Best $m$-term trigonometric approximation for the classes $B^r_{p,θ}$ of functions of low smoothness

We obtain an exact-order estimate for the best $m$-term trigonometric approximation of the Besov classes $B^r_{p,θ}$ of periodic functions of many variables of low smoothness in the space $L_q, \; 1 < p ≤ 2 < q < ∞$.

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Бібліографічні деталі
Дата:	2010
Автори:	Stasyuk, S. A., Стасюк, С. А.
Формат:	Стаття
Мова:	Російська Англійська
Опубліковано:	Institute of Mathematics, NAS of Ukraine 2010
Онлайн доступ:	https://umj.imath.kiev.ua/index.php/umj/article/view/2846
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Назва журналу:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Stasyuk, S. A. Стасюк, С. А. Стасюк, С. А.
author_facet	Stasyuk, S. A. Стасюк, С. А. Стасюк, С. А.
author_sort	Stasyuk, S. A.
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
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description	We obtain an exact-order estimate for the best $m$-term trigonometric approximation of the Besov classes $B^r_{p,θ}$ of periodic functions of many variables of low smoothness in the space $L_q, \; 1 < p ≤ 2 < q < ∞$.
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fulltext	UDK 517.51 S. A. Stasgk (Yn-t matematyky NAN Ukrayn¥, Kyev) NAYLUÇÍEE m-ÇLENNOE TRYHONOMETRYÇESKOE PRYBLYÛENYE KLASSOV Bp r ,θθ FUNKCYJ MALOJ HLADKOSTY We obtain an exact-order estimate of the best m-term trigonometric approximation of the Besov classes Bp r ,θ of periodic multivariable functions of low smoothness in the space Lq, 1 < p ≤ 2 < q < ∞. OderΩano toçnu za porqdkom ocinku najkrawoho m-çlennoho tryhonometryçnoho nablyΩennq klasiv B[sova Bp r ,θ periodyçnyx funkcij bahat\ox zminnyx malo] hladkosti u prostori Lq, 1 < < p ≤ 2 < q < ∞. V nastoqwej rabote yssleduetsq nayluçßee m-çlennoe tryhonometryçeskoe pryblyΩenye klassov Besova Bp r ,θ peryodyçeskyx funkcyj d peremenn¥x v prostranstve Lq, 1 < p ≤ 2 < q < ∞, pry d pq 11 −     < r < d p . Ustanovlennaq toçnaq po porqdku ocenka ukazannoj velyçyn¥ dopolnqet rezul\tat¥, polu- çenn¥e R. A. DeVorom y V. N. Temlqkov¥m [1]. Bolee podrobno ob πtom budet ydty reç\ v kommentaryqx k rezul\tatu rabot¥, a snaçala pryvedem neobxody- m¥e oboznaçenyq y opredelenyq. Pust\ Rd, d ≥ 1, oboznaçaet d-mernoe prostranstvo s πlementamy x = = (,,) xxd 1… y Lpd () π, 1 ≤ p ≤ ∞, πd = −[] = ∏ππ; j d 1 , — prostranstvo 2π- peryodyçeskyx po kaΩdoj peremennoj funkcyj fx() = fxxd (,,) 1…, dlq ko- tor¥x fffx p xd ==<∞ ∞ ∈ esssup() π , 1 ≤ p < ∞. Dlq fLd ∈1() π y s∈+ Z oboznaçym fxf 00 ()() =", fxfke s k ikx s j s kjjd ()() max (,) ,, = − ∈= ≤< ∑" 22 1 1 Z , s = 1, 2, … , hde (,) kx = kx11 + … + kxdd, a fkftedt dikt d "()()()(,) =−− ∫ 2π π — koπffycyent¥ Fur\e funkcyy f. PreΩde çem pryvesty opredelenye rassmatryvaem¥x klassov funkcyj sde- laem sledugwee zameçanye. V posledugwyx rassuΩdenyqx m¥ budem yspol\zo- vat\ opredelenyq klassov Bp r ,θ, 1 ≤ θ < ∞, y HB p r p r ≡∞, v tak naz¥vaemom de- kompozycyonnom vyde. S toçnost\g do absolgtn¥x postoqnn¥x πty opredele- nyq πkvyvalentn¥ ysxodn¥m, kotor¥e dan¥ v [2] y [3] sootvetstvenno dlq klas- sov Bp r ,θ y Hp r. © S. A. STASGK, 2010 104ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 1 NAYLUÇÍEE m-ÇLENNOE TRYHONOMETRYÇESKOE PRYBLYÛENYE …105 Ytak, v prynqt¥x oboznaçenyqx klass¥ Bp r ,θ, 1 < p < ∞, 1 ≤ θ ≤ ∞, r > 0, moΩno opredelyt\ sledugwym obrazom (sm., naprymer, [4]): Bfff p r B rs sp s p r , / :() , θ θθ θ θ ==⋅      ≤  ∈+ ∑21 1 Z        , 1 ≤ θ < ∞,(1) Bfff p r B s rs sp p r ,:sup() , ∞ ∈ ==⋅≤       ∞ + Z 21, θ = ∞.(2) Kak otmeçalos\ v¥ße, BH p r p r ,∞≡, hde Hp r — yzvestn¥e klass¥ S. M. Ny- kol\skoho [3]. Otmetym, çto s toçky zrenyq ocenok nekotor¥x approksymatyv- n¥x xarakterystyk klass¥ Bp r ,θ y Hp r yssledovalys\ v rabotax [1] y [5 – 8], v kotor¥x moΩno oznakomyt\sq s sootvetstvugwej byblyohrafyej. Pust\ Θm — nabor yz m d-mern¥x celoçyslenn¥x vektorov, t. e. Θm = = nk { = (,,) nn kkd 1 …, nk d ∈Z, km =} 1,. PoloΩym Pxce mn inx k m k k (,)(,) Θ= = ∑ 1 y dlq fLqd ∈() π rassmotrym velyçynu σmq P mq ffP mm ()infinf()(,) (,) =⋅−⋅ ⋅ ΘΘ Θ, kotorug naz¥vagt nayluçßym m-çlenn¥m tryhonometryçeskym pryblyΩenyem funkcyy f. Dlq funkcyonal\noho klassa FLqd ⊂() π polahaem σσ mq fF mq Ff ()sup() = ∈ .(3) Velyçyna σmf ()2 dlq funkcyj odnoj peremennoj b¥la vvedena S. B. Steçkyn¥m [9] pry formulyrovke kryteryq absolgtnoj sxodymosty orto- honal\n¥x rqdov. Perv¥e ocenky velyçyn¥ σmf ()∞ dlq nekotor¥x konkret- n¥x funkcyj b¥ly poluçen¥ R. S. Ysmahylov¥m [10]. Neskol\ko pozΩe yssle- dovanyq velyçyn σmq F() dlq tex yly yn¥x funkcyonal\n¥x klassov provo- dylys\ v rabotax mnohyx avtorov (sm., naprymer, [1, 11 – 17]), hde moΩno oznako- myt\sq s bolee podrobnoj ynformacyej po dannomu voprosu. Sformulyruem nekotor¥e utverΩdenyq, neobxodym¥e dlq dal\nejßeho yz- loΩenyq. Teorema A [4]. Pust\ zadano p∈∞ (,) 1. Suwestvugt poloΩytel\n¥e kon- stant¥ Cp 1() y Cp 2() takye, çto dlq kaΩdoj funkcyy fLpd ∈() π spra- vedlyva ocenka CpffCpf ps s p p 1 2 12 2 ()()() / ≤⋅      ≤ ∈+ ∑ Z .(4) Sootnoßenye (4) qvlqetsq analohom yzvestnoho utverΩdenyq Lyttlvuda – Pπly. ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 1 106S. A. STASGK Teorema B [3]. Pust\ txce k ikx kn jj ()(,) = ≤ ∑, hde nj∈N, jd =1,, tohda pry 1 ≤ p < q ≤ ∞ ymeet mesto neravenstvo tnt q d j pq j d p ≤− = ∏211 1 //.(5) Neravenstvo (5) ustanovleno S. M. Nykol\skym y naz¥vaetsq „neravenstvom razn¥x metryk”. Lemma A [15]. Pust\ 2 < q < ∞. Tohda dlq lgboho tryhonometryçeskoho polynoma Px N (,) Θ y dlq lgboho M < N najdetsq tryhonometryçeskyj po- lynom Px M (,) Θ, dlq kotoroho ymeet mesto ocenka PPCqNMP NMqN (,)(,)()(,) ΘΘΘ ⋅−⋅≤⋅ −1 2, pryçem ΘΘ MN ⊂. Poluçenn¥e ocenky budem formulyrovat\ v termynax porqdkov¥x sootno- ßenyj. Esly dlq poloΩytel\n¥x funkcyj µµ 11 =() n, µµ 22 =() n natural\- noho arhumenta v¥polnqetsq neravenstvo µµ 112 ()() nCn ≤, hde C10 > — ne- kotoraq postoqnnaq, to πto sootnoßenye budem zapys¥vat\ v vyde µµ 12 ()() nn !. Esly Ωe ymegt mesto sootnoßenyq µµ 12 ()() nn ! y µ2() n ! ! µ1() n, to budem pysat\ µµ 12 ()() nn ". Otmetym, çto postoqnn¥e Cj, j = 1, 2, … , kotor¥e budut vstreçat\sq v rabote, mohut zavyset\ tol\ko ot paramet- rov, opredelqgwyx klass¥, metryky Lq y razmernosty prostranstva Rd. PreΩde çem perejty neposredstvenno k formulyrovke y dokazatel\stvu osnovnoho rezul\tata rabot¥ sdelaem sledugwye zameçanyq. Osnovnaq trudnost\ pry dokazatel\stve poluçennoho namy utverΩdenyq so- stoyt v ustanovlenyy ocenok sverxu velyçyn¥ σθ mp r q B() ,. Metod, razrabotan- n¥j R. A. DeVorom y V. N. Temlqkov¥m [1] dlq dokazatel\stva ocenok sverxu velyçyn¥ σθ mp r q B() ,, ne pozvolqet poluçyt\ sootvetstvugwye ocenky sverxu πtoj velyçyn¥ v sluçae maloj hladkosty. Poπtomu m¥ yspol\zuem metod, pred- loΩenn¥j ∏. S. Belynskym [14] v odnomernom sluçae pry yssledovanyy velyçyn nayluçßeho m-çlennoho tryhonometryçeskoho pryblyΩenyq klassov Bp r ,θ. Otmetym takΩe, çto poskol\ku v formulyruemom nyΩe utverΩdenyy para- metr θ prynymaet y predel\noe znaçenye θ = ∞, sohlasno prynqt¥m oboznaçe- nyqm v πtom utverΩdenyy soderΩytsq sootvetstvugwyj rezul\tat y dlq klas- sov S. M. Nykol\skoho Hp r. Ymeet mesto sledugwaq teorema. Teorema. Pust\ 1 < p ≤ 2 < q < ∞, 1 ≤ θ ≤ ∞. Tohda pry d pq 11 −     < r < d p v¥polnqetsq porqdkovoe sootnoßenye σθ mp r q qr dpq Bm () ," −−+     2 11 .(6) Dokazatel\stvo. Poskol\ku pravaq çast\ (6) ne zavysyt ot θ, a pry 1 ≤ ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 1 NAYLUÇÍEE m-ÇLENNOE TRYHONOMETRYÇESKOE PRYBLYÛENYE …107 ≤ θ ≤ ∞ ymegt mesto vloΩenyq Bp r ,1 ⊂ Bp r ,θ ⊂ Bp r ,∞, ocenku sverxu v (6) budem ustanavlyvat\ dlq klassov Bp r ,∞, a snyzu — dlq klassov Bp r ,1. DokaΩem snaçala ocenku sverxu. Pust\ m — proyzvol\noe natural\noe çyslo, a n∈N takovo, çto 2dn < m ≤ 21 dn() +. Pust\ fBp r ∈∞,. Yzvestno [4], çto v πtom sluçae fxfx s s ()() = ∈+ ∑ Z ,(7) y pry πtom, kak vydno yz (2), fsp rs ()⋅≤−2.(8) PryblyΩagwyj polynom, dostavlqgwyj dlq f trebuemug ocenku prybly- Ωenyq, budem podbyrat\ v vyde PxfxPx msN nsqn s n s (,)()(,) / ΘΘ =+ ≤< = − ∑ ∑ 2 0 1 ,(9) hde polynom¥ Px Ns (,) Θ budut postroen¥ dlq kaΩdoho „bloka” fx s() soh- lasno lemme A, a çysla Ns podberem v vyde Ns nd s d p r qnd p r =    + −    −−     2221 2,(10) hde [] a — celaq çast\ çysla a. Ubedymsq, çto pry takom v¥bore çysel Ns polynom (9) soderΩyt po porqd- ku ne bol\ße, çem m harmonyk. Dejstvytel\no, #kkkkjd d s j s s n =…≤<= {} − = − ∑(,,):max,, 1 1 0 1 221 + + Ns nsqn ≤< ∑ /2 ! 2 2 1222 2 dndn qnd p rs d p r q n +−    + −−     −     ≤< ∑ nsqn/2 ! ! 2 2 1222 22 dndn qnd p r qnd p r q n +−    + −−     −     ! 2dnm ", hde #M oboznaçaet kolyçestvo πlementov mnoΩestva M. Takym obrazom, uçyt¥vaq razloΩenye (7), sohlasno (9), neravenstvu Myn- kovskoho, a takΩe teoreme A budem ymet\ fPfP mqsN nsqn s ()(,)()(,) / ⋅−⋅⋅−⋅      ≤< ∑ ΘΘ ! 2 2  12/ q + + fII s qnsq () / ⋅=+ ≤<∞ ∑ 2 12.(11) Dlq ocenky slahaemoho I2 vospol\zuemsq neravenstvamy Mynkovskoho, razn¥x metryk y (8). V rezul\tate poluçym ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 1 108S. A. STASGK Iff sq qns ds pq sp qn 2 2 11 2 2 ≤⋅⋅ ≤<∞ −     ∑()() // ! ≤≤<∞ ∑ s ≤ ≤ 2 11 2 −−−         ≤<∞ ∑ srd pq qns / " 22 11 −−+     qdnr dpq " m qr dpq −−+     2 11 .(12) Çtob¥ ocenyt\ slahaemoe I1, vospol\zuemsq posledovatel\no neravenstvom Mynkovskoho, lemmoj A y neravenstvom razn¥x metryk. Podstavlqq vmesto Ns yx znaçenyq yz (10), ymeem IfP sN nsqnq s 1 2 22 12 =⋅−⋅ ≤< ∑()(,) // / Θ ≤ ≤ fP sNq nsqn s ()(, / / ⋅−⋅       ≤< ∑Θ2 2 12 ! ! 2 2 2 2 12 ds s s nsqnN f() / / ⋅       ≤< ∑ ! 222112 2 2 12 dsdsp s sp nsqnN f // / / () − () ≤< ⋅       ∑ ! ! 22 22 2 12 dsprs s nsqnN / / / − ≤< ∑       ≤ 222 24 2 −−    −     ≤< ∑    dnqnd p rs d p r nsqn/      12/ " " 222 244 −−     −     dnqnd p r qnd p r = 22 11 −−+     qdnr dpq " m qr dpq −−+     2 11 .(13) Takym obrazom, podstavlqq (12) y (13) v (11), poluçaem trebuemug ocenku sverxu dlq velyçyn¥ σθ mp r q B() ,. Perejdem k dokazatel\stvu v (6) ocenky snyzu. Dlq πtoho vospol\zuemsq dvojstvenn¥m sootnoßenyem, kotoroe v¥tekaet yz bolee obweho rezul\tata S. M. Nykol\skoho (sm., naprymer, [18, s. 25]) σ π mq PL P ffxPxdx m m q d ()infsup()() () = ∈ ≤ ⊥ ′ ∫ ΘΘ 1 ,(14) hde 11 qq + ′ = 1, a Lm ⊥() Θ — mnoΩestvo funkcyj, ortohonal\n¥x podprost- ranstvu tryhonometryçeskyx polynomov s „nomeramy” harmonyk yz mnoΩest- va Θm. Pust\ m — proyzvol\noe natural\noe çyslo, a n∈N, kak y pry ustanov- lenyy ocenky sverxu, v¥berem yz uslovyq 2dn < m ≤ 21 dn() +. Rassmotrym funkcyg Fxe qn ikx kj qn jd , (,) () / , = <[] = ∑ 22 1 ,(15) na osnovanyy kotoroj postroym funkcyg Px() yz (14). ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 1 NAYLUÇÍEE m-ÇLENNOE TRYHONOMETRYÇESKOE PRYBLYÛENYE …109 Pust\ Θm — proyzvol\n¥j nabor yz m vektorov s celoçyslenn¥my koor- dynatamy. PoloΩym gxFxe qn ikx km ()() , (,) * =− ∈ ∑ Θ , hde eikx km (,) * ∈ ∑ Θ — polynom, soderΩawyj tol\ko te slahaem¥e funkcyy Fx qn,(), kotor¥e ymegt „nomera” yz Θm. Poskol\ku (sm., naprymer, [10]) eik k q j l jd (,) , ⋅ < = ∑ 2 1 " 2 1 1 dl q −    , 1 < q < ∞,(16) pry 1 < ′ q < 2 sohlasno (16) naxodym gF qqnq ′′ ≤, + eik km (,) ⋅ ∈ ∑ Θ2 ! 22 1 1 dqn q − ′     + + mdndndn "" 222 222 /// +. Otsgda sleduet, çto funkcyq Px 1() = Cee dnikxikx k km j qn j 2 2 2 2 2 1 − ∈ < −∑ [] = /(,)(,) / , Θ dd ∑         (17) s sootvetstvugwej postoqnnoj C20 > udovletvorqet uslovyqm, soderΩa- wymsq v (14). V kaçestve fx() yz (14) v¥berem funkcyg fxCFx pn qn r d p d qn ,, ()() = −−+     3 2 2, C30 >,(18) y pokaΩem, çto s nekotoroj postoqnnoj C30 > ona prynadleΩyt klassu Bp r ,1. Dejstvytel\no, ysxodq yz (1), (15) y (16), poluçaem fpnBp r , ,1 = 2 0 rs pnsp s f () , = ∞ ∑ ! 22 2 0 2 −−+     = [] ∑ qn r d p d rs qnsp s qn F() , / " " 222 2 1 1 0 2 −−+     −     = [] ∑ qn r d p d rs ds p s qn/ " 22 22 −−+     −+     qn r d p d qn r d p d = 1. Takym obrazom, podstavlqq (17) y (18) v (14), ymeem σmpnq f () , ≥ inf()() , Θm d fxPxdx pn1 π ∫ " 22 22 2 2 −−+    − − () qn r d p ddn qn Fm , " ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 1 110S. A. STASGK " 222 222 −−+    − qn r d p ddnqdn = 22 11 −−+     qdnr dpq " m qr dpq −−+     2 11 . Ocenka snyzu ustanovlena. Teorema dokazana. Prokommentyruem poluçenn¥j rezul\tat. V pervug oçered\ otmetym, çto v sluçae d = 1 sootnoßenye (6) dokazano ∏. S. Belynskym [14]. Krome toho, porqdok velyçyn¥ σθ mp r q B() ,, v çastnosty, pry 1 < p ≤ 2 < q < < ∞, 1 ≤ θ ≤ ∞ y r d p > poluçen v rabote [1] y ymeet vyd σθ mp r q r dp Bm () ," −+− 11 2.(19) Takym obrazom, sopostavlqq (6) y (19), vydym otlyçye v ocenkax pryblyΩe- nyq velyçyn¥ σθ mp r q B() , pry perexode çerez tak naz¥vaem¥j krytyçeskyj po- kazatel\ hladkosty r d p =. Predstavlqetsq ynteresn¥m takΩe sravnenye poluçennoj namy ocenky dlq σθ mp r q B() , s nayluçßym m-çlenn¥m ortohonal\n¥m tryhonometryçeskym pry- blyΩenyem klassov Bp r ,θ. Napomnym opredelenye πtoj velyçyn¥ y sformuly- ruem sootvetstvugwyj rezul\tat. Pust\ Λm d ⊂Z — koneçnoe mnoΩestvo, soderΩawee m πlementov, t. e. #Λm = m. Dlq fLqd ∈() π, 1 ≤ q ≤ ∞, poloΩym Sfxfke m m ikx k Λ Λ (,)()(,) = ∈ ∑" y rassmotrym velyçynu effSf mqq m m ⊥=⋅−⋅ ()inf()(,) ΛΛ. Esly FLqd ⊂() π — nekotor¥j klass funkcyj, to polahaem eFef mq fF mq ⊥ ∈ ⊥ = ()sup(). Velyçynu eF mq ⊥() naz¥vagt nayluçßym m-çlenn¥m ortohonal\n¥m tryhono- metryçeskym pryblyΩenyem klassa F v prostranstve Lq. Zametym, çto soh- lasno opredelenyqm σmq F() y eF mq ⊥() svqzan¥ neravenstvom σmq F() ≤ ≤ eF mq ⊥(). V [7] poluçen sledugwyj rezul\tat. Teorema V. Pust\ 1 ≤ p, q, θ ≤ ∞, (,) pq ≠ (1, 1), (,) ∞∞. Tohda pry r > > d pq 11 −    + spravedlyvo sootnoßenye ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 1 NAYLUÇÍEE m-ÇLENNOE TRYHONOMETRYÇESKOE PRYBLYÛENYE …111 eBm mp r q r dpq ⊥ −+−    + () ,θ" 11 ,(20) hde aa +={} max;0. Takym obrazom, sopostavlqq (6) y (20), vydym, çto povedenye velyçyn σθ mp r q B() , y eB mp r q ⊥() ,θ (v sm¥sle yx porqdkov¥x znaçenyj) pry v¥polnenyy uslovyj dokazannoj teorem¥ razlyçno pry m → ∞. 1.De Vore R. A., Temlyakov V. N. Nonlinear approximation by trigonometric sums // J. Fourier Anal. and Appl. – 1995. – 2, # 1. – P. 29 – 48. 2.Besov O. V. 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PryblyΩenye „plavagwej” systemoj πksponent na klassax peryodyçeskyx funkcyj s ohranyçennoj smeßannoj proyzvodnoj // Yssledovanyq po teoryy funkcyj mno- hyx vewestvenn¥x peremenn¥x. – Qroslavl\: Qroslav. un-t, 1988. – S. 16 – 33. 16.Kaßyn B. S., Temlqkov V. N. O nayluçßyx m-çlenn¥x pryblyΩenyqx y πntropyy mno- Ωestv v prostranstve L1 // Mat. zametky. – 1994. – 56, # 5. – S. 57 – 86. 17.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. 18.Kornejçuk N. P. ∏kstremal\n¥e zadaçy teoryy pryblyΩenyj. – M.: Nauka, 1976. – 320 s. Poluçeno 27.03.09, posle dorabotky — 26.10.09 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 1
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institution	Ukrains’kyi Matematychnyi Zhurnal
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language	rus English
last_indexed	2026-03-24T02:31:29Z
publishDate	2010
publisher	Institute of Mathematics, NAS of Ukraine
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spelling	umjimathkievua-article-28462020-03-18T19:38:45Z Best $m$-term trigonometric approximation for the classes $B^r_{p,θ}$ of functions of low smoothness Наилучшее $m$-членное тригонометрическое приближение классов $B^r_{p,θ}$ функций малой гладкости Stasyuk, S. A. Стасюк, С. А. Стасюк, С. А. We obtain an exact-order estimate for the best $m$-term trigonometric approximation of the Besov classes $B^r_{p,θ}$ of periodic functions of many variables of low smoothness in the space $L_q, \; 1 < p ≤ 2 < q < ∞$. Одержано точну за порядком оцінку найкращого $m$-членного тригонометричного наближення класів Бесова $B^r_{p,θ}$ в періодичних функцій багатьох змінних малої гладкості у просторі $L_q, \; 1 < p ≤ 2 < q < ∞$. Institute of Mathematics, NAS of Ukraine 2010-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2846 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 1 (2010); 104–111 Український математичний журнал; Том 62 № 1 (2010); 104–111 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2846/2444 https://umj.imath.kiev.ua/index.php/umj/article/view/2846/2445 Copyright (c) 2010 Stasyuk S. A.
spellingShingle	Stasyuk, S. A. Стасюк, С. А. Стасюк, С. А. Best $m$-term trigonometric approximation for the classes $B^r_{p,θ}$ of functions of low smoothness
title	Best $m$-term trigonometric approximation for the classes $B^r_{p,θ}$ of functions of low smoothness
title_alt	Наилучшее $m$-членное тригонометрическое приближение классов $B^r_{p,θ}$ функций малой гладкости
title_full	Best $m$-term trigonometric approximation for the classes $B^r_{p,θ}$ of functions of low smoothness
title_fullStr	Best $m$-term trigonometric approximation for the classes $B^r_{p,θ}$ of functions of low smoothness
title_full_unstemmed	Best $m$-term trigonometric approximation for the classes $B^r_{p,θ}$ of functions of low smoothness
title_short	Best $m$-term trigonometric approximation for the classes $B^r_{p,θ}$ of functions of low smoothness
title_sort	best $m$-term trigonometric approximation for the classes $b^r_{p,θ}$ of functions of low smoothness
url	https://umj.imath.kiev.ua/index.php/umj/article/view/2846
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Best $m$-term trigonometric approximation for the classes $B^r_{p,θ}$ of functions of low smoothness

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