,(I − σ h )Λf (τ ) = hniX τ∗−10h00hhi+1/2 ∫ E(|τ − t|)ei (τ ) dτ ei (τ ) f (t) dt == ∫ E(|τ − t|) −0Jihi=0 n τ∗ X −1= ∫hi+1/2 ∫ E(|τ − t|) − E(|τ 0 − t|) ehi (τ 0 ) dτ 0 ehi (τ )f (t) dt. (2.1.8)0Jihi=0Çàìåòèì, ÷òînPi=0"≤nPi=0h−1i+1/2RE(|τ − t|) − E(|τ 0 − t|)eh (τ 0 ) dτ 0 eh (τ ) ≤iiJih#1/sRE(|τ − t|) − E(|τ 0 − t|)s eh (τ 0 ) dτ 0 eh (τ )×h−1iii+1/2Jih"×nPi=0#1−1/sh−1i+1/2Rehi (τ 0 ) dτ 0 ehi (τ ).JihÏðèíèìàÿ âî âíèìàíèå ôîðìóëó (2.1.3), èìååì(I − σ h )Λf (τ )p ≤#1/sp "Xnτ∗τ∗s−10h00h≤ ∫hi+1/2 ∫ E(|τ − t|) − E(|τ − t|) ei (τ ) dτ ei (τ )|f (t)| dt ≤0"nτ∗ X≤ ∫00i=0i=0τ∗E(|τ − t|) − E(|τ 0 − t|)s ehi (τ 0 ) dτ 0 ehi (τ ) dth−1∫i+1/2#p−1×0nip/qτ∗hXs h 0−100 h×∫hi+1/2 ∫ E(|τ − t|) − E(|τ − t|) ei (τ ) dτ ei (τ )|f (t)|p dt ≤0i=0Jih39≤ ωs (E, 2hmax )s(p−1) × np/qτ∗ Xs h 000h−1×∫hi+1/2 ∫ E(|τ − t|) − E(|τ − t|) ei (τ ) dτ ei (τ )|ϕ(t)|p dt.0Jihi=0Èñïîëüçóÿ îáîáùåííîå íåðàâåíñòâî Ìèíêîâñêîãî, èìååìk(I − σ h )ΛϕkpLq (J) = k |(I − σ h )Λϕ|p kLq/p (J) ≤ ωs (E, 2hmax )s(p−1) ×p/q nτ∗ Xsh 000 h×∫|ϕ(t)|p dt ≤h−1i+1/2 ∫ ∫ |E(|τ − t|) − E(|τ − t|) ei (τ )ei (τ ) dτ dτ0i=0Jih Jih≤ ωs (E, 2hmax )s(p−1) ωbs (E; J h )sp/q kϕkpLp (J) .Èç âûâåäåííîãî íåðàâåíñòâà ïðè 1 ≤ p < ∞ ñëåäóåò îöåíêà (2.1.5).

Åñëèæå p = q = ∞, òî èç (2.1.8) ñëåäóåò(I − σ h )Λf (τ ) ≤nXτ∗0h 00 h≤h−1i+1/2 ∫ ∫ E(|τ − t|) − E(|τ − t|) dt ei (τ ) dτ ei (τ )kf kL∞ (J) ≤i=0Jih≤ ω1 (E, 2hmax )0nXehi (τ )kf kL∞ (J) = ω1 (E, 2hmax )kf kL∞ (J) .i=0Òàêèì îáðàçîì,k(I − σ h )Λf kL∞ (J) ≤ ω1 (E, 2hmax )kf kL∞ (J) .Îöåíêà (2.1.5) äîêàçàíà.Çàìå÷àÿ, ÷òî îïåðàòîð Λ(I − σ h ) ÿâëÿåòñÿ ñîïðÿæåííûì ê îïåðàòîðó(I − σ h )Λ è ó÷èòûâàÿ, ÷òî 1 − 1/q 0 + 1/p0 = 1/s, 1 − s/p0 = s/q , èìååìkΛ(I −σ h )kLp (J)→Lq (J) = k(I −σ h )ΛkLq0 (J)→Lp0 (J) ≤ ωs (E, 2hmax )s/q ωbs (E; J h )1−s/q .40Èñïîëüçóÿ îöåíêè (2.1.5), (2.1.6), è íåðàâåíñòâî kΛkLp (J)→Lq (J) ≤ Ms , èìååìkΛ(I − σ h )ΛkLp (J)→Lq (J) ≤ Ms k(I − σ h )ΛkLp (J)→Lp (J) ≤≤ Ms ω1 (E, 2hmax )1−1/p ωb1 (E; J h )1/p ,kΛ(I − σ h )ΛkLp (J)→Lq (J) ≤ Ms kΛ(I − σ h )kLq (J)→Lq (J) ≤≤ Ms ω1 (E, 2hmax )1/q ωb1 (E; J h )1−1/q .Ïåðåìíîæàÿ ïîëó÷åííûå íåðàâåíñòâà, ïðèõîäèì ê îöåíêå (2.1.7).Ñëåäñòâèå 2.1.2.

Ïóñòü 1 ≤ p ≤ q ≤ ∞, 1/s = 1 − 1/p + 1/q > 0. Òîãäàñïðàâåäëèâû îöåíêèk(I − σ h )ΛkLp (J)→Lq (J) ≤ 21/s+3/q h1/smax E(hmax /2)(1 + o(1)) ïðè hmax → 0,(2.1.9)kΛ(I − σ h )kLp (J)→Lq (J) ≤ 24/s−3/q h1/smax E(hmax /2)(1 + o(1)) ïðè hmax → 0,(2.1.10)kΛ(I − σ h )ΛkLp (J)→Lq (J) ≤ 25/(2s) Ms hmax E(hmax /2)(1 + o(1)) ïðè hmax → 0.(2.1.11)Äîêàçàòåëüñòâî.

Ïðèìåíÿÿ ñëåäóþùóþ èç (1.4.7) îöåíêóωr (E, 2hmax ) ≤ (2hmax )1/r E(hmax )(1 + o(1)) ≤ 21/r h1/rmax E(hmax /2)(1 + o(1))è îöåíêó (2.1.4), ïðèõîäèì îò (2.1.5) - (2.1.7) ê (2.1.9) - (2.1.11).Ëåììà 2.1.4. Ïóñòü f ∈ Wp1 (J), 1 ≤ p ≤ q ≤ ∞, 1/s = 1 − 1/p + 1/q .Òîãäàk(I − σ h )f kLq (J) ≤ 21/s kDf kLp (J) h1/smax .(2.1.12)41Äîêàçàòåëüñòâî. Ïóñòü q 6= ∞. Òîãäàk(I − σ h )f kLq (J) ≤ ωq (f ; J h ) ≤nPi=0≤nPi=01−p/q≤ max kDf kLp (J h )1≤i≤nikDf kqL1 (J h ) hi+1/2i=0kDf kpLp (J h )1/qi≤i0kDf kqLp (J h ) (2hi+1/2 )q/p hi+1/2inP1/q01/q≤1/s1/s21/p hmax ≤ 21/s kDf kLp (J) hmax .Ïóñòü q = ∞. Òîãäàk(I − σ h )f kL∞ (J) ≤ ω∞ (f ; J h ) ≤ max kDf kL1 (Jih ) ≤ kDf kLp (J) (2hmax )1−1/p .1≤i≤nËåììà äîêàçàíà.Ëåììà 2.1.5.

1. Ïóñòü f ∈ W11 (J). ÒîãäàkΛ(I − σ h )f kC(J) ≤ 2kDf kL1 (J) hmax E(hmax )(1 + o(1)) ïðè hmax → 0,(2.1.13)√kΛ(I − σ h )Λf kC(J) ≤ 2M2 kDΛf kL2 (J) hmax .(2.1.14)2. Ïóñòü f ∈ Wp1 (J), 1 ≤ p ≤ q ≤ ∞, 1/s = 1 − 1/p + 1/q > 0. ÒîãäàkΛ(I − σ h )f kLq (J) ≤ 2Ms kDf kLp (J) hmax ,(2.1.15)k(I − σ h )Λf kLq (J) ≤ 2kDΛf kLq (J) hmax .(2.1.16)Äîêàçàòåëüñòâî.

Èñïîëüçóÿ ôîðìóëó (2.1.2), èìååìτ∗RΛ(I − σ h )f (t) = E(|t − τ |)(I − σ h )f (τ ) dτ ≤0≤nRτ∗ P0 i=0≤n RPE(|t −i=0 J hh−1i+1/2 E(|t − τ |)|f (τ 0 ) − f (τ )|ehi (τ 0 ) dτ 0 ehi (τ ) dτ ≤Jihτ |)ehi (τ ) dτ kDf kL1 (Jih )i≤2RhRmax≤ max1≤i≤nRJihE(|t − τ |) dτnPi=0kDf kL1 (Jih ) ≤E(t) dt kDf kL1 (J) ≤ 2kDf kL1 (J) hmax E(hmax )(1 + o(1)).0Íåðàâåíñòâî (2.1.13) äîêàçàíî.42Ïîëüçóÿñü îöåíêîé (2.1.12) è ôîðìóëîé (2.1.2), èìååìkΛ(I − σ h )Λf kL1 (J)→C(J) ≤ kΛkL2 (J)→C(J) k(I − σ h )Λf kL2 (J) ≤√≤ 2M2 kDΛf kL2 (J) hmax ,kΛ(I − σ h )f kLq (J) ≤ kΛkLp (J)→Lq (J) k(I − σ h )f kLp (J)→Lp (J) ≤ 2Ms kDf kLp (J) hmax ,k(I − σ h )Λf kLq (J) ≤ 2kDΛf kLq (J) hmax .2.2ÎïåðàòîðïðîåêòèðîâàíèÿπbhèíåêîòîðûååãîñâîéñòâàÎïðåäåëèì îïåðàòîð πbh : Lp (J) → Sbh (J), 1 ≤ p ≤ ∞, ñòàâÿùèé âñîîòâåòñòâèå ôóíêöèè f ∈ Lp (J) åå ïðîåêöèþ (bπ h f )(τ ) =nPj=0πbh f (τj )ehj (τ ) íàïðîñòðàíñòâî S (J), êîòîðàÿ îïðåäåëÿåòñÿ êàê ðåøåíèå ñèñòåìû óðàâíåíèébh(bπ h f, ehi ) = (f, ehi ),0 ≤ i ≤ n.Çíà÷åíèÿ πbh f (τj ), 0 ≤ j ≤ n ìîãóò áûòü îäíîçíà÷íî íàéäåíû èç ñèñòåìûëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèénX(ehj , ehi )bπ h f (τj ) = (f, ehi ),0≤i≤n(2.2.1)j=0ñ ñèììåòðè÷íîé òðåõäèàãîíàëüíîé ìàòðèöåé, îáëàäàþùåé ñâîéñòâîì äèàãîíàëüíîãî ïðåîáëàäàíèÿ.