On one extremal problem for numerical series

Let $Γ$ be the set of all permutations of the natural series and let $α = \{α_j\}_{j ∈ ℕ},\; ν = \{ν_j\}_{j ∈ ℕ}$, and $η = {η_j}_{j ∈ ℕ}$ be nonnegative number sequences for which $$\left\| {\nu (\alpha \eta )_\gamma } \right\|_1 : = \sum\limits_{j = 1}^\infty {v _j \alpha _{\gamma (_j )} } \eta _{...

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Дата:	2005
Автори:	Radzievskaya, E. I., Radzievskii, G. V., Радзиевская, Е. И., Радзиевский, Г. В.
Формат:	Стаття
Мова:	Російська Англійська
Опубліковано:	Institute of Mathematics, NAS of Ukraine 2005
Онлайн доступ:	https://umj.imath.kiev.ua/index.php/umj/article/view/3698
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Назва журналу:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Radzievskaya, E. I. Radzievskii, G. V. Радзиевская, Е. И. Радзиевский, Г. В. Радзиевская, Е. И. Радзиевский, Г. В.
author_facet	Radzievskaya, E. I. Radzievskii, G. V. Радзиевская, Е. И. Радзиевский, Г. В. Радзиевская, Е. И. Радзиевский, Г. В.
author_sort	Radzievskaya, E. I.
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
collection	OJS
datestamp_date	2020-03-18T20:02:18Z
description	Let $Γ$ be the set of all permutations of the natural series and let $α = \{α_j\}_{j ∈ ℕ},\; ν = \{ν_j\}_{j ∈ ℕ}$, and $η = {η_j}_{j ∈ ℕ}$ be nonnegative number sequences for which $$\left\\| {\nu (\alpha \eta )_\gamma } \right\\|_1 : = \sum\limits_{j = 1}^\infty {v _j \alpha _{\gamma (_j )} } \eta _{\gamma (_j )}$$ is defined for all $γ:= \{γ(j)\}_{j ∈ ℕ} ∈ Γ$ and $η ∈ l_p$. We find $\sup _{\eta :\left\\| \eta \right\\|_p = 1} \inf _{\gamma \in \Gamma } \left\\| {\nu (\alpha \eta )_\gamma } \right\\|_1$ in the case where $1 < p < ∞$.
first_indexed	2026-03-24T02:47:18Z
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fulltext	UDK 519.658 E. Y. Radzyevskaq (Nac. un-t pyw. texnolohyj Ukrayn¥, Kyev), H. V. Radzyevskyj (Yn-t matematyky NAN Ukrayn¥, Kyev) OB ODNOJ ∏KSTREMAL\|NOJ ZADAÇE DLQ ÇYSLOVÁX RQDOV * Let Γ be a set of all permutations of positive integers, let α α= ∈{ }j j N , ν ν= ∈{ }j j N , and η η= ∈{ }j j N be nonnegative numerical sequences for which ν αη ν α ηγ γ γ( ) : ( ) ( )1 1 = = ∞∑ j j jj is defined for all γ γ: { ( )}= ∈∈j j N Γ and η ∈ l p . We establish sup inf: ( )η η γ γν αη p = ∈1 1Γ in the case where 1 < < ∞p . Nexaj Γ — mnoΩyna vsix perestanovok natural\noho rqdu, α α= ∈{ }j j N , ν ν= ∈{ }j j N i η η= ∈{ }j j N — nevid’[mni çyslovi poslidovnosti, dlq qkyx ν αη ν α ηγ γ γ( ) : ( ) ( )1 1 = = ∞∑ j j jj vyznaçeno dlq usix γ γ: { ( )}= ∈∈j j N Γ i η ∈ l p . Znajdeno sup inf: ( )η η γ γν αη p = ∈1 1Γ u vy- padku 1 < < ∞p . Dannoe soobwenye ynspyryrovano rezul\tatamy rabot [1] (hl. XI, lemma 6.1) y [2] (lemma). Dokazatel\stva sootvetstvugwyx utverΩdenyj yz [1, 2] ves\ma hro- mozdky. Tak, lemma 6.1 yz hl. XI v [1] dokaz¥vaetsq na 30 stranycax y soderΩyt 164 zanumerovann¥e formul¥. Lemma yz [2] obobwaet sootvetstvugwyj re- zul\tat yz [1], odnako y ee dokazatel\stvo zanymaet 13 stranyc. Zdes\ pryvede- no nebol\ßoe po obæemu dokazatel\stvo teorem¥, soderΩawej lemmu yz [2], a znaçyt, y lemmu 6.1 yz hl. XI v [1]. (Neobxodym¥e poqsnenyq dan¥ v konce soob- wenyq.) Otmetym, çto nekotoraq hromozdkost\ pryvedenn¥x pered formuly- rovkoj teorem¥ postroenyj svqzana ysklgçytel\no s hromozdkost\g ponqtyj y oboznaçenyj, yspol\zuem¥x v nej. Kak ob¥çno, N — mnoΩestvo cel¥x poloΩytel\n¥x (natural\n¥x) çysel. V dal\nejßem rassmatryvagtsq lyß\ posledovatel\nosty ξ ξ: { }= ∈j j N vewe- stvenn¥x çysel, a K ∞ — mnoΩestvo neotrycatel\n¥x posledovatel\nostej ξ , u kotor¥x ξ ξj j≥ +1, j ∈ N. Pust\ c0 sostoyt yz posledovatel\nostej ξ so svojstvom lim j j→∞ ξ = 0. KaΩdoj ξ ξ= ∈{ }j j N yz c0 sopostavym posledova- tel\nost\ ξ ξ∗ ∗ ∈= { }j j N, poloΩyv ξ ξ ϕj j ∗ = ( ) , hde ϕ( )⋅ — takaq perestanov- ka natural\noho rqda, çto ξ ϕ( )j j { } ∈N qvlqetsq nevozrastagwej posledova- tel\nost\g. Dlq dvux posledovatel\nostej α = { }α j j ∈N y ξ = { }ξ j j ∈N op- redelym yx proyzvedenye αξ α ξ: { }= ∈j j j N , a esly vse πlement¥ αj ≠ 0, to α α− − ∈=1 1: { }j j N y ξ α ξα/ := −1 . Vvedem banaxovo prostranstvo lr , 1 ≤ r < ∞ , sostoqwee yz posledovatel\- nostej ξ = { }ξ j j ∈N, udovletvorqgwyx uslovyg ξ r : = ξ j r j r = ∞ ∑    1 1/ < ∞ , ξ ∈ lr . * PodderΩana Hosudarstvenn¥m fondom fundamental\n¥x yssledovanyj Ukrayn¥ (proekt F7/329-2001). © E. Y. RADZYEVSKAQ, H. V. RADZYEVSKYJ , 2005 1430 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10 OB ODNOJ ∏KSTREMAL\|NOJ ZADAÇE DLQ ÇYSLOVÁX RQDOV 1431 Dalee v rabote 1 < p < ∞ , q p p: ( )= − −1 1, otnosytel\no posledovatel\no- sty α = { }α j j ∈N predpolahaem, çto vse ee πlement¥ poloΩytel\n¥, a otnosy- tel\no nenulevoj posledovatel\nosty ν = { }ν j j ∈N — çto 0 ≤ νj ≤ ν j+1 dlq vsex j ∈ N. Pust\ takΩe αν ∈ lq . Tem sam¥m α ∈ lq y, sledovatel\no, oprede- lena posledovatel\nost\ α* y α* ∈ lq . Vvedem podposledovatel\nost\ natural\n¥x çysel { }ks s∈N, poloΩyv k1 : = : = 1, a ks+1 — naybol\ßym yz çysel, dlq kotor¥x µs : = ν α j j k k j p j k k s s s s = − ∗ − = − + + ∑ ∑ 1 1 1 1 ( ) = max ( ) :k k k j j k k j p j k k s s s ≥ = ∗ − = ∑ ∑ ν α . (1) PokaΩem suwestvovanye ukazann¥x ks . Poskol\ku ν — neub¥vagwaq posledovatel\nost\, sohlasno teoreme 368 yz [3], α ν∗ q ≤ αν q < ∞ , poπto- mu dlq nekotoroj postoqnnoj c > 0 v¥polnqetsq neravenstvo α νj j c∗ ≤ pry vsex j ∈ N. Tem sam¥m ν α j j k k j p j k k s s = ∗ − = ∑ ∑ ( ) ≤ c j j k k j p j j k k s s ν α ν = ∗ − + = ∑ ∑ ( ) 1 . (2) No ( )α j p∗ − + → ∞1 pry j → ∞, a ν j c≥ >1 0 dlq vsex dostatoçno bol\ßyx j . Znaçyt, pravaq çast\ v (2) stremytsq k nulg pry k , stremqwemsq k beskoneçnosty, y poπtomu v (1) maksymum suwestvuet. Otmetym, çto { }µs s∈N — ub¥vagwaq posledovatel\nost\. Dejstvytel\no, esly b¥ πto b¥lo ne tak, to µ µs s≤ +1 dlq nekotoroho s ∈ N y tohda µs ≤ ( )α νj p j k k j j k k s s s s ∗ − = − − = −+ + ∑ ∑         2 21 1 1 , a πto protyvoreçyt v¥boru çysla ks+1. Dalee potrebuetsq posledovatel\nost\ κ κ= ∈{ }s s N s πlementamy κs : = ( ) / α νj p j k k p j j k k s s s s ∗ − = − − = −+ + ∑ ∑         1 11 1 1 . (3) PokaΩem, çto κ ∈ lq . Dejstvytel\no, sohlasno neravenstvu Hel\dera, ν j j k k s s = −+ ∑ 1 1 ≤ ( ) ( ) / / α α νj p j k k p j j q j k k q s s s s ∗ − = − ∗ = −+ + ∑ ∑         1 11 1 1 1 , a, kak uΩe otmeçalos\, α* ν ∈ lq . Tem sam¥m κ q q ≤ α ν∗ q q < ∞ . ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10 1432 E. Y. RADZYEVSKAQ, H. V. RADZYEVSKYJ Y, nakonec, çerez Γ oboznaçym mnoΩestvo vsex perestanovok natural\noho rqda, a dlq γ γ: { ( )}= ∈j j N yz Γ y posledovatel\nosty ξ ξ= ∈{ }j j N poloΩym ( ) { }: ( )ξ ξγ γ= ∈j j N . Vo vvedenn¥x oboznaçenyqx y predpoloΩenyqx dokaΩem sledugwee ut- verΩdenye. Teorema. Ymeet mesto nestrohoe neravenstvo inf ( ) γ γν αη ∈Γ 1 ≤ κ ηq p , η ∈ lp , (4) pryçem suwestvuet nenulevoj πlement η yz lp , dlq kotoroho (4) prevrawa- etsq v ravenstvo. Dokazatel\stvo. Poskol\ku ν — neub¥vagwaq posledovatel\nost\ neot- rycatel\n¥x çysel, yspol\zuq teoremu 368 yz [3] y polahaq ξ αη:= , ymeem inf ( ) γ γν αη ∈Γ 1 = ν αη( )∗ 1 = νξ∗ 1 . Analohyçno zaklgçaem, çto η p = ξ α/ p ≥ ξ α∗ ∗/ p . (5) Tem sam¥m dlq dokazatel\stva teorem¥ dostatoçno ustanovyt\ sootnoßenye νξ 1 ≤ κ ξ αq p / ∗ , ξ ∈ K ∞, ξ / α* ∈ lp , (6) y pokazat\, çto ravenstvo v nem dostyhaetsq na nekotorom nenulevom πlemente ξ yz K ∞, dlq kotoroho ξ / α* ∈ lp . Dejstvytel\no, pust\ ξ — ukazann¥j πlement yz K ∞, a λ — takaq perestanovka natural\noho rqda, çto α = ( )α λ ∗ . Tohda ( ) /ξ αλ p = ξ α/ ∗ p y neravenstvo (5) prevrawaetsq v ravenstvo na po- sledovatel\nosty ξ , ravnoj ( )ξ λ . Sledovatel\no, v (4) levaq çast\ ravna pra- voj pry η = ( ) /ξ αλ ∈ lp . V sluçae p( )ξ : = ν ξj jj k k s s = −+∑ 1 1 lemma yz soobwenyq [4] prymet vyd ν ξj j j k k s s = −+ ∑ 1 1 ≤ max ( ) ( ) ( ), ,k k k k k k p k p j j p j k k s s s s s s = … − ∗ − ∗ − ∗ = − + ++ …+ + …+           ∑ 1 1 1 1ν ν α α ξ α (7) dlq vsex ξks ≥ ξks +1 ≥ … ≥ ξks+ −1 1 ≥ 0. No funkcyq x � x p1/ , x ≥ 0, stro- ho v¥pukla kverxu, poπtomu vvydu neravenstva Yensena ( ) ( ) α ξ αj p j k k j j p j k k s s s s ∗ − = − − ∗ = −+ + ∑ ∑     1 11 1 1 ≤ ( ) / / α ξ αj p j k k p j j p j k k p s s s s ∗ − = − − ∗ = −+ + ∑ ∑               1 11 1 1 1 , pryçem ravenstvo budet lyß\ v sluçae ξks = … = ξks+ −1 1. Otsgda y yz neravenstva (7), s uçetom opredelenyj (1) y (3) çysel ks+1 y κs , ymeem ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10 OB ODNOJ ∏KSTREMAL\|NOJ ZADAÇE DLQ ÇYSLOVÁX RQDOV 1433 ν ξj j j k k s s = −+ ∑ 1 1 ≤ κ ξ αs j j p j k k p s s ∗ = −           + ∑ 1 1 1/ y, kak vydno yz yzloΩennoho v¥ße, ravenstvo dostyhaetsq lyß\ v sluçae ξks = … = ξks+ −1 1. Teper\, prynymaq vo vnymanye neravenstvo Hel\dera, poluçaem νξ 1 ≤ ≤ κ ξ αq p / ∗ , a vvydu pred¥duwyx rassuΩdenyj ravenstvo budet lyß\ v slu- çae, esly u posledovatel\nosty ξ ξ= ∈{ }j j N πlement¥ ξks = … = ξks+ −1 1 y ( )α ξj p j k k k p s s s ∗ − = −+ ∑     1 1 = c s qκ , hde c — nekotoraq neotrycatel\naq postoqnnaq. Poπtomu posledovatel\nost\ ξ / α* ∈ lp . No na osnovanyy opredelenyj (1) y (3) çysel µs y κs ξk p s = c j p j k k s q s s ( )α κ∗ − = − − + ∑     1 1 1 = c s qµ , a { }µs s∈N — ub¥vagwaq posledovatel\nost\, znaçyt, ξ ∈ K ∞. Tem sam¥m, ustanovleno neravenstvo (6), a sledovatel\no, y (4) y, krome to- ho, opysan¥ vse posledovatel\nosty ξ ∈ K ∞, dlq kotor¥x (6) prevrawaetsq v ravenstvo. Zametym, çto sohlasno neravenstvu Hel\dera y teoreme Banaxa – Ítejnhauza trebovanye αν ∈ lq qvlqetsq ne tol\ko dostatoçn¥m, no y neobxodym¥m uslo- vyem dlq koneçnosty ν αη γ( ) 1 dlq lgboj posledovatel\nosty η ∈ lp . (∏to utverΩdenye prynadleΩyt ∏. Landau, sm. teoremuS161 v [3].) V sluçae ohranyçennoj posledovatel\nosty ν utverΩdenye teorem¥ sovpa- daet s lemmoj yz rabot¥ [2], a v sluçae posledovatel\nosty ν, ravnoj posle- dovatel\nosty ν̂ : = { }ν̂ j j ∈N , ν̂ j = 0 pry j ≤ n y ν̂ j = 1 pry j > n, hde n ∈ N, y v predpoloΩenyy o nevozrastanyy posledovatel\nosty α (t. e. α = α* ) — s lemmoj 6.1 yz hl. XI monohrafyy [1]. Otmetym, çto v πtom sluçae podavlqgwee çyslo postroenyj, v¥polnenn¥x pered formulyrovkoj teorem¥, sleduet opustyt\. Naprymer, çysla ks moΩno poloΩyt\ takymy: k1 : = 1, k2 ravno naybol\ßemu yz çysel, dlq kotoroho v¥raΩenye ( )k n j p j k − −     − = − − ∑1 1 1 1 α dostyhaet svoeho maksymuma po k > n, a vse ostavßyesq ks+2 : = k s2 + , s ∈ N. Pry takom v¥bore çysel ks spravedlyv¥ sootnoßenyq ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10 1434 E. Y. RADZYEVSKAQ, H. V. RADZYEVSKYJ ( )k n j p j k 2 1 1 1 1 2 − −     − = − − ∑ α > αk p 2 ≥ αk p 2 1+ ≥ … , dostatoçn¥e, çtob¥ na zaklgçytel\nom πtape dokazatel\stva teorem¥ dlq ne- strohoho neravenstva νξ 1 ≤ κ ξ αq p/ suwestvovala otlyçnaq ot nulevoj posledovatel\nost\ ξ yz konusa K ∞ y ξ / α ∈ lp , dlq kotoroj πto neravenstvo prevrawalos\ b¥ v ravenstvo. 1. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj: VS2St. – Kyev: Yn-t matematyky NAN Ukray- n¥, 2002. – T. 2. – 468 s. 2. Íydliç A. L. Najkrawi n-çlenni nablyΩennq Λ -metodamy v prostorax S p ϕ // Ekstre- mal\ni zadaçi teori] funkcij ta sumiΩni pytannq: Pr. In-tu matematyky NAN Ukra]ny. – 2003. – 46. – S.S283 – 306. 3. Xardy H. H., Lyttl\vud DΩ. E., Polya H. Neravenstva. – M.: Yzd-vo ynostr. lyt., 1948. – 456Ss. 4. Radzyevskaq E. Y., Radzyevskyj H. V. Ob odnoj πkstremal\noj zadaçe dlq polunorm¥ na prostranstve l1 s vesom // Ukr. mat. Ωurn. – 2005. – 57, # 7. – S. 1002 – 1006. Poluçeno 01.11.2004 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
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spelling	umjimathkievua-article-36982020-03-18T20:02:18Z On one extremal problem for numerical series Об одной экстремальной задаче для числовых рядов Radzievskaya, E. I. Radzievskii, G. V. Радзиевская, Е. И. Радзиевский, Г. В. Радзиевская, Е. И. Радзиевский, Г. В. Let $Γ$ be the set of all permutations of the natural series and let $α = \{α_j\}_{j ∈ ℕ},\; ν = \{ν_j\}_{j ∈ ℕ}$, and $η = {η_j}_{j ∈ ℕ}$ be nonnegative number sequences for which $$\left\\| {\nu (\alpha \eta )_\gamma } \right\\|_1 : = \sum\limits_{j = 1}^\infty {v _j \alpha _{\gamma (_j )} } \eta _{\gamma (_j )}$$ is defined for all $γ:= \{γ(j)\}_{j ∈ ℕ} ∈ Γ$ and $η ∈ l_p$. We find $\sup _{\eta :\left\\| \eta \right\\|_p = 1} \inf _{\gamma \in \Gamma } \left\\| {\nu (\alpha \eta )_\gamma } \right\\|_1$ in the case where $1 < p < ∞$. Нехай $Γ$ — множина всіх перестановок натурального ряду, $α = \{α_j\}_{j ∈ ℕ},\; ν = \{ν_j\}_{j ∈ ℕ}$ і $η = {η_j}_{j ∈ ℕ}$ — невід'ємні числові послідовності, для яких $$\left\\| {\nu (\alpha \eta )_\gamma } \right\\|_1 : = \sum\limits_{j = 1}^\infty {v _j \alpha _{\gamma (_j )} } \eta _{\gamma (_j )}$$ визначаю для усіх $γ:= \{γ(j)\}_{j ∈ ℕ} ∈ Γ$ і $η ∈ l_p$. Знайдено $\sup _{\eta :\left\\| \eta \right\\|_p = 1} \inf _{\gamma \in \Gamma } \left\\| {\nu (\alpha \eta )_\gamma } \right\\|_1$ у випадку $1 < p < ∞$. Institute of Mathematics, NAS of Ukraine 2005-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3698 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 10 (2005); 1430–1434 Український математичний журнал; Том 57 № 10 (2005); 1430–1434 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3698/4121 https://umj.imath.kiev.ua/index.php/umj/article/view/3698/4122 Copyright (c) 2005 Radzievskaya E. I.; Radzievskii G. V.
spellingShingle	Radzievskaya, E. I. Radzievskii, G. V. Радзиевская, Е. И. Радзиевский, Г. В. Радзиевская, Е. И. Радзиевский, Г. В. On one extremal problem for numerical series
title	On one extremal problem for numerical series
title_alt	Об одной экстремальной задаче для числовых рядов
title_full	On one extremal problem for numerical series
title_fullStr	On one extremal problem for numerical series
title_full_unstemmed	On one extremal problem for numerical series
title_short	On one extremal problem for numerical series
title_sort	on one extremal problem for numerical series
url	https://umj.imath.kiev.ua/index.php/umj/article/view/3698
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On one extremal problem for numerical series

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