On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight
Let $α=\{α_j\}_{j∈N}$ be a nondecreasing sequence of positive numbers and let $l_{1,α}$ be the space of real sequences $ξ=\{ξ_j\}_{j∈N}$ for which $∥ξ∥_{1,α} := ∑^{∞}_{j=1}α_j|ξ_j| < +∞$. We associate every sequence $ξ$ from $l_{1,α}$ with a sequence $ξ^∗ = \{|ξ_{φ(j)}|\}_{j∈N}$, where $ϕ(·)$...
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2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509785703579648 |
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| author | Radzievskaya, E. I. Radzievskii, G. V. Радзієвськая, О. І. Радзієвський, Г. В. |
| author_facet | Radzievskaya, E. I. Radzievskii, G. V. Радзієвськая, О. І. Радзієвський, Г. В. |
| author_sort | Radzievskaya, E. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:01:15Z |
| description | Let $α=\{α_j\}_{j∈N}$ be a nondecreasing sequence of positive numbers and let $l_{1,α}$ be the space of real sequences $ξ=\{ξ_j\}_{j∈N}$ for which $∥ξ∥_{1,α} := ∑^{∞}_{j=1}α_j|ξ_j| < +∞$. We associate every sequence $ξ$ from $l_{1,α}$ with a sequence $ξ^∗ = \{|ξ_{φ(j)}|\}_{j∈N}$, where $ϕ(·)$ is a permutation of the natural series such that $|ξ_{φ(j)}| ⩾ |ξ_{φ(j+1)}|,\; j ∈ ℕ$.
If $p$ is a bounded seminorm on $l_{1,α}$ and $\omega _m :\; = \left\{ {\underbrace {1, \ldots ,1}_m,\;0,\;0,\; \ldots } \right\}$, then
$$\mathop {\sup }\limits_{\xi \ne 0,\;\xi \ne 1_{1,\alpha } } \frac{{p\left( {\xi *} \right)}}{{\left\| \xi \right\|_{1,\alpha } }} = \mathop {\sup }\limits_{m \in \mathbb{N}} \frac{{p\left( {\omega _m } \right)}}{{\sum {_{s = 1}^m } \alpha _s }}.$$
Using this equality, we obtain several known statements. |
| first_indexed | 2026-03-24T02:46:37Z |
| format | Article |
| fulltext |
UDK 519.658
E. Y. Radzyevskaq (Nac. un-t pyw. texnolohyj, Kyev),
H. V. Radzyevskyj (Yn-t matematyky NAN Ukrayn¥, Kyev)
OB ODNOJ ∏KSTREMAL|NOJ ZADAÇE
DLQ POLUNORMÁ NA PROSTRANSTVE l1 S VESOM*
Let α = α j j
{ } ∈ N
be a nondecreasing sequence of positive numbers, l1, α be the space of real
sequences ξ = ξ j j
{ } ∈ N
, satisfying ξ α1, : =
j j j=
∞∑ 1
α ξ < + ∞. We associate each sequence ξ
from l1,α with a sequence ξ* = ξϕ( )j j{ } ∈ N
, where ϕ( )⋅ is such permutation of the natural series
that ξϕ( )j ≥ ξϕ( )j +1 , j ∈ N . If p is a bounded seminorm on l1, α and a sequence ωm : =
: =
1 1 0 0, , , , ,… …
m
��� , then
sup
( )
sup
( )
,
*
,,ξ ξ αα
ξ
ξ
ω
α≠ ∈ ∈ =
=
∑0 1 11l
p p
m
m
s
m
sN
.
This equality implies a number of the known statements.
Nexaj α = α j j
{ } ∈ N
— nespadna poslidovnist\ dodatnyx çysel, l1, α — prostir dijsnyx posli-
dovnostej ξ = ξ j j
{ } ∈ N
, dlq qkyx ξ α1, : =
j j j=
∞∑ 1
α ξ < + ∞. KoΩnij poslidovnosti ξ z l1, α
postavymo u vidpovidnist\ poslidovnist\ ξ* = ξϕ( )j j{ } ∈ N
, de ϕ( )⋅ — taka perestanovka na-
tural\noho rqdu, wo ξϕ( )j ≥ ξϕ( )j +1 , j ∈ N . Qkwo p — obmeΩena pivnorma na l1, α i posli-
dovnist\ ωm : =
1 1 0 0, , , , ,… …
m
��� , to
sup
( )
sup
( )
,
*
,,ξ ξ αα
ξ
ξ
ω
α≠ ∈ ∈ =
=
∑0 1 11l
p p
m
m
s
m
sN
.
Z ci[] rivnosti vyvodyt\sq nyzka vidomyx tverdΩen\.
Nastoqwee soobwenye ynspyryrovano rezul\tatamy rabot [1] (hl..III, lem-
ma.15.2), [2] (lemma, prymer 4), [3] (lemm¥ 2 y 2′), [4] (hl..XI, lemm¥ 5.1 y 5.1′),
[5] (lemma 1). V nem pryveden¥ lemma, teorema y sformulyrovan¥ try
sledstvyq yz nee, dopolnqgwye sootvetstvugwye utverΩdenyq yz [1 – 5]. Sut\
πtyx dopolnenyj poqsnena posle dokazatel\stva lemm¥ y formulyrovok
sledstvyj.1 – 3.
Kak ob¥çno, N y R — πto, sootvetstvenno, mnoΩestva cel¥x poloΩytel\-
n¥x (natural\n¥x) y dejstvytel\n¥x çysel. Budem rassmatryvat\ lyß\ posle-
dovatel\nosty ξ = ξ j j{ } ∈N
vewestvenn¥x çysel, a c0 — mnoΩestvo posledo-
vatel\nostej ξ, dlq kotor¥x lim
j
j
→∞
ξ = 0. KaΩdoj posledovatel\nosty ξ =
= ξ j j{ } ∈N
yz c0 sopostavym posledovatel\nost\ ξ* = ξ j j
*{ } ∈N
, polahaq ξ j
* =
= ξϕ( )j , hde ϕ( )⋅ — takaq perestanovka natural\noho rqda, çto posledova-
tel\nost\ ξϕ( )j j{ } ∈N
qvlqetsq nevozrastagwej.
* PodderΩana Hosudarstvenn¥m fondom fundamental\n¥x yssledovanyj Ukrayn¥ (proekt
F7/329-2001).
© E. Y. RADZYEVSKAQ, H. V. RADZYEVSKYJ, 2005
1002 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
OB ODNOJ ∏KSTREMAL|NOJ ZADAÇE DLQ POLUNORMÁ NA PROSTRANSTVE … 1003
Dlq neub¥vagwej posledovatel\nosty poloΩytel\n¥x çysel α = α j j{ } ∈N
vvedem banaxovo prostranstvo l1,α , sostoqwee yz posledovatel\nostej ξ, dlq
kotor¥x
ξ α ξα1
1
, := < + ∞
=
∞
∑
j
j j , ξ α∈l1, .
Poskol\ku l1,α ⊂ c0 , dlq vsex πlementov ξ yz l1,α opredelena posledova-
tel\nost\ ξ*
y, vvydu teorem¥ 368 yz [6],
ξ
α
*
,1
≤ ξ α1, , ξ α∈l1, . (1)
Zadann¥j na l1,α vewestvennoznaçn¥j funkcyonal naz¥vaetsq polunor-
moj, esly p( )ξ η+ ≤ p( )ξ + p( )η y p( )λξ = λ ξp( ) dlq vsex λ ∈ R y ξ, η ∈
∈ l1,α . Yz πtoho opredelenyq sleduet, çto p( )ξ ≥ 0 dlq vsex ξ ∈ l1,α . Esly Ωe
suwestvuet postoqnnaq c > 0, dlq kotoroj p( )ξ ≤ c ξ α1, pry ξ ∈ l1,α , to
polunorma p naz¥vaetsq ohranyçennoj.
Teorema. Pust\ p — ohranyçennaq polunorma na l1,α , a posledovatel\-
nost\ ωm : = 1 1 0 0, , , , ,… …
m
��� . Tohda
sup
( )
sup
( )
,
*
,,ξ ξ αα
ξ
ξ
ω
α≠ ∈ ∈ =
=
∑0 1 11l
p p
m
m
s
m
sN
. (2)
Otmetym, çto ohranyçennost\ obeyx çastej v (2) qvlqetsq sledstvyem ohra-
nyçennosty polunorm¥ p, poskol\ku v¥polnqetsq neravenstvo (1) y ω αm 1, =
=
s
m
s=∑ 1
α . Yz posledneho ravenstva, v çastnosty, sleduet, çto levaq çast\ v (2)
bol\ße yly ravna pravoj.
Dokazatel\stvo teorem¥ osnovano na ee koneçnomernom varyante. Pry πtom
çerez Kn
oboznaçen konus v R
n
, sostoqwyj yz vektorov ξ : = ξ j j
n{ } =1
s koor-
dynatamy ξ1 ≥ ξ2 ≥ … ≥ ξn ≥ 0. Zadadym takΩe vektor¥ es : = δ j s j
n
,{ } =1
, hde
s = 1, … , n, a δ j s, — symvol Kronekera, y poloΩym ωm
n( ) : =
s
m
se=∑ 1
, m = 1, …
… , n. V πtyx oboznaçenyqx spravedlyvo sledugwee utverΩdenye.
Lemma. Pust\ p — polunorma v R
n
, a α = α j j
n{ } =1
— proyzvol\n¥j vek-
tor s poloΩytel\n¥my koordynatamy. Tohda
p( )ξ ≤ max
, ,
( )
m n
m
n
s
m
s j
n
j j
p
= …
= =
( )
∑ ∑
1
1 1
ω
α
α ξ , ξ ∈Kn
, (3)
pryçem suwestvugt nenulev¥e vektor¥ ξ y z Kn
, dlq kotor¥x (3) prevra-
waetsq v ravenstvo.
Dokazatel\stvo. Pust\ maksymum v (3) dostyhaetsq na yndekse m0 ∈
∈ 1, ,…{ }n . Tohda (3) prevrawaetsq v ravenstvo na vektore ξ : = ωm
n
0
( ) ∈ Kn
.
Dalee, razlahaq proyzvol\n¥j vektor ξ yz Kn
po vektoram
s
m
s m
n
=
−
∑( )1
1
α ω( )
,
m = 1, … , n, ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
1004 E. Y. RADZYEVSKAQ, H. V. RADZYEVSKYJ
ξ λ ω
α
λ
α
= =
= = = = =
∑ ∑ ∑ ∑ ∑m
n
m
m
n
s
m
s j
n
m j
n
m
s
m
s
je
1
1
1
1
( )
.
Poskol\ku ξ =
j
n
j je=∑ 1
ξ ∈ Kn
, yz vtoroho predstavlenyq vektora ξ sleduet,
çto λm ≥ 0, m = 1, … , n, y
j
n
j j
j
n
j
m j
n
m
s
m
s m
n
m
= = = = =
∑ ∑ ∑ ∑ ∑=
=
1 1
1
1
α ξ α λ
α
λ .
No p — polunorma y poπtomu
p( )ξ ≤
m
n
m
m
n
s
m
s
p
= =
∑ ∑
( )
1
1
λ
ω
α
( )
≤ max
, ,
( )
m n
m
n
s
m
s
p
= …
=
( )
∑1
1
ω
α m
n
m
=
∑
1
λ ,
otkuda y poluçaem neravenstvo (3).
V sluçae p( )ξ : =
j
n
j j=∑ 1
ν ξ , hde ν j j
n{ } =1
∈ Kn
, utverΩdenye lemm¥ sov-
padaet s lemmoj 15.2 yz hl..III monohrafyy [1].
Dokazatel\stvo teorem¥. Oboznaçym çerez K∞
mnoΩestvo neotryca-
tel\n¥x posledovatel\nostej ξ = ξ j j{ } ∈N
, u kotor¥x ξ j ≥ ξ j +1, j ∈N . Tohda
sohlasno neravenstvu (1) verxngg hran\ v levoj çasty ravenstva (2) dostatoçno
brat\ lyß\ po posledovatel\nostqm ξ ≠ 0 y ξ ∈ l1,α ∩ K∞
, çto y budem pred-
polahat\ v dal\nejßem.
KaΩdoj posledovatel\nosty ξ = ξ j j{ } ∈N
sopostavym vektor ξ( )n : = {ξ1, …
… , ξn , 0, 0, … }. Esly ξ ∈ l1,α , to vektor¥ ξ( )n
sxodqtsq po norme prostran-
stva l1,α k ξ y poπtomu
lim ( )( )
n
np p
→∞
( ) =ξ ξ , ξ α∈l1, . (4)
Funkcyonal ξ ξ( ) ( )n np� ( ) udovletvorqet uslovyqm lemm¥, na osnovanyy ko-
toroj
p nξ( )( ) ≤ max
, ,m n
m
s
m
s j
n
j j
p
= …
= =
( )
∑ ∑
1
1 1
ω
α
α ξ ≤ sup
m
m
s
m
s
p
∈
=
( )
∑N
ω
α
1
ξ α1, , ξ α∈ ∞l1, ∩ K .
Otsgda y yz ravenstva (4) sleduet, çto pravaq çast\ v (2) moΩet b¥t\ lyß\
bol\ße yly ravna levoj. Spravedlyvost\ obratnoho neravenstva b¥la pokazana
neposredstvenno posle formulyrovky teorem¥.
Pryvedem try sledstvyq yz teorem¥.
Po neotrycatel\noj posledovatel\nosty çysel ν = ν j j{ } ∈N
, udovletvorq-
gwej uslovyg
sup
j
j
j∈
< + ∞
N
ν
α
, (5)
y dlq q ∈ 1; + ∞[ ] vvedem polunorm¥ pq,ν, zadann¥e ravenstvamy
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
OB ODNOJ ∏KSTREMAL|NOJ ZADAÇE DLQ POLUNORMÁ NA PROSTRANSTVE … 1005
pq
j
j
q
j
q
q
,
/
( ) :ν ξ ν ξ=
=
∞
∑
1
1
, 1 ≤ q < + ∞,
p
j
j j∞
∈
=, ( ) : supν ξ ν ξ
N
.
Vvydu uslovyq (5) y teorem¥ 19 yz [6] polunorm¥ pq,ν ohranyçen¥ na pro-
stranstve l1,α . Prymenqq k nym teoremu, poluçaem takoe utverΩdenye.
Sledstvye 1. Spravedlyvo ravenstvo
sup
( )
sup
,
,
*
,
/
,ξ ξ
ν
αα
ξ
ξ
ν
α≠ ∈ ∈
=
=
=
( )∑
∑0 1
1
1
111
pq
m
s
m
s
q
q
s
m
sN
,
pryçem dlq q = + ∞ polahaem
s
m
s
q
q
=∑( )1
1
ν
/
:= max , ,ν ν1 …{ }m .
Esly v sledstvyy 1 sçytat\ posledovatel\nost\ α = α j j{ } ∈N
s lim
j j→∞
α =
= + ∞, a ν — ohranyçennoj y neub¥vagwej posledovatel\nost\g y 1 ≤ q < + ∞,
to utverΩdenye sledstvyq 1 sovpadaet s lemmoj 1 yz [5]. Sleduet otmetyt\, çto
dokazatel\stvo πtoj lemm¥ v [5] ves\ma obæemnoe y zanymaet 12 stranyc Ωur-
nal\noho teksta. V svog oçered\ lemma 1 yz [5] soderΩyt lemmu 2 yz [3] yly
lemmu 5.1 yz hl..XI v [4], esly poloΩyt\ v nej posledovatel\nost\ ν ravnoj
posledovatel\nosty
ˆ ˆν ν= { } ∈j j N
, ν̂ j = 0 , j ≤ r, ν̂ j = 1, j > r, (6)
dlq r ∈ N.
Pust\ [[ ]]β — celaq çast\ β ∈ R. Dlq r > 0 y 1 < q < + ∞ vvedem çysla
rq : = r
q −
1
, d q r
r
r r
r
r r
q
q
q
q
q
q
( ; ) : max
( )
;
( )/ /
=
+
+
+ +
1 11
1
. (7)
Zametym, çto pry razlyçn¥x znaçenyqx q y r maksymum v opredelenyy (7)
velyçyn¥ d q r( ; ) moΩet dostyhat\sq lybo na pervom, lybo na vtorom v¥raΩe-
nyy yz ee opredelenyq. V çastnosty,
d q r
q
q
r
q
( ; ) :
/
= −
−1 1 1 1
, esly
r
q −
∈
1
N .
Krome toho,
lim ( ; )
( )/
/
r
q
q
d q r r
q
q→∞
−
−
= −1 1
1 11
, 1 < q < + ∞.
Dlq α = α j j{ } ∈N
s α j = 1 pry j ∈ N polahaem l1 : = l1,α .
Vo vvedenn¥x oboznaçenyqx yz sledstvyq 1 v¥tekaet takoe utverΩdenye.
Sledstvye 2. Pust\ r ∈ N, a posledovatel\nost\ ν̂ zadana sootnoße-
nyqmy (6). Tohda
sup
( )
, ,
( ) , , ,
( ; ), , .
,
, ˆ
*
ξ ξ
ν ξ
ξ≠ ∈
−=
=
+ < ≤ + ∞ ≤ −
< < + ∞ > −
0 1
1
1
1 1
1 1 1
1 1
1
p
q
r q r q
d q r q r q
q
V sluçae 1 < q < + ∞ sledstvye 2 sovpadaet s lemmoj 5.1′ yz hl..XI v [4].
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
1006 E. Y. RADZYEVSKAQ, H. V. RADZYEVSKYJ
Sledstvye 3. Pust\ α = j j{ } ∈N
, a pry r ∈ N posledovatel\nost\ ν̂
zadana sootnoßenyqmy (6). Tohda
sup
( )
,
, ˆ
*
,,ξ ξ
ν
αα
ξ
ξ≠ ∈
=
+0
1
11
1
2 1l
p
r
.
Ravenstvo, ustanovlennoe v πtom sledstvyy, sovpadaet s tem, çto dokaz¥va-
etsq vo vtoroj çasty prymera 4 yz [2], hde podsçet v¥polnen do ukazanyq
konkretnoho çysla. Odnako sootvetstvugwee çyslo v [2] ukazano neverno.
1. Hoxberh Y. C., Krejn M. H. Vvedenye v teoryg lynejn¥x nesamosoprqΩenn¥x operatorov v
hyl\bertovom prostranstve. – M.: Nauka, 1965. – 448 s.
2. Stepanec A. Y. Approksymacyonn¥e xarakterystyky prostranstv S p
ϕ // Ukr. mat. Ωurn. –
2001. – 53, # 3. – S. 392 – 416.
3. Stepanec A. Y. Approksymacyonn¥e xarakterystyky prostranstv S p
ϕ v razn¥x metrykax
// Tam Ωe. – # 8. – S..1121 – 1146.
4. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj: V 2 t. – Kyev: Yn-t matematyky NAN Ukray-
n¥, 2002. – T. 2. – 468 s.
5. Stepanec\ O. I., Íydliç A. L. Najkrawi n-çlenni nablyΩennq Λ-metodamy u prostorax S p
ϕ
// Ukr. mat. Ωurn. – 2003. – 55, # 8. – S. 1107 – 1126.
6. Xardy H. H., Lyttl\vud DΩ. E., Polya H. Neravenstva. – M.: Yzd-vo ynostr. lyt., 1948. –
456 s.
Poluçeno 18.11.2003
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
|
| id | umjimathkievua-article-3659 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:46:37Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c5/f0244e12acc831f4c06be3b1c4b4c7c5.pdf |
| spelling | umjimathkievua-article-36592020-03-18T20:01:15Z On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight Об одной экстремальной задаче для полунормы на пространстве $l_1$ с весом Radzievskaya, E. I. Radzievskii, G. V. Радзієвськая, О. І. Радзієвський, Г. В. Let $α=\{α_j\}_{j∈N}$ be a nondecreasing sequence of positive numbers and let $l_{1,α}$ be the space of real sequences $ξ=\{ξ_j\}_{j∈N}$ for which $∥ξ∥_{1,α} := ∑^{∞}_{j=1}α_j|ξ_j| < +∞$. We associate every sequence $ξ$ from $l_{1,α}$ with a sequence $ξ^∗ = \{|ξ_{φ(j)}|\}_{j∈N}$, where $ϕ(·)$ is a permutation of the natural series such that $|ξ_{φ(j)}| ⩾ |ξ_{φ(j+1)}|,\; j ∈ ℕ$. If $p$ is a bounded seminorm on $l_{1,α}$ and $\omega _m :\; = \left\{ {\underbrace {1, \ldots ,1}_m,\;0,\;0,\; \ldots } \right\}$, then $$\mathop {\sup }\limits_{\xi \ne 0,\;\xi \ne 1_{1,\alpha } } \frac{{p\left( {\xi *} \right)}}{{\left\| \xi \right\|_{1,\alpha } }} = \mathop {\sup }\limits_{m \in \mathbb{N}} \frac{{p\left( {\omega _m } \right)}}{{\sum {_{s = 1}^m } \alpha _s }}.$$ Using this equality, we obtain several known statements. Нехай $α=\{α_j\}_{j∈N}$ — нeспадна послідовність додатних чисел, $l_{1,α}$ — простір дійсних послідовностей $ξ=\{ξ_j\}_{j∈N}$, для яких $∥ξ∥_{1,α} := ∑^{∞}_{j=1}α_j|ξ_j| < +∞$. Кожній послідовності $ξ$ з $l_{1,α}$ поставимо у відповідність послідовність $ξ^∗ = \{|ξ_{φ(j)}|\}_{j∈N}$, де $ϕ(·)$ — така перестановка натурального ряду, що $|ξ_{φ(j)}| ⩾ |ξ_{φ(j+1)}|,\; j ∈ ℕ$. Якщо р — обмежена півнорма на $l_{1,α}$ і послідовність $\omega _m :\; = \left\{ {\underbrace {1, \ldots ,1}_m,\;0,\;0,\; \ldots } \right\}$, то . $$\mathop {\sup }\limits_{\xi \ne 0,\;\xi \ne 1_{1,\alpha } } \frac{{p\left( {\xi *} \right)}}{{\left\| \xi \right\|_{1,\alpha } }} = \mathop {\sup }\limits_{m \in \mathbb{N}} \frac{{p\left( {\omega _m } \right)}}{{\sum {_{s = 1}^m } \alpha _s }}.$$ З цієї рівності виводиться низка відомих тверджень. Institute of Mathematics, NAS of Ukraine 2005-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3659 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 7 (2005); 1002–1006 Український математичний журнал; Том 57 № 7 (2005); 1002–1006 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3659/4048 https://umj.imath.kiev.ua/index.php/umj/article/view/3659/4049 Copyright (c) 2005 Radzievskaya E. I.; Radzievskii G. V. |
| spellingShingle | Radzievskaya, E. I. Radzievskii, G. V. Радзієвськая, О. І. Радзієвський, Г. В. On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight |
| title | On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight |
| title_alt | Об одной экстремальной задаче для полунормы на пространстве $l_1$ с весом |
| title_full | On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight |
| title_fullStr | On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight |
| title_full_unstemmed | On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight |
| title_short | On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight |
| title_sort | on one extremal problem for a seminorm on the space $l_1$ with weight |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3659 |
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