Approximative characteristics of the isotropic classes of periodic functions of many variables

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov $(B_{p,θ}^r)$ and Nukol’skii $(H_p^r )$ classes of periodic functions of many variables in the metric of $L_q , 1 ≤ p, q ≤ ∞$. We also establish the orders of the best approximations of functions fro...

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Дата:	2009
Автори:	Romanyuk, A. S., Романюк, А. С.
Формат:	Стаття
Мова:	Російська Англійська
Опубліковано:	Institute of Mathematics, NAS of Ukraine 2009
Онлайн доступ:	https://umj.imath.kiev.ua/index.php/umj/article/view/3036
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Назва журналу:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Romanyuk, A. S. Романюк, А. С. Романюк, А. С.
author_facet	Romanyuk, A. S. Романюк, А. С. Романюк, А. С.
author_sort	Romanyuk, A. S.
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
collection	OJS
datestamp_date	2020-03-18T19:43:50Z
description	Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov $(B_{p,θ}^r)$ and Nukol’skii $(H_p^r )$ classes of periodic functions of many variables in the metric of $L_q , 1 ≤ p, q ≤ ∞$. We also establish the orders of the best approximations of functions from the same classes in the spaces $L_1$ and $L_{∞}$ by trigonometric polynomials with the corresponding spectrum.
first_indexed	2026-03-24T02:35:03Z
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fulltext	UDK 517. 51 A. S. Romangk (Yn-t matematyky NAN Ukrayn¥, Kyev) APPROKSYMATYVNÁE XARAKTERYSTYKY YZOTROPNÁX KLASSOV PERYODYÇESKYX FUNKCYJ MNOHYX PEREMENNÁX Exact-order estimates of the best orthogonal trigonometric approximations of the Besov classes Bp r ,θ and the Nikol's'kyi classes Hp r of periodic multivariable functions in the metric Lq , 1 ≤ p, q ≤ ∞, are obtained. Orders of the best approximations of functions from these classes in the spaces L1 and L∞ by trigonometric polynomials with the corresponding spectrum are also established. OderΩano toçni za porqdkom ocinky najkrawyx ortohonal\nyx tryhonometryçnyx nablyΩen\ klasiv B[sova Bp r ,θ i Nikol\s\koho Hp r periodyçnyx funkcij bahat\ox zminnyx u metryci Lq , 1 ≤ p, q ≤ ∞. Vstanovleno takoΩ porqdky najkrawyx nablyΩen\ funkcij z cyx Ωe klasiv u prostorax L1 i L∞ tryhonometryçnymy polinomamy z vidpovidnym spektrom. 1. Vvedenye. V nastoqwej rabote yssledugtsq nekotor¥e vopros¥ pryblyΩe- nyq yzotropn¥x klassov O. V. Besova B p r ,θ [1] y S. M. Nykol\skoho Hp r [2] pe- ryodyçeskyx funkcyj mnohyx peremenn¥x. Rassmatryvaem¥e approksymatyv- n¥e xarakterystyky πtyx klassov funkcyj budut opredelen¥ v sootvetstvug- wyx çastqx rabot¥, a snaçala pryvedem neobxodym¥e v dal\nejßem oboznaçe- nyq y opredelenyq. Pust\ Rd , d ≥ 1, oboznaçaet d-mernoe prostranstvo s πlementamy x = = x1( , … , xd ) y Lp d( )T , Td = j d =∏ −[ ) 1 π π; � prostranstvo 2π -peryodyçes- kyx po kaΩdoj peremennoj funkcyj f x( ), dlq kotor¥x f p = ( ) ( ) / 2 1 π − ∫       d p p d f x dx T < ∞, 1 ≤ p < ∞, f ∞ = ess sup ( ) x d f x ∈T < ∞. Oboznaçym çerez V tm( ) , m ∈N , t ∈R, qdro Valle Pussena vyda V tm( ) = 1 + 2 k m kt = ∑ 1 cos + 2 k m m m k m kt = + ∑ −    1 2 2 cos . Tohda mnohomernoe qdro V xm( ) , m ∈N , x d∈R , opredelym sohlasno formule V xm( ) = j d m jV x = ∏ 1 ( ). Pust\ Vm � operator, kotor¥j zadaet svertku funkcyj f x( ) s mnohomer- n¥m qdrom V xm( ) , t. e. V fm = f Vm∗ = V f xm( , ). Takym obrazom, V f xm( , ) � kratnaq summa Valle Pussena funkcyy f x( ). Dlq f ∈ L d 1( )T poloΩym σ0( , )f x = V f x1( , ) , σs f x( , ) = V f xs2 ( , ) – V f xs2 1− ( , ), s = 1, 2, … . © A. S. ROMANGK, 2009 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 513 514 A. S. ROMANGK PreΩde çem pryvesty opredelenye rassmatryvaem¥x klassov funkcyj, sde- laem sledugwee zameçanye. V posledugwyx rassuΩdenyqx nam budet udobno pol\zovat\sq opredelenyqmy klassov B p r ,θ y Hp r v tak naz¥vaemom dekompozy- cyonnom vyde. S toçnost\g do absolgtn¥x postoqnn¥x πty opredelenyq πkvy- valentn¥ ysxodn¥m, kotor¥e dan¥ v [1] y [2] sootvetstvenno dlq klassov B p r ,θ y Hp r . Ytak, v prynqt¥x oboznaçenyqx klass¥ B p r ,θ , 1 ≤ p, θ ≤ ∞, r > 0, moΩno opredelyt\ sledugwym obrazom (sm., naprymer, [3]): Bp r ,θ = f x f f xB s sr s pp r( ): ( , ) , / θ θ θ θ σ=       ≤        ∈ + ∑ Z 2 1 1 , 1 ≤ θ < ∞, (1) Bp r ,∞ = f x f f xB s sr s pp r( ): sup ( , ) , ∞ + = ≤      ∈Z 2 1σ , θ = ∞. Otmetym, çto v sluçae 1 < p < ∞ moΩno zapysat\ πkvyvalentn¥e opredele- nyq klassov B p r ,θ , 1 ≤ θ < ∞, y B p r ,∞ ≡ Hp r , yspol\zuq v (1) vmesto σs f x( , ) �bloky� rqda Fur\e funkcyy f x( ). Dlq f ∈ L d 1( )T y s ∈ +Z vvedem oboznaçenyq f x( )( )0 = ˆ( )f 0 y f xs( )( ) = 2 2 1 1s j sk j d i k xf k e − ≤ < = ∑ max , ( , )ˆ( ) , s = 1, 2, … , hde (k, x) = k x1 1 + … + k xd d y ˆ( )f k = ( ) ( ) ( , )2π − −∫d i k y d f y e dy T � koπffycyent¥ Fur\e funkcyy f x( ). Takym obrazom, pry 1 < p < ∞, r > 0 (s toçnost\g do absolgtn¥x postoqn- n¥x) Bp r ,θ = f x f f xB s sr s pp r( ): ( ) , ( ) / θ θ θ θ =       ≤        ∈ + ∑ Z 2 1 1 , 1 ≤ θ < ∞, (2) Bp r ,∞ = f x f f xB s sr s pp r( ): sup ( ) , ( )∞ + = ≤      ∈Z 2 1 , θ = ∞. Otmetym, çto s uvelyçenyem parametra θ klass¥ B p r ,θ rasßyrqgtsq, t. e. pry 1 ≤ θ ≤ ′θ ≤ ∞ ymegt mesto vloΩenyq B p r ,1 � B p r ,θ � B p r , ′θ � B p r ,∞ ≡ Hp r . Takym obrazom, poskol\ku v formulyruem¥x nyΩe utverΩdenyqx parametr θ prynymaet y predel\noe znaçenye θ = ∞, sohlasno prynqt¥m oboznaçenyqm v πtyx utverΩdenyqx soderΩatsq sootvetstvugwye rezul\tat¥ y dlq klassov S. M. Nykol\skoho Hp r . 2. Nayluçßye ortohonal\n¥e tryhonometryçeskye pryblyΩenyq . Sna- çala opredelym approksymatyvnug xarakterystyku, kotorug budem yssledo- vat\ v dannom punkte. ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 APPROKSYMATYVNÁE XARAKTERYSTYKY YZOTROPNÁX KLASSOV … 515 Pust\ Λ ⊂ Zd � koneçnoe mnoΩestvo, soderΩawee m πlementov, t. e. Λ = m. Dlq f ∈ Lq d( )T , 1 ≤ q ≤ ∞, poloΩym S f xΛ( , ) = k i k xf k e ∈ ∑ Λ ˆ( ) ( , ) y rassmotrym velyçynu e fm q ⊥( ) = inf ( ) ( , ) Λ Λf x S f x q− . Esly F � Lq d( )T � nekotor¥j klass funkcyj, to polahaem e Fm q ⊥( ) = sup ( ) f F m qe f ∈ ⊥ . (3) Velyçynu e Fm q ⊥( ) naz¥vagt nayluçßym ortohonal\n¥m tryhonometryçes- kym pryblyΩenyem klassa F v prostranstve Lq . Nayluçßye ortohonal\n¥e tryhonometryçeskye pryblyΩenyq nekotor¥x klassov funkcyj mnohyx pere- menn¥x yssledovalys\ v rabotax [4, 5], v kotor¥x moΩno oznakomyt\sq s soot- vetstvugwej byblyohrafyej. V πtom punkte osnovnoe vnymanye sosredotoçeno na poluçenyy toçn¥x po porqdku ocenok velyçyn e Bm p r q ⊥( ),θ , 1 ≤ p, q ≤ ∞. Parallel\no budut ustanov- len¥ porqdky pryblyΩenyq funkcyj yz klassov B p r ,θ yx çastn¥my summamy Fur\e po sootvetstvugwym oblastqm. V nekotor¥x sluçaqx dlq dokazatel\- stva ocenok snyzu velyçyn e Bm p r q ⊥( ),θ budem yspol\zovat\ yzvestn¥e ocenky nayluçßyx m-çlenn¥x tryhonometryçeskyx pryblyΩenyj funkcyj yz klassov Bp r ,θ . Dlq formulyrovky sootvetstvugweho rezul\tata pryvedem neobxodym¥e oboznaçenyq y opredelenyq. Pust\ f ∈ Lq d( )T y k j j m{ } =1 � proyzvol\n¥j nabor d-mern¥x vektorov k j = k j 1( , … , kd j ) s celoçyslenn¥my koordynatamy. Tohda velyçyna e fm q( ) = inf ( ) , , k c j m j i k x q j j j f x c e− = ( )∑ 1 , hde cj � proyzvol\n¥e çysla, naz¥vaetsq nayluçßym m-çlenn¥m tryhonomet- ryçeskym pryblyΩenyem funkcyy f x( ). Dlq funkcyonal\noho klassa F � � Lq d( )T polahaem e Fm q( ) = sup ( ) f F m qe f ∈ . Zametym, çto, kak sleduet yz opredelenyj, dlq e Fm q( ) y e Fm q ⊥( ) spraved- lyvo sootnoßenye e Fm q( ) ≤ e Fm q ⊥( ) . (4) Velyçyna e fm( )2 dlq funkcyj odnoj peremennoj vvedena S. B. Steçkyn¥m [6] pry formulyrovke kryteryq absolgtnoj sxodymosty ortohonal\n¥x rqdov. Zatem velyçyn¥ e fm q( ) y e Fm q( ) , 1 ≤ q ≤ ∞, dlq funkcyj kak odnoj peremen- noj, tak y mnohyx peremenn¥x, yssledovalys\ vo mnohyx rabotax. S sootvetst- ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 516 A. S. ROMANGK vugwej byblyohrafyej moΩno oznakomyt\sq, naprymer, v rabote [7]. Dlq ys- sleduem¥x namy klassov funkcyj B p r ,θ v [8] poluçen sledugwyj rezul\tat. Teorema A. Pust\ 1 ≤ p, q, θ ≤ ∞ y r p q( , ) = d p q p q q p d p d 1 1 1 2 1 2 −    ≤ ≤ ≤ ≤ ≤ ≤ ∞              + , , ; — yly v druhyx sluçaqx.max Tohda dlq r > r p q( , ) e Bm p r q,θ( ) � m r d p q− + − { }( )+/ / max / ; /1 1 1 2 , hde a+ = max ;a 0{ }. Pry poluçenyy ocenok sverxu velyçyn e Bm p r q ⊥( ),θ budem yspol\zovat\ nera- venstvo, ustanovlennoe S. M. Nykol\skym [2] y naz¥vaemoe �neravenstvom raz- n¥x metryk�. Dlq udobstva takΩe sformulyruem eho. Teorema B. Pust\ n = n1( , … , nd ), nj ∈N , j = 1, d , y t x( ) = k n k i k x j j c e ≤ ∑ ( , ) . Tohda pry 1 ≤ q < p ≤ ∞ ymeet mesto neravenstvo t p ≤ 2 1 1 1d j d j q p qn t = −∏ / / . Otmetym, çto v sluçae d = 1 , p = ∞ sootvetstvugwee neravenstvo dokazal DΩekson [9]. Vse poluçenn¥e rezul\tat¥ budem formulyrovat\ v termynax porqdkov¥x sootnoßenyj. Pry πtom dlq funkcyj ν1( )n y ν2( )n zapys\ ν1 ! ν2 oznaça- et, çto suwestvuet postoqnnaq C1 > 0 takaq, çto ν1( )n ≤ C1 ν2( )n . Sootnoße- nye ν1 � ν2 ravnosyl\no tomu, çto v¥polnen¥ porqdkov¥e neravenstva ν1 ! ! ν2 y ν1 " ν2 . Otmetym, çto postoqnn¥e Ci , i = 1, 2, … , vxodqwye v opre- delenyq funkcyj y v porqdkov¥e sootnoßenyq, mohut zavyset\ tol\ko ot tex parametrov, kotor¥e soderΩatsq v opredelenyqx klassov, ot metryky y razmer- nosty prostranstva Rd . Ymeet mesto sledugwaq teorema. Teorema 1. Pust\ 1 ≤ p, q, θ ≤ ∞, (p, q) ≠ (1, 1), (∞, ∞). Tohda pry r > > d p1/( – 1/q)+ spravedlyvo sootnoßenye e Bm p r q ⊥( ),θ � m r d p q− + −( )+/ / /1 1 . (5) Dokazatel\stvo. Snaçala ustanovym v (5) ocenku sverxu. Pry πtom v sylu vloΩenyq Bp r ,θ � Hp r , 1 ≤ θ < ∞ , neobxodymug ocenku dostatoçno poluçyt\ dlq velyçyn¥ e Hm p r q ⊥( ) . Rassmotrym posledovatel\no neskol\ko sootnoßenyj meΩdu parametramy p y q. Pust\ snaçala ymeet mesto sluçaj 1 ≤ p < q ≤ ∞. Tohda po zadannomu m ∈N podberem çyslo n m( ) yz sootnoßenyq 2n � m d1/ y rassmotrym dlq f Hp r∈ pryblyΩenye ee kubyçeskoj summoj Fur\e S f xn( , ) vyda ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 APPROKSYMATYVNÁE XARAKTERYSTYKY YZOTROPNÁX KLASSOV … 517 S f xn( , ) = s n sf x = ∑ 0 ( )( ) . Pust\ q0 � nekotoroe çyslo, udovletvorqgwee uslovyg p < q0 < q. Po- skol\ku dlq f Hp r∈ , 1 ≤ p ≤ ∞, v¥polneno sootnoßenye σs pf x( , ) ≤ 2−sr ( sm. (1)), sohlasno neravenstvu Mynkovskoho y neravenstvu razn¥x metryk Ny- kol\skoho moΩem zapysat\ e fm q ⊥( ) ! f x S f xn q( ) ( , )− = s n s q f x = + ∞ ∑ 1 ( )( ) ≤ s n s q f x = + ∞ ∑ 1 ( )( ) ! ! s n sd q q s q f x = + ∞ −( )∑ 1 1 12 0 0 / / ( )( ) � s n sd q q s qf x = + ∞ −( )∑ 1 1 12 0 0 / / ( , )σ ! ! s n sd q q sd p q s pf x = + ∞ −( ) −( )∑ 1 1 1 1 12 20 0/ / / / ( , )σ = s n sd p q s pf x = + ∞ −( )∑ 1 1 12 / / ( , )σ ≤ ≤ s n sd r d p q = + ∞ − − +( )∑ 1 1 12 / / / ! 2 1 1− − +( )nd r d p q/ / / � m r d p q− + −/ / /1 1 . V sluçae 1 < p = q < ∞ v sylu neravenstva Mynkovskoho y sootnoßenyq f xs p( )( ) ≤ 2−sr , f Hp r∈ (sm. (2)), budem ymet\ e fm q ⊥( ) ! f x S f xn p( ) ( , )− = s n s p f x = + ∞ ∑ 1 ( )( ) ≤ s n s p f x = + ∞ ∑ 1 ( )( ) ≤ ≤ s n sr = + ∞ −∑ 1 2 ! 2−nr � m r d− / . (6) Ocenka sverxu velyçyn¥ e Hm p r q ⊥( ) v sluçae 1 < q < p < ∞ sleduet yz (6) sohlasno vloΩenyg H Hp r q r⊂ , t. e. e Hm p r q ⊥( ) ≤ e Hm q r q ⊥( ) � m r d− / . (7) Pry 1 < q < ∞ y p = ∞ sootvetstvugwaq ocenka velyçyn¥ e Hm r q ⊥ ∞( ) qvlqetsq sledstvyem (7) v sylu vloΩenyq B Br p r ∞ ⊂, ,θ θ . Nakonec, v sluçae q = 1 y 1 < p ≤ ∞ yskomaq ocenka velyçyn¥ e Hm p r⊥( )1 sleduet yz (7) sohlasno neravenstvu ⋅ 1 < ⋅ q , q > 1, y vloΩenyg H Hr p r ∞ ⊂ . Ocenky sverxu vo vsex sluçaqx teorem¥ dokazan¥. Perexodq k ustanovlenyg v (5) ocenky snyzu, zametym, çto v sluçaqx 1 ≤ p ≤ ≤ q ≤ 2, 1 ≤ q ≤ p ≤ ∞ y (p, q) ≠ (1, 1), (∞, ∞) yskom¥e ocenky sledugt yz teo- rem¥ A sohlasno (4). Poπtomu rassmotrym te yz ostavßyxsq sluçaev, dlq koto- r¥x velyçyn¥ e Bm p r q ⊥( ),θ y e Bm p r q( ),θ ymegt razn¥e porqdky. Predvarytel\no zametym, çto v sylu vloΩenyq B Bp r p r , ,1 ⊂ θ, 1 < θ ≤ ∞ , dostatoçno poluçyt\ so- otvetstvugwye ocenky snyzu dlq velyçyn¥ e Bm p r q ⊥( ),1 . Ytak, pust\ snaçala v¥polnen¥ sootnoßenyq 2 ≤ p < q < ∞ yly 1 < p ≤ 2 < < q < ∞. Dlq f Bp r∈ ,1 rassmotrym pryblyΩagwyj polynom ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 518 A. S. ROMANGK S f xΛ( , ) = k i k xf k e ∈ ∑ Λ ˆ( ) ( , ), hde Λ ⊂ Zd y Λ = m. Tohda v sylu sledstvyq D 1.2 (sm. [10, s. 392]) moΩem zapysat\ f x S f x q( ) ( , )− Λ = sup ( ) ( , ) ( ) :g g q d f x S f x g x dx ′ ≤ ∫ −( ) 1 T Λ , (8) hde 1 q + 1 ′q = 1. Çtob¥ vospol\zovat\sq sootnoßenyem (8), postroym sootvetstvugwye funkcyy. Po zadannomu m podberem n ∈N yz neravenstv 2 2( )n d− ≤ m < < 2 1( )n d− y rassmotrym funkcyg F xn( ) = j d k ik x j n n j je = = − ∏ ∑ + 1 2 2 11 . Prynymaq vo vnymanye, çto k ik x pj n n j je = −+ ∑ 2 2 11 � 2 1 1n p( / )− , 1 < p < ∞, (9) zapys¥vaem Fn Bp r ,1 = s sr n s p F x∑ 2 ( ) ( )( ) = 2 1( )n r n pF+ � � 2 2 1 1nr nd p( / )− = 2 1 1nd r d p( / / )+ − . (10) Yz (10) delaem zaklgçenye, çto funkcyq f x1( ) = C F xnd r d p n2 1 12− + −( / / ) ( ) pry nadleΩawem v¥bore postoqnnoj C2 > 0 prynadleΩyt klassu B p r ,1. Dalee, poskol\ku sohlasno (9) Fn q′ � 2nd q/ , 1 < q < ∞, funkcyq g x1( ) = C F xnd q n3 2− / ( ) s sootvetstvugwej postoqnnoj C3 > 0 udovletvorqet uslovyg g q1 ′ ≤ 1. Takym obrazom, prymenyv sootnoßenye (8) k funkcyqm f x1( ) y g x1( ) , bu- dem ymet\ f x S f x q1 1( ) ( , )− Λ " 2 2 21 1− + − − −nd r d p nd q nd m( / / ) / ( ) > > 2 2 1 1 2 1 1 1− + + − −    nd r d q p nd d ( / / / ) � 2 1 1− − +nd r d p q( / / / ) � m r d p q− + −/ / /1 1 . Pust\ teper\ ymeet mesto sluçaj 1 ≤ p < ∞ y q = ∞. Po zadannomu m pod- berem çyslo n(m) ∈ N tak, çtob¥ v¥polnqlys\ sootnoßenyq 2nd � m y 2nd ≥ ≥ 4m, y poloΩym ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 APPROKSYMATYVNÁE XARAKTERYSTYKY YZOTROPNÁX KLASSOV … 519 vn x+1( ) = j d j jV x V xn n = ∏ + −( ) 1 2 21 ( ) ( ) . Rassmotrym funkcyg f x2( ) = C xnd r d p n4 1 1 12− + − + ( / / ) ( )v , C4 > 0. Poskol\ku (sm., naprymer, [11, s. 66]) vn p+1 � 2 1 1nd p( / )− , 1 ≤ p ≤ ∞, (11) lehko ubedyt\sq, çto funkcyq f x2( ) s sootvetstvugwej postoqnnoj C 4 > 0 prynadleΩyt klassu Bp r ,1. Dejstvytel\no, sohlasno (1) moΩem zapysat\ f Bp r2 1, = s sr s pf x∑ 2 2σ ( , ) � 2 21 1 1 1 − + − + + nd r d p n r n p ( / / ) ( ) v � � 2 21 1 1 1− − −nd p nd p( / ) ( / ) = 1. Dalee, pust\ S f xΛ( , )2 � çastnaq summa, sostoqwaq yz m proyzvol\n¥x harmonyk rqda Fur\e funkcyy f x2( ) . Tohda, prynymaq vo vnymanye, çto vn + ∞1 � 2nd , ymeem f x S f x2 2( ) ( , )− ∞Λ ≥ f x2( ) ∞ – S f xΛ( , )2 ∞ " " 2 21 1− + − −nd r d p nd m( / / ) ( ) � 2 21 1− + −nd r d p nd( / / ) = 2 1− −nd r d p( / / ) � � m r d p− +/ /1 . Nakonec, pust\ p = 1 y q ∈ (1, ∞). Zametym, çto v sluçae q ∈ 1 2,( ] yskomaq ocenka snyzu velyçyn¥ e Bm r q ⊥( ),1 1 sleduet yz teorem¥ A. Poπtomu ostalos\ po- luçyt\ yskomug ocenku pry q ∈ (2, ∞). S πtoj cel\g vospol\zuemsq sootnoße- nyem (8). V kaçestve funkcyj f x( ) y g x( ) yz (8) v¥berem funkcyy f x2( ) pry p = 1 y g x2( ) = C xnd q n5 12− + / ( )v , C5 > 0, sootvetstvenno. Pry πtom, po-preΩnemu, predpolahagtsq v¥polnenn¥my soot- noßenyq 2nd � m y 2nd ≥ 4m. Poskol\ku v sylu (11) g q2 ′ � 2 1 1 1 − − ′ + ′ nd q n q ( / ) v � 2 21 1 1 1− − ′ − ′nd q nd q( / ) ( / ) = 1, pry sootvetstvugwem v¥bore postoqnnoj C5 > 0 funkcyq g x2( ) udovletvorq- et uslovyg sootnoßenyq (8) dlq funkcyy g x( ). Takym obrazom, prymenyv πto sootnoßenye k funkcyqm f x2( ) pry p = 1 y g x2( ), poluçym e fm q ⊥( )2 " 2 2 1 2 2− − + −( )nr nd q n m/ v " 2 2 21 1− + −−( )nd r d q nd n d( / / ) ( ) � � 2 21− +nd r d q nd( / / ) = 2 1 1− − +nd r d q( / / ) � m r d q− + −/ /1 1 . Ocenky snyzu vo vsex sluçaqx teorem¥ ustanovlen¥. Teorema dokazana. V zaklgçenye πtoho punkta pryvedem nekotor¥e kommentaryy k poluçenno- ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 520 A. S. ROMANGK mu rezul\tatu, sopostavyv eho s ocenkamy blyzkyx, v opredelennom sm¥sle, ap- proksymatyvn¥x xarakterystyk klassov B p r ,θ . Pust\ m ∈N . Oboznaçym K m( ) = k k k k m j dd j d= … ≤ [ ] + ={ }( , , ): , ,/ 1 1 1 1 y rassmotrym dlq f ∈ L d 1( )T ee kubyçeskug summu Fur\e vyda S f xK m( )( , ) = k K m i k xf k e ∈ ∑ ( ) ( , )ˆ( ) . Tohda, kak sleduet yz dokazatel\stva teorem¥ 1, v rassmotrenn¥x v nej sluçaqx moΩem zapysat\ e Bm p r q ⊥( ),θ � sup ( ) ( , ) , ( ) f B K m q p r f x S f x ∈ − θ . Pust\ f ∈ L d 1( )T y ˆ ( )f k l l ( ){ } = ∞ 1 � koπffycyent¥ Fur\e ˆ( )f k k d{ } ∈Z funk- cyy f x( ), uporqdoçenn¥e v porqdke nevozrastanyq yx modulej, t. e. ˆ ( )f k 1( ) ≥ ˆ ( )f k 2( ) ≥ … . Oboznaçym dlq f ∈ Lq d( )T G f xm( , ) = l m i k l xf k l e = ( )∑ ( ) 1 ˆ ( ) ( ), y, esly F � Lq d( )T � nekotor¥j klass funkcyj, poloΩym G Fm q( ) = sup ( ) ( , ) f F m qf x G f x ∈ − . Lehko vydet\, çto sohlasno opredelenyqm velyçyn e Fm q ⊥( ) y G Fm q( ) ymeet mesto neravenstvo e Fm q ⊥( ) ≤ G Fm q( ) . V [12] poluçeno sledugwee utverΩdenye. Teorema V. Pust\ 1 ≤ p, q, θ ≤ ∞. Tohda G Bm p r q( ),θ � m q r d p m q r d p r d p r d p q − + −( ) + − + { }− + ≤ ≤ > −    ≤ ≤ ∞ >             / / / / max / ; / / , , , , , max ; . 1 1 2 1 1 2 1 1 2 1 1 2 2 1 1 2 (12) Takym obrazom, sopostavlqq teoremu 1 s ocenkamy (12), vydym, çto suwest- vugt sootnoßenyq meΩdu parametramy p, q y r, dlq kotor¥x e Bm p r q ⊥( ),θ � G Bm p r q( ),θ , a takΩe takye, dlq kotor¥x velyçyn¥ e Bm p r q ⊥( ),θ y G Bm p r q( ),θ ymegt razn¥e porqdky. 3. Nayluçßye pryblyΩenyq funkcyj yz klassov Bp r ,θθ polynomamy yz Tm . Pust\ Tm � mnoΩestvo tryhonometryçeskyx polynomov t x( ) vyda ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 APPROKSYMATYVNÁE XARAKTERYSTYKY YZOTROPNÁX KLASSOV … 521 t x( ) = k K m k i k xc e ∈ ∑ ( ) ( , ). Dlq f ∈ Lq d( )T , 1 ≤ q ≤ ∞, oboznaçym E fm q( ) = inf t T q m f t ∈ − y, esly F � Lq d( )T � nekotor¥j funkcyonal\n¥j klass, poloΩym E Fm q( ) = sup ( ) f F m qE f ∈ . Yzvestno (sm., naprymer, [3], lemma 3), çto dlq f ∈ Lq d( )T , 1 < q < ∞, spra- vedlyvo sootnoßenye f x S f xK m q ( ) ( , )( )− � E fm q( ) . Sledovatel\no, porqdkov¥e ocenky velyçyn E Bm p r q( ),θ pry 1 < q < ∞ rea- lyzugtsq s pomow\g pryblyΩenyq funkcyj f x( ) yz klassov B p r ,θ yx çast- n¥my summamy Fur\e SK m( )(f, x). Takym obrazom, predstavlqetsq ynteresn¥m poluçyt\ ocenky velyçyn E Bm p r q( ),θ v sluçaqx q = 1, ∞. Ymeet mesto sledugwee utverΩdenye. Teorema 2. Pust\ r > 0, 1 ≤ θ ≤ ∞. Tohda pry p = 1, ∞ E Bm p r p( ),θ � m r d− / . (13) Dokazatel\stvo. Ocenku sverxu v (13) ustanovym pry θ = ∞, t. e. dlq klassov Hp r . Po zadannomu m ∈N podberem çyslo n(m) ∈ N yz sootnoßenyq 2n � m d1/ y rassmotrym dlq f Hp r∈ pryblyΩagwyj polynom vyda t f xn( , ) = s n s f x = ∑ 0 σ ( , ) . Poskol\ku dlq f Hp r∈ v¥polneno sootnoßenye σs pf x( , ) ≤ 2−sr , p = = 1, ∞, sohlasno neravenstvu Mynkovskoho budem ymet\ f x t f xn p( ) ( , )− = s n s p f x = + ∞ ∑ 1 σ ( , ) ≤ s n s pf x = + ∞ ∑ 1 σ ( , ) ≤ ≤ s n sr = + ∞ −∑ 1 2 ! 2−nr � m r d− / . Ocenka sverxu v (13) ustanovlena. Sootvetstvugwaq ocenka snyzu sleduet yz teorem¥ A sohlasno neravenstvu E Bm p r p( ),θ " e Bm p r p( ),θ . Teorema dokazana. V sledugwem utverΩdenyy rassmotrym sluçaj 1 ≤ p < ∞, q = ∞, hde porqd- ky velyçyn E Bm p r( ),θ ∞ y e Bm p r( ),θ ∞ razlyçn¥. Teorema 3. Pust\ 1 ≤ p < ∞, r > d p . Tohda pry 1 ≤ θ ≤ ∞ E Bm p r( ),θ ∞ � m r d p− +/ /1 . (14) ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 522 A. S. ROMANGK Dokazatel\stvo. Ocenka sverxu v (14) sleduet yz rezul\tata pryblyΩe- nyq funkcyj f B p r∈ ,θ yx çastn¥my summamy Fur\e SK m( )(f, x) sohlasno soot- noßenyg E fm( )∞ ! f x S f xK m( ) ( , )( )− ∞. Sootvetstvugwaq ocenka poluçena pry dokazatel\stve teorem¥ 1. Perexodq k ustanovlenyg v (14) ocenky snyzu, zametym, çto pry πtom dosta- toçno rassmotret\ sluçaj θ = 1 . Po zadannomu m ∈N podberem çyslo n ∈N yz neravenstv 2 2d n( )− ≤ m < 2 1d n( )− , t. e. 2nd � m. Rassmotrym funkcyg f x2( ) = C xnd r d p n4 1 1 12− + − + ( / / ) ( )v , C4 > 0, kotoraq, kak pokazano pry dokazatel\stve teorem¥ 1, prynadleΩyt klas- su B p r ,1. Dalee, pust\ t x∗( ) = k K m k i k xc e ∈ ∗∑ ( ) ( , ) � polynom nayluçßeho pryblyΩenyq funkcyy f x2( ) v metryke prostranstva L d ∞( )T . Tohda, s odnoj storon¥, sohlasno sootnoßenyg meΩdu çyslamy m y n mo- Ωem zapysat\ ( ),f t n2 1−( )∗ +v = ( , )f n2 1v + � 2 1 1 1 2 2− + − + nd r d p n ( / / ) v � � 2 21 1− + −nd r d p nd( / / ) = 2 1− −nd r d p( / / ) � m r d p− +/ /1 , (15) a s druhoj � v sylu neravenstva Hel\dera y ocenky (11) budem ymet\ ( ),f t n2 1−( )∗ +v ≤ f t n2 1 1 − ∗ ∞ +v � f t2 − ∗ ∞ = E fm( )2 ∞ . (16) Sopostavlqq (15) y (16), poluçaem ocenku E fm( )2 ∞ " m r d p− +/ /1 . Ocenka snyzu, a vmeste s nej y teorema dokazan¥. Zameçanye 1. Sopostavlqq sootnoßenye (sm. teoremu A pry q = ∞) e Bm p r( ),θ ∞ � m r d p− + −( )+/ / /1 1 2 , 1 ≤ p ≤ ∞, 1 ≤ θ ≤ ∞, s ocenkamy (13) y (14), vydym, çto pry 1 ≤ p < ∞, r > max / ; /d p d 2{ } y r > 0 pry p = ∞ ymeet mesto sootnoßenye E Bm p r( ),θ ∞ � e B mm p r p( ), / θ ∞ ∗1 , hde p∗ = max ;p 2{ }. V zaklgçenye sformulyruem utverΩdenye, kotoroe qvlqetsq sledstvyem yzvestn¥x rezul\tatov. Teorema 4. Pust\ 1 ≤ θ ≤ ∞, 1 < p ≤ ∞. Tohda pry r > 0 spravedlyva ocenka E Bm p r( ),θ 1 � m r d− / . (17) ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 APPROKSYMATYVNÁE XARAKTERYSTYKY YZOTROPNÁX KLASSOV … 523 Dokazatel\stvo. Ocenka sverxu v (17) sleduet yz rezul\tata pryblyΩe- nyq funkcyj yz klassa B p r ,θ yx çastn¥my summamy Fur\e SK m( )(f, x). Sootvet- stvugwaq ocenka poluçena pry dokazatel\stve teorem¥ 1. Ocenka snyzu v (17) sleduet yz teorem¥ A. Zameçanye 2. Sopostavlqq (13) y (17) s sootvetstvugwymy utverΩdenyqmy teorem¥ A, vydym, çto pry 1 ≤ p, θ ≤ ∞ y r > 0 spravedlyv¥ sootnoßenyq E Bm p r( ),θ 1 � e Bm p r( ),θ 1 � m r d− / . 1. Besov O. V. Yssledovanyq odnoho semejstva funkcyonal\n¥x prostranstv v svqzy s teore- mamy vloΩenyq y prodolΩenyq // Tr. Mat. yn-ta AN SSSR. � 1961. � 60. � S. 42 � 61. 2. Nykol\skyj S. M. Neravenstva dlq cel¥x funkcyj koneçnoj stepeny y yx prymenenye v teoryy dyfferencyruem¥x funkcyj mnohyx peremenn¥x // Tam Ωe. � 1951. � 38. � S. 244 � 278. 3. Lyzorkyn P. Y. Obobwenn¥e hel\derov¥ prostranstva B p r , ( ) θ y yx sootnoßenyq s prost- ranstvamy Soboleva L p r( ) // Syb. mat. Ωurn. � 1968. � 9, # 5. � S. 1127 � 1152. 4. Romangk A. S. PryblyΩenye klassov peryodyçeskyx funkcyj mnohyx peremenn¥x // Mat. zametky. � 2002. � 70, # 1. � S. 109 � 121. 5. Romangk A. S. Bylynejn¥e y tryhonometryçeskye pryblyΩenyq klassov Besova B p r ,θ pe- ryodyçeskyx funkcyj mnohyx peremenn¥x // Yzv. RAN. Ser. mat. � 2006. � 70, # 2. � S. 69 � 98. 6. Steçkyn S. B. Ob absolgtnoj sxodymosty ortohonal\n¥x rqdov // Dokl. AN SSSR. � 1955. � 102, # 2. � S. 37 � 40. 7. Romangk A. S. Nayluçßye M-çlenn¥e tryhonometryçeskye pryblyΩenyq klassov Besova peryodyçeskyx funkcyj mnohyx peremenn¥x // Yzv. RAN. Ser. mat. � 2003. � 67, # 2. � S. 61 � 100. 8. De Vore R. A., Temlyakov V. N. Nonlinear approximation by trigonometric sums // J. Fourier Anal. and Appl. – 1995. – 2, # 1. – P. 29 – 48. 9. Jakson D. Certain problem of closest approximation // Bull. Amer. Math. Soc. (2). – 1933. – 39, # 12. – P. 889 – 906. 10. Kornejçuk N. P. Toçn¥e konstant¥ v teoryy pryblyΩenyq. � M.: Nauka, 1987. � 424 s. 11. Temlqkov V. N. PryblyΩenye funkcyj s ohranyçennoj smeßannoj proyzvodnoj // Tr. Mat. yn-ta AN SSSR. � 1986. � 178. � S. 1 � 112. 12. Temlyakov V. N. Greedy algorithm and m-term trigonometric approximation // Constr. Approxim. – 1998. – 14. – P. 569 – 587. Poluçeno 04.06.08 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
id	umjimathkievua-article-3036
institution	Ukrains’kyi Matematychnyi Zhurnal
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language	rus English
last_indexed	2026-03-24T02:35:03Z
publishDate	2009
publisher	Institute of Mathematics, NAS of Ukraine
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spelling	umjimathkievua-article-30362020-03-18T19:43:50Z Approximative characteristics of the isotropic classes of periodic functions of many variables Аппроксимативные характеристики изотропных классов периодических функций многих переменных Romanyuk, A. S. Романюк, А. С. Романюк, А. С. Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov $(B_{p,θ}^r)$ and Nukol’skii $(H_p^r )$ classes of periodic functions of many variables in the metric of $L_q , 1 ≤ p, q ≤ ∞$. We also establish the orders of the best approximations of functions from the same classes in the spaces $L_1$ and $L_{∞}$ by trigonometric polynomials with the corresponding spectrum. Одержано точні за порядком оцінки найкращих ортогональних тригонометричних наближень класів Бесова $(B_{p,θ}^r)$ і Нікольського $(H_p^r )$ періодичних функцій багатьох змінних у метриці $L_q , 1 ≤ p, q ≤ ∞$. Встановлено також порядки найкращих наближень функцій з цих же класів у просторах $L_{1}$ і $L_{∞}$ тригонометричними поліномами з відповідним спектром. Institute of Mathematics, NAS of Ukraine 2009-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3036 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 4 (2009); 513-523 Український математичний журнал; Том 61 № 4 (2009); 513-523 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3036/2821 https://umj.imath.kiev.ua/index.php/umj/article/view/3036/2822 Copyright (c) 2009 Romanyuk A. S.
spellingShingle	Romanyuk, A. S. Романюк, А. С. Романюк, А. С. Approximative characteristics of the isotropic classes of periodic functions of many variables
title	Approximative characteristics of the isotropic classes of periodic functions of many variables
title_alt	Аппроксимативные характеристики изотропных классов периодических функций многих переменных
title_full	Approximative characteristics of the isotropic classes of periodic functions of many variables
title_fullStr	Approximative characteristics of the isotropic classes of periodic functions of many variables
title_full_unstemmed	Approximative characteristics of the isotropic classes of periodic functions of many variables
title_short	Approximative characteristics of the isotropic classes of periodic functions of many variables
title_sort	approximative characteristics of the isotropic classes of periodic functions of many variables
url	https://umj.imath.kiev.ua/index.php/umj/article/view/3036
work_keys_str_mv	AT romanyukas approximativecharacteristicsoftheisotropicclassesofperiodicfunctionsofmanyvariables AT romanûkas approximativecharacteristicsoftheisotropicclassesofperiodicfunctionsofmanyvariables AT romanûkas approximativecharacteristicsoftheisotropicclassesofperiodicfunctionsofmanyvariables AT romanyukas approksimativnyeharakteristikiizotropnyhklassovperiodičeskihfunkcijmnogihperemennyh AT romanûkas approksimativnyeharakteristikiizotropnyhklassovperiodičeskihfunkcijmnogihperemennyh AT romanûkas approksimativnyeharakteristikiizotropnyhklassovperiodičeskihfunkcijmnogihperemennyh

Approximative characteristics of the isotropic classes of periodic functions of many variables

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