On one extremal problem for positive series
The approximation properties of the spaces $S^p_{\varphi}$ introduced by Stepanets’ were studied in a series of works of Stepanets’ and his disciples. In these works, problems related to the determination of exact values of $n$-term approximations of $q$-ellipsoids in these spaces were reduced to so...
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| Дата: | 2005 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3718 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509849755844608 |
|---|---|
| author | Stepanets, O. I. Shydlich, A. L. Степанець, О. І. Шидліч, А. Л. |
| author_facet | Stepanets, O. I. Shydlich, A. L. Степанець, О. І. Шидліч, А. Л. |
| author_sort | Stepanets, O. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:02:57Z |
| description | The approximation properties of the spaces $S^p_{\varphi}$ introduced by Stepanets’ were studied in a series of works of Stepanets’ and his disciples. In these works, problems related to the determination of exact values of $n$-term approximations of $q$-ellipsoids in these spaces were reduced to some extremal problems for series with terms that are products of elements of two nonnegative sequences one of which is fixed and the other varies on a certain set.
Since solutions of these extremal problems may be of independent interest, in the present work we develop a new method for finding these solutions that enables one to obtain the required result in a substantially shorter and more transparent way. |
| first_indexed | 2026-03-24T02:47:39Z |
| format | Article |
| fulltext |
UDK 517.5
O. I. Stepanec\, A. L. Íydliç (In-t matematyky NAN Ukra]ny, Ky]v)
PRO ODNU EKSTREMAL|NU ZADAÇU
DLQ DODATNYX RQDIV
In the series of works, O. I. Stepanets’ and his successors study approximation properties of the spaces
S p
ϕ introduced by Stepanets’. Problems of finding exact values of n-term approximations of q-
ellipsoids in the spaces considered are reduced to some extremal problems for series with terms that are
determined as a product of elements of two nonnegative sequences one of which is fixed and another
varies on certain set.
Since solutions of these extremal problems may be of an independent interest, in the present paper,
the authors propose a new method of finding these solutions ,which leads to the required result by
essentially shorter and more transparent way.
U cykli robit O. I. Stepancq ta joho poslidovnykiv vyvçagt\sq aproksymacijni vlastyvosti vve-
denyx nym prostoriv S p
ϕ . Pry c\omu zadaçi, pov’qzani iz znaxodΩennqm toçnyx znaçen\ n-çlen-
nyx nablyΩen\ q-elipso]div u cyx prostorax, zvodqt\sq do pevnyx ekstremal\nyx zadaç dlq rq-
div iz çlenamy, wo [ dobutkom elementiv dvox nevid’[mnyx poslidovnostej, odna z qkyx [ fikso-
vanog, a inßa varig[t\sq na pevnij mnoΩyni.
ZvaΩagçy na te, wo rozv’qzky cyx ekstremal\nyx zadaç moΩut\ skladaty i samostijnyj
interes, u danij roboti zaproponovano novyj metod znaxodΩennq ]x rozv’qzkiv, qkyj pryvodyt\
do mety sutt[vo korotßym i prozorißym ßlqxom.
Vstup. Nexaj M — mnoΩyna vsix poslidovnostej m = { } =
∞mk k 1 nevid’[mnyx
çysel, mk ≥ 0 ∀k ∈ N, takyx, wo | m | =df
mkk =
∞∑ 1
≤ 1.
Nexaj, dali, r — deqke dodatne çyslo i Ar — mnoΩyna vsix nezrostagçyx
poslidovnostej α = { } =
∞αk k 1 dodatnyx çysel, αk > 0 ∀k ∈ N, dlq qkyx
lim
k
k
→∞
α = 0. Pry c\omu u vypadku, koly r ∈ ( 0, 1 ), vymaha[t\sq, wob
αk
r
k
1 1
1
/ ( − )
=
∞∑ < ∞.
Poznaçymo çerez γn = { k1 , k2 , … , kn } dovil\nyj nabir iz n riznyx natural\-
nyx çysel i poklademo dlq bud\-qkyx m ∈ M , α ∈ Ar i n ∈ N
�n ( m ) = �n ( α, r, m ) =df
α α
γ γ
k k
r
k
k k
r
k
m m
n n=
∞
∈
∑ ∑−
1
sup , (1)
�n = �n ( α, r ) =df
sup
m∈M
�n ( m ) =
sup
m∈M
�n ( α, r, m ). (2)
Z oznaçennq mnoΩyn M i A r vyplyva[, wo rqd u pravij çastyni spivvidno-
ßennq (1) zbiha[t\sq, i tomu velyçyny �n ( α, r, m ) i �n ( α, r ) magt\ zmist.
U cykli robit O. I. Stepancq ta joho poslidovnykiv (dyv., napryklad, [1 – 19])
vyvçagt\sq aproksymacijni vlastyvosti vvedenyx nym prostoriv S p
ϕ . Pry c\omu
zadaçi, pov’qzani iz znaxodΩennqm toçnyx znaçen\ najkrawyx n-çlennyx nably-
Ωen\ q-elipso]div u cyx prostorax, bazugt\sq na nastupnyx tverdΩennqx.
Teorema 1. Nexaj α ∈ Ar
, r ≥ 1, i �n r( )α, — velyçyna, qka vyznaça[t\sq
rivnistg (2). Todi dlq bud\-qkoho natural\noho n isnu[ çyslo s* > n take,
wo
�n ( α, r ) = ( − )
−
=
−
∑s n k
r
k
s
r
* /
*
α 1
1
. (3)
© O. I. STEPANEC|, A. L. ÍYDLIÇ, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12 1677
1678 O. I. STEPANEC|, A. L. ÍYDLIÇ
Çyslo s*
vyznaça[t\sq rivnistg
sup / * /
*
s n
k
r
k
s r
k
r
k
s
r
s n s n
>
−
=
−
−
=
−
( − )
= ( − )
∑ ∑α α1
1
1
1
. (4)
Pry c\omu toçnu verxng meΩu u pravij çastyni spivvidnoßennq (2) realizu[ po-
slidovnist\ m* = { } =
∞mk k
*
1, v qkij
m k s
k s
k
k
r
i
r
i
s
*
/ / *
*
*
, , ,
, .
=
∈ [ ]
>
−
=
−
∑α α1 1
1
1
1
0
(5)
Teorema 2. Nexaj α ∈ Ar
, r ∈ ( 0, 1 ), i �n r( )α, — velyçyna, qka vyzna-
ça[t\sq rivnistg (2). Todi pry koΩnomu n ∈ N spravdΩu[t\sq rivnist\
�n = �n ( α, r ) = ( − )
+
( − ) −
=
( − )
( − )
= +
∞
−
∑ ∑s n r
k
r
k
s
r r
k
r
k s
r
* / /
/
/
*
*
1 1 1
1
1
1 1
1
1
α α , (6)
u qkij s*
, s* > n, — najbil\ße natural\ne çyslo, qke zadovol\nq[ umovu
s – n ≤ α αs
r
k
r
k
s
1 1
1
/ /−
=
∑ dlq vsix s ∈ ( n, s*
]. (7)
Take çyslo s*
zavΩdy isnu[. Toçnu verxng meΩu u pravij çastyni spivvidnoßen-
nq (2) realizu[ poslidovnist\ m* = { } =
∞mk k
*
1 z M , v qkij
m s n k s
k s
k
k
r r
i
r
i
s
r
n
r
k
r
n
r
*
/ * / /
/
/ *
/ / *
*
, , ,
, .
= ( − ) ( ) ∈ [ ]
>
− ( − ) −
=
( − )
( − )
( − ) ( − )
∑α α
α
1 1 1 1
1
1 1
1 1
1 1 1 1
1�
�
(8)
Ci teoremy dovedeno vidpovidno v [2, 4]. ZvaΩagçy na te, wo ci tverdΩennq
moΩut\ maty i samostijnyj interes, u danij roboti navodqt\sq novi dovedennq
zhadanyx teorem, qki [ znaçno korotßymy ta prozorißymy.
1. Dovedennq teoremy 1. Spoçatku vstanovymo nastupne tverdΩennq.
TverdΩennq 1. Nexaj α ∈ Ar
, r > 0, i M ′ — mnoΩyna vsix poslidovnos-
tej m z mnoΩyny M , dlq qkyx çysla αk k
rm ne zrostagt\. Todi spravdΩu-
[t\sq rivnist\
� =
sup
m∈ ′M
� ( m ). (9)
Dovedennq. PokaΩemo, wo dlq bud\-qko] poslidovnosti m ∈ M znajdet\sq
poslidovnist\ m′ ∈ M ′ taka, wo �n m( ′) ≥ �n m( ) i | m′ | ≤ | m |. Z ci[g metog
poklademo ′ =
m
m
k
k k
r
k
r
α
α
1/
, k ∈ N, de αk k
rm — perestanovka poslidovnosti
αk k
rm za nezrostannqm (za oznaçennqm mnoΩyn Ar i M taka perestanovka zav-
Ωdy isnu[). Todi oçevydno, wo �n ( m′ ) = �n ( m ). Krim toho, na pidstavi teoremy
368 iz monohrafi] [20]
| m′ | =
α
α
k k
r
k
r
k
k
k
m
m
≤
=
∞
=
∞
∑ ∑
1
1 1
/
= | m |,
tobto poslidovnist\ m′ naleΩyt\ mnoΩyni M ′.
TverdΩennq 1 dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
PRO ODNU EKSTREMAL|NU ZADAÇU DLQ DODATNYX RQDIV 1679
Zhidno z oznaçennqm mnoΩyny M ′ dlq bud\-qko] poslidovnosti m ∈ M ′
sup
γ γ α
n
n
k k
r
k
m∈∑ = αk k
r
k
n
m=∑ 1
, a tomu �n m( ) = αk k
r
k n
m= +
∞∑ 1
. Zvidsy vyply-
va[, wo pry vidßukanni velyçyn �n dostatn\o obmeΩytysq poslidovnostqmy
m ∈ M ′, dlq qkyx çysla αk k
rm [ rivnymy pry vsix k ∈ [ 1, n + 1 ]. Oskil\ky pry
c\omu
α αk k
r
k
k
n r
k
r
k
n r
m m=
=
+
−
=
+ −
∑ ∑
1
1
1
1
1
/ , k ∈ [ 1, n + 1 ],
to
�n ( m ) = ( )( + ) −
+ +
=
+
−
=
+ −
+ +
= +
∞
∑ ∑ ∑n n m m mk
k
n r
k
r
k
n r
n n
r
k k
r
k n
1
1
1
1
1
1
2 2
3
α α α/
.
Poklademo
x1 = mk
k
n
=
+
∑
1
1
, c = mk
k
n
=
+
∑
1
2
,
(10)
a1 = αk
r
k
n r
−
=
+ −
∑
1
1
1
/ , a2 = αn + 2 , b = ( n + 1 ) – n
i na vidrizku [ 0, c ] rozhlqnemo funkcig f ( x ) = a1 b xr + a2 ( c – x )
r
za umovy, wo
a1 xr ≥ a2 ( c – x )
r
, qku moΩna zapysaty u vyhlqdi c ≥ x ≥ a c a ar r r
2
1
1
1
2
1 1/ / /( + )− =df
=df
x0
. Vnaslidok toho, wo f ′′ ( x ) > 0 dlq bud\-qkyx x ∈ [ 0, c ], [dyna krytyçna
toçka ci[] funkci], qka ma[ vyhlqd x* = ca a b ar r r
2
1 1
1
1 1
2
1 1 1/ / /( − ) ( − ) ( − ) −( )( ) + , [ toç-
kog minimumu.
Zvidsy vyplyva[, wo dana funkciq dosqha[ svoho najbil\ßoho na vidrizku
[ x0
, c ] znaçennq na odnomu z kinciv c\oho vidrizka. Tomu
f ( x ) ≤ max { f ( x0 ), f ( c ) } ∀x ∈ [ x0
, c ]. (11)
Oskil\ky vnaslidok oznaçennq mnoΩyny M ′ x1 ∈ [ x0
, c ], to, vraxovugçy poz-
naçennq v (10), otrymu[mo
� ( m ) = f ( x1 ) + αk k
r
k n
m
= +
∞
∑
3
≤ max { f ( x0 ), f ( c ) } + αk k
r
k n
m
= +
∞
∑
3
=
= m mk
k
n r
n n k k
r
k n=
+
+ +
= +
∞
∑ ∑
{ } +
1
2
1 2
3
max ,ξ ξ α , ξs = ( − )
−
=
−
∑s n k
r
k
s r
α 1
1
/
.
Dali, poklademo
x1 = mk
k
n
=
+
∑
1
2
, c = mk
k
n
=
+
∑
1
3
, a1 = αk
r
k
l r
−
=
−
∑
1
1
/ , a2 = αn + 3 , b = l – n,
de l — natural\ne çyslo, dlq qkoho ξl = max ,{ }+ +ξ ξn n1 2 . Z toho, wo m ∈ M ′,
vyplyva[
a x m mr
k
k
n r
k
r
k
l r
n n
r
1 1
1
2
1
1
3 3≥
≥
=
+
−
=
−
+ +∑ ∑α α/ = a2 ( c – x1 )
r
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1680 O. I. STEPANEC|, A. L. ÍYDLIÇ
tobto toçka x1 znovu naleΩyt\ vidrizku [ x0
, c ]. Zvidsy na pidstavi (11) otry-
mu[mo
�( )m ≤ m mk
k
n r
n n k k
r
k n=
+
+ +
= +
∞
∑ ∑
{ } +
1
2
1 2
3
max ,ξ ξ α = f ( x1 ) + αk k
r
k n
m
= +
∞
∑
4
≤
≤ max { f ( x0 ), f ( c ) } + αk k
r
k n
m
= +
∞
∑
4
=
= m mk
k
n r
n n n k k
r
k n=
+
+ + +
= +
∞
∑ ∑
{ } +
1
3
1 2 3
4
max , ,ξ ξ ξ α .
Provodqçy analohiçni mirkuvannq dali, vstanovlg[mo, wo dlq dovil\no] posli-
dovnosti m ∈ M ′
� ( m ) ≤ | m |
r sup
s n
s
>
ξ ≤ sup /
s n
k
r
k
s r
s n
>
−
=
−
( − )
∑α 1
1
. (12)
Zhidno z oznaçennqm mnoΩyny Ar
, αk → 0 pry k → ∞ , i v danomu vypadku
r ≥ 1, tomu
lim /
s
k
r
k
s r
s n
→∞
−
=
−
( − )
∑α 1
1
= 0.
Zvidsy vyplyva[, wo znajdet\sq prynajmni odne natural\ne çyslo s*
, qke zado-
vol\nq[ spivvidnoßennq (4). Takym çynom, z uraxuvannqm (12) ma[mo
�n ≤ ( − )
−
=
−
∑s n k
r
k
s
r
* /
*
α 1
1
, (13)
i dlq zaverßennq dovedennq teoremy zalyßylosq pokazaty, wo v c\omu spivvid-
noßenni stroho] nerivnosti buty ne moΩe. Z ci[g metog zauvaΩymo, wo posli-
dovnist\ m* = { } =
∞mk k
*
1 vyhlqdu (5) naleΩyt\ mnoΩyni M ′ i
�n ( m*
) = ( − )
−
=
−
∑s n k
r
k
s
r
* /
*
α 1
1
.
Ob’[dnugçy ostann[ spivvidnoßennq zi spivvidnoßennqm (13), zaverßu[mo dove-
dennq teoremy 1.
2. Dovedennq teoremy 2. Na pidstavi tverdΩennq 1 dlq vidßukannq velyçy-
ny �n i v c\omu vypadku dostatn\o obmeΩytysq mnoΩynog M ′ vsix poslidov-
nostej m z mnoΩyny M , dlq qkyx çysla αk k
rm ne zrostagt\.
Dlq bud\-qko] poslidovnosti m ∈ M ′
�n ( m ) = αk k
r
k n
m
= +
∞
∑
1
= ν αk k k
r
k
m
=
∞
∑
1
, νk =df
0 1
1
, , ,
, .
k n
k n
∈ [ ]
>
(14)
Teper zaznaçymo, wo nerivnist\ (7) rivnosyl\na nerivnostqm
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
PRO ODNU EKSTREMAL|NU ZADAÇU DLQ DODATNYX RQDIV 1681
( − − )
≤−
=
− −
∑s n k
r
k
s
s
r1 1
1
1 1
1α α/ / ,
(15)
( − − )
≤ ( − )
−
=
− −
−
=
−
∑ ∑s n s nk
r
k
s
k
r
k
s
1 1
1
1 1
1
1
1
α α/ /
,
i oskil\ky αk → ∞ pry k → ∞, to
lim lim/ /
s
k
r
k
s
s
k
r
k
s
s n s
→∞
−
=
−
→∞
−
=
−
( − )
≤
∑ ∑α α1
1
1
1
1
1
= 0.
Tomu zavΩdy znajdet\sq najbil\ße çyslo s* = s*
( α, n ) take, wo pry vsix s ∈
∈ ( n, s*
] budut\ vykonuvatysq nerivnosti (15), a otΩe, i (7).
TverdΩennq 2. Nexaj M ′′ — mnoΩyna vsix poslidovnostej m z mnoΩy-
ny M ′, dlq qkyx çysla αk k
rm [ rivnymy pry vsix k ∈ [ 1, s*
]. Todi spravd-
Ωu[t\sq rivnist\
�n =
sup
m∈ ′′M
�n ( m ). (16)
Dovedennq. Qk i pry dovedenni tverdΩennq 1, pokaΩemo, wo dlq koΩno]
poslidovnosti m ∈ M ′ znajdet\sq poslidovnist\ m′′ ∈ M ′ taka, wo | m′′ | ≤
≤ | m | i
�n ( m′′ ) ≥ �n ( m ). (17)
Z ci[g metog rozhlqnemo poslidovnist\ m ′′ taku, wo pry k > s* ′′mk = mk , a
pry k ∈ [ 1, s*
] znaçennq ′′mk vyznaçagt\sq spivvidnoßennqm
α αk k
r
k
k
s
r
k
r
k
s
r
m m( ′′) =
=
−
=
−
∑ ∑
1
1
1
* *
/
. (18)
Dlq ci[] poslidovnosti
| m′′ | = α αk
r
k
s
k
k
s
k
r
k
s
k
k s
m m−
= =
−
=
−
= +
∞
∑ ∑ ∑ ∑
+1
1 1
1
1
1
1
/ /
* * *
*
= | m |. (19)
Dlq dovedennq nerivnosti (17) dostatn\o pokazaty, wo
ν α ν αk k k
r
k
s
k
k
s
k
k
s
r
k
r
k
s
r
m m
= = =
−
=
−
∑ ∑ ∑ ∑≤
1 1 1
1
1
* * * *
/
. (20)
PokaΩemo spoçatku, wo
ν α ν α αk k k
r
k
s
k
k
s
i
r
i
s
j
r r
j
r
j
s
m m
= =
−
=
−
( − )
=
∑ ∑ ∑ ∑≤
1 1
1
1
1
1
1
* * * *
/ /
. (21)
Todi, zastosuvavßy do ostann\o] sumy nerivnist\ Hel\dera, otryma[mo (20). Pry
c\omu vykorysta[mo pryjom, qkyj bulo zaproponovano dlq dovedennq vidomo]
nerivnosti Çebyßova dlq intehraliv u monohrafi] [21, s. 246]. Oskil\ky
α ν α αk
r
k
s
k k
r
i
r
i
s
s n−
=
−
=
−
∑ ∑− ( − )
1
1
1 1
1
1
/ / * /
* *
= 0,
to rivnist\
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1682 O. I. STEPANEC|, A. L. ÍYDLIÇ
ν α ν α αk k k
r
k
s
k
k
s
i
r
i
s
j
r r
j
r
j
s
m m
= =
−
=
−
( − )
=
∑ ∑ ∑ ∑−
1 1
1
1
1
1
1
* * * *
/ /
=
= α α ν α αk
r
k
s
k k
r
k k
r
i
r
i
s
m c s n−
=
−
=
−
∑ ∑( − ) − ( − )
1
1
1 1
1
1
/ / * /
* *
(22)
vykonu[t\sq dlq bud\-qkoho dijsnoho çysla c.
Pokladagçy c* = αn n
rm+ +1 1, na pidstavi (7) ta oznaçennq mnoΩyny M ′, dlq
vsix k ∈ [ 1, s*
] ma[mo
( − ) − ( − )
−
=
−
∑α ν α αk k
r
k k
r
i
r
i
s
m c s n* / * /
*
1 1
1
1
≤ 0,
zvidky vyplyva[ spivvidnoßennq (21), a razom z nym, na pidstavi oznaçennq posli-
dovnosti m′′, i nerivnist\ (17).
Z oznaçennq mnoΩyny M ′′ vyplyva[, wo dlq koΩno] poslidovnosti m ∈ M ′′
pry vsix k ∈ [ 1, s*
]
α αk k
r
k
k
s
r
k
r
k
s
r
m m=
=
−
=
−
∑ ∑
1
1
1
* *
/
. (23)
Tomu
�n ( m ) = ( − )
+
=
−
=
−
= +
∞
∑ ∑ ∑s n m mk
k
s
r
k
r
k
s
r
k k
r
k s
* /
* *
*1
1
1 1
α α . (24)
Zastosovugçy do vyrazu v pravij çastyni (24) nerivnist\ Hel\dera, ma[mo
�n ( m ) ≤ m s nk
k
r
r
k
r
k
s
r r
k
r
k s
r
=
∞
( − ) −
=
( − )
( − )
= +
∞
−
∑ ∑ ∑
( − )
+
1
1 1 1
1
1
1 1
1
1
* / /
/
/
*
*
α α .
Zvidsy vnaslidok toho, wo | m | ≤ 1, otrymu[mo potribnu ocinku zverxu velyçy-
nyNN�n .
Rozhlqnemo teper poslidovnist\ m*
, qka vyznaça[t\sq spivvidnoßennqm (8),
i pokaΩemo, wo m* ∈ M ′′. Spravdi, vnaslidok (8) pry koΩnomu k ∈ [ 1, s*
]
α αk k
r r r
k
r
k
s
r r
n
r rm s n( ) = ( − )
( − ) −
=
( − )
( − )∑* * / /
/
/
*
1 1
1
1
1� , (25)
a pry k > s*
α αk k
r
k
r
n
r rm( ) = ( − ) ( − )* / /1 1 1� , (26)
tobto poslidovnist\ αk k
rm( )*
na promiΩku [ 1, s*
] [ stalog, a na promiΩku
[ s*, ∞ ) — ne zrosta[. Tomu dostatn\o pokazaty, wo
α α
s s
r
s s
rm m* * * *
* *( ) ≥ ( )+ +1 1
, (27)
abo
( − )
≥( − ) −
=
( − )
+
( − )∑s n r r
k
r
k
s
r r
s
r* / /
/
/
*
*
1 1
1
1
1
1 1α α .
Ostannq nerivnist\ vyplyva[ z oznaçennq çysla s*
. Takym çynom, poslidovnist\
m*
naleΩyt\ mnoΩyni M ′′. Krim toho, vnaslidok (25) – (27)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
PRO ODNU EKSTREMAL|NU ZADAÇU DLQ DODATNYX RQDIV 1683
�n m( )* = ( − )
+
( − ) −
=
( − )
( − )
= +
∞
−
∑ ∑s n r
k
r
k
s
r r
k
r
k s
r
* / /
/
/
*
*
1 1 1
1
1
1 1
1
1
α α .
1. Stepanec A. Y. Approksymacyonn¥e xarakterystyky prostranstv S p
ϕ // Ukr. mat. Ωurn. –
2001. – 53, # 3. – S. 392 – 416.
2. Stepanec A. Y. Approksymacyonn¥e xarakterystyky prostranstv S p
ϕ v razn¥x metrykax
// Tam Ωe. – # 8. – S. 1121 – 1146.
3. Vojcexivs\kyj V. R. Nerivnosti typu DΩeksona pry nablyΩenni funkcij z prostoru S p
sumamy Zyhmunda // Teoriq nablyΩennq funkcij ta sumiΩni pytannq: Pr. In-tu matematyky
NAN Ukra]ny. – 2002. – 35. – S. 33 – 46.
4. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj: V 2 ç. // Pr. In-tu matematyky NAN Ukra]ny.
– 2002. – 40, ç. II. – S. 333 – 368.
5. Stepanec A. Y. Approksymacyonn¥e xarakterystyky prostranstv S p
// Teoriq nablyΩen\
ta harmonijnyj analiz: Pr. Ukr. mat. konhresu-2001. – Ky]v: In-t matematyky NAN Ukra]ny,
2002. – S. 208 – 226.
6. Stepanec A. Y., Serdgk A. S. Prqm¥e y obratn¥e teorem¥ teoryy pryblyΩenyj funkcyj
v prostranstve S p
// Ukr. mat. Ωurn. – 2002. – 54, # 1. – S. 106 – 124.
7. Vakarçuk S. B. O nekotor¥x πkstremal\n¥x zadaçax teoryy approksymacyy v prostran-
stvax S p
( 1 ≤ p < ∞ ) // VoroneΩ. zym. mat. ßkola „Sovremenn¥e metod¥ teoryy funkcyj y
smeΩn¥e problem¥” (VoroneΩ, 26 qnv. – 2 fevr. 2003 h.). – VoroneΩ: VoroneΩ. un-t, 2003. –
S. 47 – 48.
8. Vojcexivs\kyj V. R. Popereçnyky deqkyx klasiv z prostoru S p
// Ekstremal\ni zadaçi teo-
ri] funkcij ta sumiΩni pytannq: Pr. In-tu matematyky NAN Ukra]ny. – 2003. – 46. – S. 17 –
26.
9. Rukasov V. Y. Nayluçßye n-çlenn¥e pryblyΩenyq v prostranstvax s nesymmetryçnoj
metrykoj // Ukr. mat. Ωurn. – 2003. – 55, # 6. – S. 806 – 816.
10. Serdgk A. S. Popereçnyky v prostori S p
klasiv funkcij, wo oznaçagt\sq modulqmy ne-
perervnosti ]x ψ-poxidnyx // Ekstremal\ni zadaçi teori] funkcij ta sumiΩni pytannq: Pr. In-
tu matematyky NAN Ukra]ny. – 2003. – 46. – S. 229 – 248.
11. Stepanec A. Y. ∏kstremal\n¥e zadaçy teoryy pryblyΩenyj v lynejn¥x prostranstvax //
Ukr. mat. Ωurn. – 2003. – 55, # 10. – S. 1392 – 1423.
12. Stepanec A. Y., Rukasov V. Y. Prostranstva S p
s nesymmetryçnoj metrykoj // Tam Ωe. –
2002. – 54, # 2. – S. 264 – 277.
13. Stepanec A. Y., Rukasov V. Y. Nayluçßye „sploßn¥e” n-çlenn¥e pryblyΩenyq v pros-
transtvax S p
ϕ // Tam Ωe. – 2003. – 55, # 5. – S. 663 – 670.
14. Stepanec A. Y., Íydlyç A. L. Nayluçßye n-çlenn¥e pryblyΩenyq Λ -metodamy v pros-
transtvax S p
ϕ // Tam Ωe. – # 8. – S. 1107 – 1126.
15. Íydliç A. L. Najkrawi n-çlenni nablyΩennq Λ-metodamy v prostorax S p
ϕ // Ekstremal\-
ni zadaçi teori] funkcij ta sumiΩni pytannq: Pr. In-tu matematyky NAN Ukra]ny. – 2003. –
46. – S. 283 – 306.
16. Vakarçuk S. B. Neravenstvo typa DΩeksona y toçn¥e znaçenyq popereçnykov klassov fun-
kcyj v prostranstvax S p
, 1 ≤ p < ∞ // Ukr. mat. Ωurn. – 2004. – 56, # 5. – S. 595 – 605.
17. Stepanec A. Y. Nayluçßye pryblyΩenyq q-πllypsoydov v prostranstvax S p
ϕ
µ,
// Tam Ωe.
– # 10. – S. 1378 – 1383.
18. Íydliç A. L. Pro nasyçennq linijnyx metodiv pidsumovuvannq rqdiv Fur’[ u prostorax S p
ϕ
// Tam Ωe. – # 1. – S. 133 – 138.
19. Stepanec A. Y. Nayluçßye n-çlenn¥e pryblyΩenyq s ohranyçenyqmy // Tam Ωe. – 2005. –
57, # 4. – S. 533 – 553.
20. Xardy H. H., Lyttl\vud D. E., Polya H. Neravenstva. – M.: Yzd-vo ynostr. lyt., 1948. –
456Ns.
21. Mitrimovic D. S., Pecaric J Eˆ . . , Fink A. M. Classical and new inequalities in analysis. –
Dordrecht: Kluwer, 1993. – 740 p.
OderΩano 23.06.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
|
| id | umjimathkievua-article-3718 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:47:39Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3b/c5e2958d094b95578bfc1faa061cfc3b.pdf |
| spelling | umjimathkievua-article-37182020-03-18T20:02:57Z On one extremal problem for positive series Про одну екстремальну задачу для додатних рядів Stepanets, O. I. Shydlich, A. L. Степанець, О. І. Шидліч, А. Л. The approximation properties of the spaces $S^p_{\varphi}$ introduced by Stepanets’ were studied in a series of works of Stepanets’ and his disciples. In these works, problems related to the determination of exact values of $n$-term approximations of $q$-ellipsoids in these spaces were reduced to some extremal problems for series with terms that are products of elements of two nonnegative sequences one of which is fixed and the other varies on a certain set. Since solutions of these extremal problems may be of independent interest, in the present work we develop a new method for finding these solutions that enables one to obtain the required result in a substantially shorter and more transparent way. У циклі робіт O. I. Степанця та його послідовників вивчаються апроксимаційні властивості введених ним просторів $S^p_{\varphi}$. При цьому задачі, пов'язані із знаходженням точних значень $n$-членних наближень $q$-еліпсоїдів у цих просторах, зводяться до певних екстремальних задач для рядів із членами, що є добутком елементів двох невід'ємних послідовностей, одна з яких є фіксованою, а інша варіюється на певній множині. Зважаючи на те, що розв'язки цих екстремальних задач можуть складати і самостійний інтерес, у даній роботі запропоновано новий метод знаходження їх розв'язків, який приводить до мети суттєво коротшим і прозорішим шляхом. Institute of Mathematics, NAS of Ukraine 2005-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3718 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 12 (2005); 1677–1683 Український математичний журнал; Том 57 № 12 (2005); 1677–1683 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3718/4161 https://umj.imath.kiev.ua/index.php/umj/article/view/3718/4162 Copyright (c) 2005 Stepanets O. I.; Shydlich A. L. |
| spellingShingle | Stepanets, O. I. Shydlich, A. L. Степанець, О. І. Шидліч, А. Л. On one extremal problem for positive series |
| title | On one extremal problem for positive series |
| title_alt | Про одну екстремальну задачу для додатних рядів |
| title_full | On one extremal problem for positive series |
| title_fullStr | On one extremal problem for positive series |
| title_full_unstemmed | On one extremal problem for positive series |
| title_short | On one extremal problem for positive series |
| title_sort | on one extremal problem for positive series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3718 |
| work_keys_str_mv | AT stepanetsoi ononeextremalproblemforpositiveseries AT shydlichal ononeextremalproblemforpositiveseries AT stepanecʹoí ononeextremalproblemforpositiveseries AT šidlíčal ononeextremalproblemforpositiveseries AT stepanetsoi proodnuekstremalʹnuzadačudlâdodatnihrâdív AT shydlichal proodnuekstremalʹnuzadačudlâdodatnihrâdív AT stepanecʹoí proodnuekstremalʹnuzadačudlâdodatnihrâdív AT šidlíčal proodnuekstremalʹnuzadačudlâdodatnihrâdív |