On some extremal problems of different metrics for differentiable functions on the axis

For an arbitrary fixed segment $[α, β] ⊂ R$ and given $r ∈ N, A_r, A_0$, and $p > 0$, we solve the extremal problem $$∫^{β}_{α} \left|x^{(k)}(t)\right|^qdt → \sup,\; q⩾p,\; k=0,\; q⩾1,\; 1 ⩽ k ⩽ r−1,$$ on the set of all functions $x ∈ L^r_{∞}$ such that $∥x (r)∥_{∞} ≤ A_r$ and $L(x)_p ≤ A_0$...

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Datum:	2009
Hauptverfasser:	Kofanov, V. A., Кофанов, В. А.
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Sprache:	Russisch Englisch
Veröffentlicht:	Institute of Mathematics, NAS of Ukraine 2009
Online Zugang:	https://umj.imath.kiev.ua/index.php/umj/article/view/3057
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Назва журналу:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Kofanov, V. A. Кофанов, В. А. Кофанов, В. А.
author_facet	Kofanov, V. A. Кофанов, В. А. Кофанов, В. А.
author_sort	Kofanov, V. A.
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
collection	OJS
datestamp_date	2020-03-18T19:44:24Z
description	For an arbitrary fixed segment $[α, β] ⊂ R$ and given $r ∈ N, A_r, A_0$, and $p > 0$, we solve the extremal problem $$∫^{β}_{α} \left\|x^{(k)}(t)\right\|^qdt → \sup,\; q⩾p,\; k=0,\; q⩾1,\; 1 ⩽ k ⩽ r−1,$$ on the set of all functions $x ∈ L^r_{∞}$ such that $∥x (r)∥_{∞} ≤ A_r$ and $L(x)_p ≤ A_0$, where $$L(x)p := \left\{\left( ∫^b_a \|x(t)\|^p dt\right)^{1/ p} : a,b ∈ R,\; \|x(t)\| > 0,\; t ∈ (a,b)\right\}$$ In the case where $p = ∞$ and $k ≥ 1$, this problem was solved earlier by Bojanov and Naidenov.
first_indexed	2026-03-24T02:35:24Z
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fulltext	UDK 517. 5 V. A. Kofanov (Dnepropetr. nac. un-t) O NEKOTORÁX ∏KSTREMAL\|NÁX ZADAÇAX RAZNÁX METRYK DLQ DYFFERENCYRUEMÁX FUNKCYJ NA OSY For any fixed interval α β,[ ] � R, given r ∈N and Ar , A0 , p > 0, we solve the extremal problem α β ∫ x t dtk q( ) ( ) → sup, q ≥ p, k = 0, q ≥ 1, 1 ≤ k ≤ r – 1, on the set of all functions x Lr∈ ∞ such that x r( ) ∞ ≤ Ar , L x p ( ) ≤ A0 , where L x p ( ) : = a b p p x t dt a b x t t a b∫        ∈ > ∈( ) : , , ( ) , ( , ) /1 0R      . For the case of p = ∞, k ≥ 1, this problem was earlier solved by B. Bojanov and N. Naidenov. Dlq dovil\noho fiksovanoho vidrizka α β,[ ] � R ta zadanyx r ∈N , Ar , A0 , p > 0 rozv’qzano ekstremal\nu zadaçu α β ∫ x t dtk q( ) ( ) → sup, q ≥ p, k = 0, q ≥ 1, 1 ≤ k ≤ r – 1, na mnoΩyni vsix funkcij x Lr∈ ∞ takyx, wo x r( ) ∞ ≤ Ar , L x p ( ) ≤ A0 , de L x p ( ) : = a b p p x t dt a b x t t a b∫        ∈ > ∈( ) : , , ( ) , ( , ) /1 0R      . U vypadku p = ∞, k ≥ 1 cq zadaça bula rozv’qzana raniße B. Boqnovym i N. Najd\onovym. 1. Vvedenye. Pust\ G oboznaçaet dejstvytel\nug os\ R yly koneçn¥j otre- zok α β,[ ] � R. Budem rassmatryvat\ prostranstva L Gp( ), 0 < p ≤ ∞, vsex yz- merym¥x funkcyj x : G → R takyx, çto x L Gp ( ) < ∞, hde x L Gp ( ) : = G p p t G x t dt p x t ∫     < < ∞ ∈ ( ) , , sup ( ) /1 esly 0 vrai ,, .esly p = ∞       Dlq r ∈N y p, s ∈ 0, ∞( ] oboznaçym çerez Lp s r , prostranstvo vsex funkcyj x : R → R, dlq kotor¥x x r( – )1 lokal\no absolgtno neprer¥vna, x Lp∈ ( )R y x r( ) ∈ Ls( )R . Budem pysat\ x p vmesto x Lp ( )R y Lr ∞ vmesto Lr ∞ ∞, . V nastoqwej stat\e yzuçagtsq nekotor¥e modyfykacyy yzvestnoj πkstre- mal\noj zadaçy x k q ( ) → sup, 0 ≤ k ≤ r – 1, q ≥ 1, (1) na mnoΩestve vsex funkcyj x Lp s r∈ , takyx, çto x r s ( ) ≤ Ar , x p ≤ A0 . (2) © V. A. KOFANOV, 2009 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 765 766 V. A. KOFANOV Yzvestno (sm., naprymer, [1, s. 47]), çto zadaça (1) πkvyvalentna naxoΩdenyg toçnoj konstant¥ C v neravenstve typa Kolmohorova – Nadq x k q ( ) ≤ C x xp r s α α( ) 1− , 0 ≤ k ≤ r – 1, (3) dlq funkcyj x Lp s r∈ , , hde α = (r – k + 1 / q – 1 / s) / (r + 1 / p – 1 / s). Lyß\ v neskol\kyx sluçaqx toçnaq konstanta v neravenstve (3) yzvestna dlq vsex r ∈N . Podrobnoe opysanye sluçaev, dlq kotor¥x yzvestna toçnaq kon- stanta v neravenstve (3), moΩno najty v rabotax [1 – 3]. Dlq proyzvol\noho otrezka α β,[ ] � R B. Boqnov¥m y N. Najdenov¥m [4] re- ßena zadaça α β ∫ ( )Φ x t dtk( ) ( ) → sup, 1 ≤ k ≤ r – 1, na mnoΩestve vsex funkcyj x Lr∈ ∞, udovletvorqgwyx (2) s p = s = ∞, hde Φ — neprer¥vno dyfferencyruemaq funkcyq na 0, ∞[ ) , poloΩytel\naq na (0, ∞) takaq, çto Φ( )t / t ne ub¥vaet y Φ( )0 = 0. Budem rassmatryvat\ klass W neprer¥vn¥x, neotrycatel\n¥x y v¥pukl¥x funkcyj Φ na 0, ∞[ ) takyx, çto Φ( )0 = 0. Dlq p > 0 poloΩym L x p( ) : = sup : , , ( ) , ( , ),x a b x t t a bL a bp[ ] ∈ > ∈{ }R 0 . Funkcyonal¥ takoho typa yzuçalys\ v rabote [5]. Otmetym, çto L x( )∞ = x ∞ y L x( )′ 1 ≤ 2 x ∞ . V nastoqwej rabote reßen¥ sledugwye modyfykacyy zadaçy B. Boqnova y N. Najdenova: α β ∫ ( )Φ x t dtp( ) → sup, Φ ∈W , p > 0, y α β ∫ ( )Φ x t dtk( ) ( ) → sup, Φ ∈W , 1 ≤ k ≤ r – 1, na klasse vsex funkcyj x Lr∈ ∞, udovletvorqgwyx uslovyqm x r( ) ∞ ≤ Ar , L x p( ) ≤ A0 vmesto uslovyj (2) s s = p = ∞. 2. Vspomohatel\n¥e utverΩdenyq. Symvolom ϕr t( ) , r ∈N , oboznaçym r- j 2π -peryodyçeskyj yntehral s nulev¥m srednym znaçenyem na peryode ot funkcyy ϕ0( )t = sgn sin t. Dlq λ > 0 poloΩym ϕλ, ( )r t : = λ−r ϕ λr t( ) . Peresta- novku funkcyy x , x ∈ L a b1 ,[ ], budem oboznaçat\ symvolom r (x, t) (sm., na- prymer, [6], § 1.3). PoloΩym r (x, t) = 0 dlq t ≥ b – a. Zametym, çto esly funkcyq x Lr∈ ∞ udovletvorqet uslovyg L x p( ) < ∞ dlq nekotoroho p > 0 y x t( ) > 0, t ∈ (a, b), pryçem a = – ∞ yly b = + ∞, to x t( ) → 0, esly t → – ∞ yly t → + ∞. V πtom sluçae budem polahat\ x( )−∞ = 0 y x( )+∞ = 0. ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 O NEKOTORÁX ∏KSTREMAL\|NÁX ZADAÇAX RAZNÁX METRYK … 767 Lemma 1. Pust\ r ∈N , Ar , p > 0, a ynterval (a, b), – ∞ ≤ a < b ≤ ∞, y funkcyq x Lr∈ ∞ takov¥, çto x r( ) ∞ ≤ Ar , L x p( ) ≤ ∞, x a( ) = x b( ) = 0, x t( ) > 0, t ∈ (a, b). Pust\ takΩe λ > 0 udovletvorqet uslovyg L x p( ) ≤ A Lr r p( ),ϕλ . (4) Tohda dlq lgboj funkcyy Φ ∈W y proyzvol\noho yzmerymoho mnoΩestva E � (a, b), µE ≤ π / λ, v¥polnen¥ neravenstva a b px t dt∫ ( )Φ ( ) ≤ 0 π λ λϕ / , ( )∫ ( )Φ A t dtr r p (5) y E px t dt∫ ( )Φ ( ) ≤ m m r r p A t dt − + ∫ ( ) Θ Θ Φ ϕλ, ( ) , Θ = µE 2 , (6) hde m — toçka lokal\noho maksymuma splajna ϕλ,r . Krome toho, esly – ∞ < a < b < ∞, to 1 b a x t dt a b p − ( )∫Φ ( ) ≤ λ π ϕ π λ λ 0 / , ( )∫ ( )Φ A t dtr r p . (7) Dokazatel\stvo. Zafyksyruem proyzvol\nug funkcyg x Lr∈ ∞ y ynter- val (a, b), udovletvorqgwye uslovyqm lemm¥ 1. Snaçala dokaΩem neraven- stvo x ∞ ≤ Ar rϕλ, ∞ . (8) PredpoloΩym, çto (8) ne v¥polneno. Tohda suwestvuet ω < λ takoe, çto x ∞ = Ar rϕω, ∞ . (9) Pust\ t0 ∈R udovletvorqet uslovyg ϕω,r ∞ = ϕω, ( )r t0 (10) y c — naybol\ßyj nul\ splajna ϕω,r takoj, çto c < t0 . Zafyksyruem proyz- vol\noe ε > 0. Najdetsq toçka tε ∈ ( , )c t0 , dlq kotoroj ϕω ε, ( )r t = ϕω,r ∞ – ε. PoloΩym δ : = t0 – tε . Qsno, çto δ → 0 pry ε → 0. Dlq dostatoçno maloho ε > > 0 opredelym funkcyg ψε( )t na c[ , c + π ω/ ] sledugwym obrazom: ψε( )t : = ϕ δ δ ϕ δ π ω δ δ π ω δ π ω ω ω , , ( ), , , ( ), , / , , , / , / . r r t t c t t t t c t c c c c − ∈ +[ ] + ∈ + −[ ] ∈ +[ ] + − +[ ]        esly esly esly 0 0 0 ∪ ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 768 V. A. KOFANOV Oçevydno, çto ψε( )t0 = ϕω,r ∞ – ε y ψε( )t → ϕω, ( )r t , t ∈ c[ , c + π ω/ ], pry ε → 0. Poskol\ku L x p( ) < ∞, yz (9) y (10) sleduet suwestvovanye takoho sdvy- ha x tε( ) : = x (t + τε ), çto ′x tε( )0 = 0 y x tε( )0 ≥ Ar rϕ εω, ∞ −( ) = A tr ψε( )0 . (11) Zametym, çto funkcyq x udovletvorqet v sylu (9) uslovyqm teorem¥ srav- nenyq Kolmohorova [7]. Sohlasno πtoj teoreme yz (11) sleduet neravenstvo x tε( ) ≥ A tr ψε( ), t ∈ c c+ + −[ ]δ π ω δ, / . Znaçyt, L x p( ) = L x p( )ε ≥ Ar L c cp ψε δ π ω δ+ + −[ ], / . Ustremlqq ε k nulg, poluçaem L x p( ) ≥ A Lr r p( ),ϕω > A Lr r p( ),ϕλ , çto protyvoreçyt uslovyg (4). Tem sam¥m neravenstvo (8) dokazano. DokaΩem teper\ neravenstvo (5). Oboznaçym çerez x suΩenye funkcyy x na (a, b), a çerez ϕ suΩenye splajna Ar ϕλ,r na c[ , c + π λ/ ], hde c — nul\ splajna ϕλ,r . V sylu teorem¥ Xardy – Lyttlvuda (sm., naprymer, [6], utverΩdenye 1.3.11) dlq dokazatel\stva (5) dostatoçno pokazat\, çto 0 ξ ∫ ( )r x t dtp, ≤ 0 ξ λϕ∫ ( )r t dtr p , , , ξ > 0. (12) Na osnovanyy (8) y uslovyq x(a) = x(b) = 0 lemm¥O1 dlq lgboho z ∈ 0( , x L a b∞ )( , ) suwestvugt toçky ti ∈ (a, b), i = 1, … , m, m ≥ 2, y dve toçky yj ∈ ∈ (c , c + π λ/ ) takye, çto z = x ti( ) = ϕ ( )yj . (13) Po teoreme sravnenyq Kolmohorova ′x ti( ) ≤ ′ϕ ( )yj . (14) Poπtomu esly toçky θ1 y θ2 v¥bran¥ tak, çto z = r x( , )θ1 = r( , )ϕ θ2 , to sohlasno teoreme o proyzvodnoj perestanovky (sm., naprymer, [6], predlo- Ωenye 1.3.2) ′r x( , )θ1 = i m ix t = − − ∑ ′        1 1 1 ( ) ≤ j jy = − − ∑ ′        1 2 1 1 ϕ ( ) = ′r ( , )ϕ θ2 . Otsgda sleduet, çto raznost\ ∆( )t : = r x t( , ) – r t( , )ϕ menqet znak ne bolee od- noho raza (s – na + ). To Ωe samoe verno y dlq raznosty ∆ p t( ) : = r x tp( , ) – – r tp( , )ϕ . Rassmotrym yntehral ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 O NEKOTORÁX ∏KSTREMAL\|NÁX ZADAÇAX RAZNÁX METRYK … 769 I( )ξ : = 0 ξ ∫ ∆ p t dt( ) . Qsno, çto I( )0 = 0. PoloΩym M : = max b{ – a, π λ/ } . Tohda vsledstvye (4) dlq lgboho ξ ≥ M I( )ξ = L x p p( ) – A Lr p r p p( ),ϕλ ≤ 0. Krome toho, proyzvodnaq ′I t( ) = ∆ p t( ) menqet znak ne bolee odnoho raza (s – na +O). Sledovatel\no, I( )ξ ≤ 0 dlq vsex ξ ≥ 0. Takym obrazom, neravenstva (12) y (5) dokazan¥. DokaΩem teper\ (6). Zametym, çto ϕ qvlqetsq funkcyej sravnenyq dlq funkcyy x , t.Oe. yz (13) sleduet (14). Dokazatel\stvo (5) b¥lo osnovano ymenno na πtom fakte y na neravenstve (4). Poπtomu, yspol\zuq (5) vmesto (4), takymy Ωe rassuΩdenyqmy moΩno dokazat\ neravenstvo 0 ξ ∫ ( )r x t dtpΦ( , ) ≤ 0 ξ ϕ∫ ( )r t dtpΦ( , ) , ξ > 0. Otsgda sleduet ocenka E px t dt∫ ( )Φ ( ) ≤ 0 µE pr x t dt∫ ( )Φ( , ) ≤ 0 µ ϕ E pr t dt∫ ( )Φ( , ) = = m m r r p A t dt − + ∫ ( ) Θ Θ Φ ϕλ, ( ) , çto y dokaz¥vaet (6). Ostalos\ dokazat\ (7). Pust\ – ∞ < a < b < + ∞. V¥berem d ∈ (a, b) tak, çto a d px t dt∫ ( )Φ ( ) = d b px t dt∫ ( )Φ ( ) : = I. Tohda vsledstvye (5) suwestvuet y ∈ [0, π / (2λ)], dlq kotoroho I = c c y r r p A t dt + ∫ ( )Φ ϕλ, ( ) , hde c — nul\ splajna ϕλ,r . Otsgda v sylu teorem¥ sravnenyq Kolmohorova v¥tekagt neravenstva d – a ≥ y, b – d ≥ y. Sledovatel\no, a b px t dt∫ ( )Φ ( ) = a d px t dt∫ ( )Φ ( ) + d b px t dt∫ ( )Φ ( ) ≤ ≤ d a y A t dt c c y r r p− ( ) + ∫ Φ ϕλ, ( ) + b d y A t dt c c y r r p− ( ) + ∫ Φ ϕλ, ( ) = = ( ) ( ),b a y A t dt c c y r r p − ( ) + ∫1 Φ ϕλ . Netrudno vydet\, çto funkcyq ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 770 V. A. KOFANOV 1 y A t dt c c y r r p + ∫ ( )Φ ϕλ, ( ) ne ub¥vaet na [0, π / (2λ)]. Poπtomu a b px t dt∫ ( )Φ ( ) ≤ ( ) ( ) /( ) ,b a A t dt c c r r p − ( ) + ∫2 2 λ π ϕ π λ λΦ = = ( ) ( ) / ,b a A t dt c c r r p − ( ) + ∫λ π ϕ π λ λΦ , çto πkvyvalentno (7). Lemma 1 dokazana. Polahaq Φ( )t = tq p/ , q ≥ p, poluçaem takoe sledstvye. Sledstvye 1. V uslovyqx lemm¥ 1 L x q( ) ≤ A Lr r q( ),ϕλ , q ≥ p. V çastnosty, x ∞ ≤ Ar rϕλ, ∞ . Lemma 2. Pust\ r ∈N , Ar , p > 0. PredpoloΩym, çto funkcyq x Lr∈ ∞ ymeet nuly y udovletvorqet uslovyg x r( ) ∞ ≤ Ar , L x p( ) < ∞, a λ > 0 v¥brano tak, çto L x p( ) ≤ A Lr r p( ),ϕλ . Esly t0 — nul\ funkcyy x, a c — nul\ splajna ϕλ,r , to dlq proyzvol\noj funkcyy Φ ∈W y lgboho ξ ∈ 0, /π λ( ] t t px t dt 0 0 + ∫ ( ) ξ Φ ( ) ≤ c c r r p A t dt + ∫ ( ) ξ λϕΦ , ( ) (15) y t t px t dt 0 0 − ∫ ( ) ξ Φ ( ) ≤ c c r r p A t dt − ∫ ( ) ξ λϕΦ , ( ) . Dokazatel\stvo. Perexodq k sdvyhu x(⋅ + τ), esly nuΩno, moΩno sçy- tat\, çto t0 = c. DokaΩem (15) (vtoroe neravenstvo lemm¥O2 dokaz¥vaetsq analohyçno). Po- loΩym ϕ( )t : = Ar ϕλ, ( )r t . Pry dokazatel\stve lemm¥O1 b¥lo ustanovleno, çto splajn ϕ qvlqetsq funkcyej sravnenyq dlq funkcyy x, t.Oe. yz x( )ξ = = ϕ η( ) sleduet ′x ( )ξ ≤ ′ϕ η( ) . Otsgda ymeem x t( ) ≤ ϕ( )t , t ∈ c( , c + + π / ( )2λ ). Esly poslednee neravenstvo v¥polneno dlq vsex t ∈ (c , c + ξ), to (15) oçevydno. Poπtomu moΩno predpoloΩyt\, çto raznost\ ∆( )t : = x t( ) – ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 O NEKOTORÁX ∏KSTREMAL\|NÁX ZADAÇAX RAZNÁX METRYK … 771 – ϕ( )t menqet znak na (c, c + ξ). Pry πtom ona ymeet ne bolee odnoj peremen¥ znaka (s – na +) na (c, c + π / λ), tak kak ϕ qvlqetsq funkcyej sravnenyq dlq x. Qsno, çto to Ωe samoe spravedlyvo dlq raznosty ∆Φ( )t : = Φ x t p( )( ) – – Φ ϕ( )t p( ). Pust\ toçka d ∈ (c, c + π / λ) takova, çto ∆( )t ≤ 0, t ∈ (c, d), y ∆( )t ≥ 0, t ∈ (d, c + π / λ). Tohda ∆Φ( )t ≤ 0, t ∈ (c, d), y ∆Φ( )t ≥ 0, t ∈ (d, c + π / λ). Rassmotrym dva sluçaq: 1) x t( ) > 0, t ∈ (c, c + ξ), 2) x (t) ymeet nul\ na (c, c + ξ). PoloΩym I tΦ( ) : = c c t u du + ∫ ∆Φ( ) . DokaΩem neravenstvo I tΦ( ) ≤ 0, t ∈ (0, π / λ), kotoroe πkvyvalentno (15). Snaçala predpoloΩym, çto x t( ) > 0, t ∈ (c, c + ξ). Sohlasno predpoloΩe- nyg d < c + ξ. Poπtomu x t( ) ≥ ϕ( )t > 0, t ∈ (d, c + π / λ), y, sledovatel\no, x t( ) > 0, t ∈ (c, c + π / λ). No tohda sohlasno neravenstvu (5) IΦ( / )π λ ≤ 0. Kro- me toho, IΦ( )0 = 0 y proyzvodnaq ′I tΦ( ) = ∆Φ( )c t+ menqet znak na (0, π / λ) ne bolee odnoho raza (s – na +). Takym obrazom, I tΦ( ) ≤ 0, t ∈ (0, π / λ). PredpoloΩym teper\, çto x t( ) ymeet nul\ na (c, c + ξ). PoloΩym c1 : = : = sup t{ ∈ (c , c + π / λ) : x t( ) = 0} . Qsno, çto x c( )1 = 0 y x t( ) ≤ ϕ( )t , t ∈ ( , )c c1 . Sledovatel\no, c px t dt γ ∫ ( )Φ ( ) ≤ c pt dt γ ϕ∫ ( )Φ ( ) , γ ∈[ ]c c, 1 . (16) Esly c + ξ ≤ c1, to (15) sleduet yz (16). Pust\ teper\ c1 < c + ξ. Tohda x t( ) > > 0, t ∈ (c1, c + π / λ). V πtom sluçae (15) uΩe dokazano. Poπtomu, polahaq t0 : = : = c1 y prymenqq (15) s c + ξ – c1 vmesto ξ, poluçaem c c px t dt 1 + ∫ ( ) ξ Φ ( ) ≤ c c c pt dt 2 1+ − ∫ ( ) ξ ϕΦ ( ) ≤ c c pt dt 1 + ∫ ( ) ξ ϕΦ ( ) . (17) Poslednee neravenstvo sleduet yz oçevydnoho sootnoßenyq inf ( ) ( , / )a c a a pt dt ∈ − + ∫ ( ) π λ δ δ ϕΦ = c c pt dt + ∫ ( ) δ ϕΦ ( ) , δ π λ ≤ . Sklad¥vaq (17) y (16) s γ = c1, poluçaem (15). LemmaO2 dokazana. Lemma 3. Pust\ r ∈N , Ar , p > 0 y funkcyq x Lr∈ ∞ takov¥, çto x r( ) ∞ ≤ Ar , L x p( ) < ∞, a λ > 0 udovletvorqet uslovyg L x p( ) ≤ A Lr r p( ),ϕλ . Tohda dlq lgboj funkcyy Φ ∈W y proyzvol\noho otrezka [a, b] � R , b – – a ≤ π / λ, v¥polneno neravenstvo a b px t dt∫ ( )Φ ( ) ≤ m m r r p A t dt − + ∫ ( ) Θ Θ Φ ϕλ, ( ) , Θ = b a− 2 , (18) ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 772 V. A. KOFANOV hde m — toçka lokal\noho maksymuma splajna ϕλ,r . V çastnosty, a b px t dt∫ ( )Φ ( ) ≤ 0 π λ λϕ / , ( )∫ ( )Φ A t dtr r p . Dokazatel\stvo. Esly x t( ) > 0 dlq t ∈ (a, b), to (18) sleduet yz nera- venstva (6). Poπtomu predpoloΩym, çto x t( ) ymeet nul\ t0 ∈ (a, b). Tohda so- hlasno lemme 2 a t px t dt 0 ∫ ( )Φ ( ) ≤ c t a c r r p A t dt + − − + ∫ ( ) π λ π λ λϕ / ( ) / , ( ) 0 Φ (19) y t b px t dt 0 ∫ ( )Φ ( ) ≤ c c b t r r p A t dt + − ∫ ( )0 Φ ϕλ, ( ) , (20) hde c — nul\ splajna ϕλ,r . Sklad¥vaq (19) y (20), poluçaem (18), tak kak sup ( ), µ δ λϕ E E r r p A t dt = ∫ ( )Φ = m m r r p A t dt − + ∫ ( ) δ δ λϕ / / , ( ) 2 2 Φ , δ π λ ≤ . Lemma 3 dokazana. 3. Osnovn¥e rezul\tat¥. Zafyksyruem proyzvol\n¥j otrezok α β,[ ] � R , r ∈N y Ar , A0 , p > 0. Napomnym konstrukcyg πkstremal\noj funkcyy ϕ α β, ,[ ] r v zadaçe B. Boqnova y N. Najdenova [4]. Dlq πtoho snaçala v¥berem λ > > 0, udovletvorqgwee ravenstvu A0 = A Lr r p( ),ϕλ , (21) zatem predstavym dlynu otrezka α β,[ ] v vyde β – α = n π λ + 2Θ, Θ ∈( )0 2, /( )π λ , (22) hde n ∈N yly n = 0. Teper\ poloΩym ϕ α β, , ( )[ ] r t : = A tr rϕ τλ, ( )+ , (23) hde τ v¥brano tak, çto ϕ αα β, , ( )[ ] +r Θ = ϕ βα β, , ( )[ ] −r Θ = Ar rϕλ, ∞ . Qsno, çto ϕ α β, ,[ ] r ∈ Lr ∞ y ϕ α β, , ( ) [ ] ∞r r = Ar , L r p ϕ α β, ,[ ]( ) = A0 . Teorema 1. Pust\ r ∈N , A0 , Ar , p > 0, Φ ∈W , α β,[ ] � R. Tohda sup ( ) : , , ( )( ) α β ∫ ( ) ∈ ≤ ≤    ∞ ∞ Φ x t dt x L x A L x Ap r r r p 0      = α β α βϕ∫ [ ]( )Φ , , ( )r p t dt . Dokazatel\stvo. Zafyksyruem proyzvol\nug funkcyg x Lr∈ ∞ takug, çto ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 O NEKOTORÁX ∏KSTREMAL\|NÁX ZADAÇAX RAZNÁX METRYK … 773 x r( ) ∞ ≤ Ar , L x p( ) ≤ A0 . Sohlasno (21) L x p( ) ≤ A Lr r p( ),ϕλ . PoloΩym ak : = α + k π / λ, k = 0, 1, … , n. Po lemmeO3 a a p k k x t dt + ∫ ( ) 1 Φ ( ) ≤ 0 π λ λϕ / , ( )∫ ( )Φ A t dtr r p , k = 0, 1, … , n – 1, y a p n x t dt β ∫ ( )Φ ( ) ≤ m m r r p A t dt − + ∫ ( ) Θ Θ Φ ϕλ, ( ) , hde m — toçka lokal\noho maksymuma splajna ϕλ,r , a Θ opredeleno v (22). Takym obrazom, α β ∫ ( )Φ x t dtp( ) ≤ n A t dtr r p 0 π λ λϕ / , ( )∫ ( )Φ + m m r r p A t dt − + ∫ ( ) Θ Θ Φ ϕλ, ( ) = = α β α βϕ∫ [ ]( )Φ , , ( )r p t dt . Ravenstvo zdes\ dostyhaetsq dlq x = ϕ α β, ,[ ] r . Teorema 1 dokazana. Pust\ q ≥ p. Polahaq Φ( )t = tq p/ , poluçaem takoe sledstvye. Sledstvye 2. V uslovyqx teorem¥ 1 dlq lgboho q ≥ p > 0 sup ( ) : , , ( )( ) α β ∫ ∈ ≤ ≤      ∞ ∞ x t dt x L x A L x Aq r r r p 0   = α β α βϕ∫ [ ], , ( )r q t dt . Dlq [α, β] � R, k, r ∈N , k < r, rassmotrym funkcyg ϕ α β, , , ( )[ ] r k t : = ϕ τα β, , ( )[ ] +r kt , τk : = π λ4 1 1 1+ −( )+( )k , hde ϕ α β, ,[ ] r opredelena ravenstvom (23). Qsno, çto ϕ α β, , , ( ) ( )[ ] r k k t = ϕ α β, , ( )[ ] −r k t . Krome toho, ϕ α β, , ,[ ] r k ∈ Lr ∞ y ϕ α β, , , ( ) [ ] ∞r k r = Ar , L r k p ϕ α β, , ,[ ]( ) = A 0 . Teorema 2. Pust\ k, r ∈N , k < r, A 0 , Ar , p > 0, Φ ∈W , α β,[ ] � R. Tohda sup ( ) : , , ( )( ) ( ) α β ∫ ( ) ∈ ≤ ≤  ∞ ∞ Φ x t dt x L x A L x Ak r r r p 0       = = α β α βϕ∫ [ ]( )Φ , , , ( ) ( )r k k t dt . ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 774 V. A. KOFANOV Dokazatel\stvo. Zafyksyruem proyzvol\nug funkcyg x Lr∈ ∞ takug, çto x r( ) ∞ ≤ Ar , L x p( ) ≤ A 0 . Sohlasno (21) L x p( ) ≤ A Lr r p( ),ϕλ y na osnovanyy sledstvyq 1 x ∞ ≤ Ar rϕλ, ∞ . Otsgda v sylu teorem¥ Kolmohorova sleduet neravenstvo x i( ) ∞ ≤ Ar r iϕλ, − ∞ , i = 1, … , r – 1. Sledovatel\no, L x k( )( )1 ≤ 2 1x k( )− ∞ ≤ 2 1Ar r kϕλ, + − ∞ = A Lr r k( ),ϕλ − 1. Poπtomu, prymenqq teoremuO1 s p = 1 k funkcyy x k( ) ∈ Lr k ∞ − , poluçaem α β ∫ ( )Φ x t dtk p( )( ) ≤ α β α βϕ∫ [ ] −( )Φ , , ( )r k p t dt = α β α βϕ∫ [ ]( )Φ , , , ( ) ( )r k k p t dt . Ravenstvo zdes\ dostyhaetsq dlq funkcyy x = ϕ α β, , ,[ ] r k . Zameçanye 1. V sluçae p = ∞ teoremaO2 b¥la dokazana B. Boqnov¥m y N.ONajdenov¥m [4]. Sledstvye 3. V uslovyqx teorem¥ 2 dlq lgb¥x q ≥ 1 y p > 0 sup ( ) : , , ( )( ) ( ) α β ∫ ∈ ≤ ≤    ∞ ∞ x t dt x L x A L x Ak q r r r p 0      = α β α βϕ∫ [ ], , , ( ) ( )r k k q t dt . Sledugwaq teorema utoçnqet teorem¥ 1 y 2 dlq funkcyj, ymegwyx nuly, y dlq peryodyçeskyx funkcyj. Teorema 3. Pust\ r ∈N , p > 0 , Φ ∈W . Tohda dlq lgboho otrezka [α, β] � R y proyzvol\noj funkcyy x Lr∈ ∞ takoj , çto L x p( ) < ∞ y x( )α = = x( )β = 0, v¥polneno neravenstvo 1 β α α β − ( )∫ Φ x t dtp( ) ≤ 1 0 1 1 1 π ϕ ϕ π ∫             + ∞ +Φ L x L x t dtp r p r r p r p r p r p ( ) ( ) ( ) / ( ) / / . (24) V çastnosty, dlq lgboho q ≥ p 1 1 β α α β −        ∫ x t dtq q ( ) / ≤ 1 0 1 π ϕ ϕ π ∫            r q q p r p r r t dt L x L ( ) ( ) ( ) / ++ ∞ +1 1 1 / ( ) / / p r p r px .(25) Krome toho, esly q ≥ 1 y k = 1, … , r – 1, to dlq proyzvol\noho otrezka [a , b] � R y lgboj funkcyy x Lr∈ ∞, udovletvorqgwej uslovyg x ak( )( ) = = x bk( )( ) = 0, v¥polnqetsq neravenstvo ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 O NEKOTORÁX ∏KSTREMAL\|NÁX ZADAÇAX RAZNÁX METRYK … 775 1 1 β α α β −        ∫ x t dtk q q ( ) / ( ) ≤ 1 0 1 π ϕ ϕ π ∫ − ∞ ∞ −            r k q q r r k r t dt x x( ) / (rr k r) / ∞ . (26) Dokazatel\stvo. Zafyksyruem proyzvol\n¥j otrezok [ α, β ] � R y funkcyg x Lr∈ ∞ takug, çto L x p( ) < ∞, x( )α = x( )β = 0. PoloΩym Ar : = : = x r( ) ∞ y v¥berem λ > 0, udovletvorqgwee uslovyg L x p( ) = A Lr r p( ),ϕλ = A Lr r p r pλ ϕ− −1/ ( ) , t.OOe. λ−1 = L x A L p r r p r p( ) ( ) / ϕ     + 1 1 . (27) Rassmotrym mnoΩestvo vsex otrezkov a bj j,[ ] � [α, β] takyx, çto x aj( ) = x bj( ) = 0, x t( ) > 0, t a bj j∈( , ) . Qsno, çto α β ∫ ( )Φ x t dtp( ) = j a b p j j x t dt∑ ∫ ( )Φ ( ) y j j jb a∑ −( ) ≤ β – α. Zametym, çto funkcyq x na kaΩdom yz otrezkov ( , )a bj j udovletvorqet vsem uslovyqm lemm¥O1. Poπtomu, ocenyvaq yntehral¥ a b p j j x t dt∫ ( )Φ ( ) s pomow\g neravenstva (7) y prynymaq vo vnymanye opredelenye ϕλ, ( )r t : = λ−r ϕ λr t( ), po- luçaem α β ∫ ( )Φ x t dtp( ) ≤ j j j r r p b a A t dt∑ ∫− ( )( ) ( ) / , λ π ϕ π λ λ 0 Φ ≤ ≤ ( ) ( ) / ,β α λ π ϕ π λ λ− ( )∫ 0 Φ A t dtr r p = ( ) ( )β α π λ ϕ π − ( )∫ −1 0 Φ A s dsr r r p . Otsgda sleduet (24), esly uçest\, çto Ar : = x r( ) ∞ , a λ opredeleno ravenst- vom (27). Polahaq Φ( )t = tq p/ v (24), poluçaem (25). Ostalos\ dokazat\ (26). Zafyksyruem proyzvol\noe k = 1, … , r – 1, otrezok [a, b] � R y funkcyg x Lr∈ ∞, udovletvorqgwug uslovyg x ak( )( ) = x bk( )( ) = = 0. Prymenqq neravenstvo (25) s p = 1 k funkcyy x Lk r k( ) ∈ ∞ − , dlq q ≥ 1 ymeem 1 1 β α α β −      ∫ x t dtk q q ( ) / ( ) ≤ 1 0 1 1 1 1 1 1 π ϕ ϕ π ∫ − − − − + ∞ − +      ( )     r k q q k r k r k r k r r kt dt L x L x( ) ( ) / ( ) ( ) . (28) ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6 776 V. A. KOFANOV Uçyt¥vaq oçevydn¥e sootnoßenyq L x k( )( )1 ≤ 2 1x k( )− ∞ , L r k( )ϕ − 1 = 2 1ϕr k− + ∞ y ocenyvaq x k( )− ∞ 1 (v sluçae k > 1) s pomow\g neravenstva Kolmohorova x k( )− ∞ 1 ≤ ϕ ϕr k r r k r r k rx x− + ∞ ∞ ∞ − + ∞ −    1 1 1 ( ) , yz (28) poluçaem (26). TeoremaO3 dokazana. Zameçanye 2. Neravenstvo (26) s b – a = 2π dlq 2π-peryodyçeskoj funk- cyy x Lr∈ ∞ transformyruetsq v yzvestnoe neravenstvo Lyhuna [8] x k Lq ( ) ,0 2π[ ] ≤ ϕ ϕπr k L r r k r r k r q x x− [ ] ∞ ∞ − ∞    0 2, ( ) / . 1. Kornejçuk N. P., Babenko V. F., Kofanov V. A., Pyçuhov S. A. Neravenstva dlq proyzvodn¥x y yx pryloΩenyq. – Kyev: Nauk. dumka, 2003. – 590 s. 2. Babenko V. F. Yssledovanyq dnepropetrovskyx matematykov po neravenstvam dlq proyz- vodn¥x peryodyçeskyx funkcyj y yx pryloΩenyqm // Ukr. mat. Ωurn. – 2000. – 52, # 1. – S.O5 – 29. 3. Kwong M. K., Zettl A. Norm inequalities for derivatives and differences // Lect. Notes Math. – Berlin: Springer, 1992. – 1536. – 150 p. 4. Bojanov B., Naidenov N. An extension of the Landau – Kolmogorov inequality. Solution of a prob- lem of Erdos // J. Anal. Math. – 1999. – 78. – P. 263 – 280. 5. Pinkus A., Shisha O. Variations on the Chebyshev and Lq theories of best approximation // J. Approxim. Theory. – 1982. – 35, # 2. – P. 148 – 168. 6. Kornejçuk N. P., Babenko V. F., Lyhun A. A. ∏kstremal\n¥e svojstva polynomov y splajnov. – Kyev: Nauk. dumka, 1992. – 304 s. 7. Kolmohorov A. N. O neravenstvax meΩdu verxnymy hranqmy posledovatel\n¥x proyzvod- n¥x funkcyy na beskoneçnom yntervale // Yzbr. trud¥. Matematyka, mexanyka. – M.: Nauka, 1985. – S. 252 – 263. 8. Ligun A. A. Inequalities for upper bounds of functionals // Anal. Math. – 1976. – 2 , # 1. – P. 11 – 40. Poluçeno 22.09.08 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6
id	umjimathkievua-article-3057
institution	Ukrains’kyi Matematychnyi Zhurnal
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language	rus English
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publishDate	2009
publisher	Institute of Mathematics, NAS of Ukraine
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spelling	umjimathkievua-article-30572020-03-18T19:44:24Z On some extremal problems of different metrics for differentiable functions on the axis O некоторых экстремальных задачах разных метрик для дифференцируемых функций на оси Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. For an arbitrary fixed segment $[α, β] ⊂ R$ and given $r ∈ N, A_r, A_0$, and $p > 0$, we solve the extremal problem $$∫^{β}_{α} \left\|x^{(k)}(t)\right\|^qdt → \sup,\; q⩾p,\; k=0,\; q⩾1,\; 1 ⩽ k ⩽ r−1,$$ on the set of all functions $x ∈ L^r_{∞}$ such that $∥x (r)∥_{∞} ≤ A_r$ and $L(x)_p ≤ A_0$, where $$L(x)p := \left\{\left( ∫^b_a \|x(t)\|^p dt\right)^{1/ p} : a,b ∈ R,\; \|x(t)\| > 0,\; t ∈ (a,b)\right\}$$ In the case where $p = ∞$ and $k ≥ 1$, this problem was solved earlier by Bojanov and Naidenov. Для довільного фіксованого відрізка $[α, β] ⊂ R$ та заданих $r ∈ N, A_r, A_0$, $p > 0$ розв'язано екстремальну задачу $$∫^{β}_{α} \left\|x^{(k)}(t)\right\|^qdt → \sup,\; q⩾p,\; k=0,\; q⩾1,\; 1 ⩽ k ⩽ r−1,$$ на множині всіх функцій $x ∈ L^r_{∞}$ таких, що $∥x (r)∥_{∞} ≤ A_r$, $L(x)_p ≤ A_0$, де $$L(x)p := \left\{\left( ∫^b_a \|x(t)\|^p dt\right)^{1/ p} : a,b ∈ R,\; \|x(t)\| > 0,\; t ∈ (a,b)\right\}$$ У випадку $p = ∞$, $k ≥ 1$ ця задача була розв'язана раніше В. Вояновим і Н. Найдьоновим. Institute of Mathematics, NAS of Ukraine 2009-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3057 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 6 (2009); 765-776 Український математичний журнал; Том 61 № 6 (2009); 765-776 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3057/2863 https://umj.imath.kiev.ua/index.php/umj/article/view/3057/2864 Copyright (c) 2009 Kofanov V. A.
spellingShingle	Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. On some extremal problems of different metrics for differentiable functions on the axis
title	On some extremal problems of different metrics for differentiable functions on the axis
title_alt	O некоторых экстремальных задачах разных метрик для дифференцируемых функций на оси
title_full	On some extremal problems of different metrics for differentiable functions on the axis
title_fullStr	On some extremal problems of different metrics for differentiable functions on the axis
title_full_unstemmed	On some extremal problems of different metrics for differentiable functions on the axis
title_short	On some extremal problems of different metrics for differentiable functions on the axis
title_sort	on some extremal problems of different metrics for differentiable functions on the axis
url	https://umj.imath.kiev.ua/index.php/umj/article/view/3057
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On some extremal problems of different metrics for differentiable functions on the axis

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