Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis
We investigate exact constants in Jackson-type inequalities in the space $L_2$ for the approximation of functions on an axis by the subspace of entire functions of exponential type.
Збережено в:
| Дата: | 2009 |
|---|---|
| Автори: | , , , |
| 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/3004 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509016426283008 |
|---|---|
| author | Doronin, V. G. Ligun, A. A. Доронин, В. Г. Лигун, А. А. Доронин, В. Г. Лигун, А. А. |
| author_facet | Doronin, V. G. Ligun, A. A. Доронин, В. Г. Лигун, А. А. Доронин, В. Г. Лигун, А. А. |
| author_sort | Doronin, V. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:07Z |
| description | We investigate exact constants in Jackson-type inequalities in the space $L_2$ for the approximation of functions on an axis by the subspace of entire functions of exponential type. |
| first_indexed | 2026-03-24T02:34:24Z |
| format | Article |
| fulltext |
UDK 517.5
A. A. Lyhun (DneprodzerΩ. texn. un-t),
V. H. Doronyn (Dnepropetr. nac. un-t)
TOÇNÁE KONSTANTÁ V NERAVENSTVAX TYPA
DÛEKSONA DLQ L2 -APPROKSYMACYY NA PRQMOJ
We investigate exact constants in the Jackson-type inequalities in the space L2 for the approximation
of functions on the straight line by a subspace of entire functions of exponential type.
Provedeno doslidΩennq toçnyx konstant u nerivnostqx typu DΩeksona u prostori L2 dlq na-
blyΩennq funkcij na prqmij pidprostorom cilyx funkcij eksponencial\noho typu.
Pust\ L
2
� prostranstvo vewestvennoznaçn¥x funkcyj f, opredelenn¥x y
yzmerym¥x na (– , )∞ ∞ , kotor¥e udovletvorqgt uslovyg
f 2 =
–
( )
∞
∞
∫ f x 2 < ∞;
L r
2, r ≥ 0, � mnoΩestvo vsex funkcyj f, u kotor¥x (r – 1)-q proyzvodnaq na
osy lokal\no absolgtno neprer¥vna y f r( ) ∈ L2 (esly r ne celoe, to f r( ) �
proyzvodnaq v sm¥sle Vejlq).
Oboznaçym çerez Eσ klass cel¥x funkcyj πksponencyal\noho typa s poka-
zatelem ≥ σ,
Bσ = L2 I Eσ ,
A fσ( ) = inf –f g g Bσ σ σ∈{ } (1)
� pryblyΩenye funkcyy f L∈
2
mnoΩestvom Bσ .
Pust\
ω p f t( ; ) = sup ( )∆η ηp f t⋅ ≤{ } (2)
� p-j yntehral\n¥j modul\ hladkosty funkcyy f, hde ∆η
p f x( ) � raznost\ po-
rqdka p funkcyy f v toçke x ßahom η.
Kak ob¥çno,
F f( ; )ω = l.i.m. exp(– ) ( )
–A A
A
i t f t dt
→∞
∫1
2π
ω (3)
� transformacyq Fur\e funkcyy f.
Neravenstva vyda
A fσ( ) ≤ ℵ ( )
σ
ω δ σr p
rf ( ); (4)
naz¥vagt neravenstvamy typa DΩeksona, a naymen\ßug konstantu v nyx
ℵ = ℵσ δ, , ( )r p = sup
( )
; /( )
f L
f
r
p
r
r
A f
f∈
≠
( )2
const
σ
ω δ σ
σ (5)
� toçnoj.
© A. A. LYHUN , V. H. DORONYN, 2009
92 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
TOÇNÁE KONSTANTÁ V NERAVENSTVAX TYPA DÛEKSONA DLQ .… 93
Problema naxoΩdenyq toçn¥x konstant v neravenstvax typa DΩeksona v
prostranstve L
2
yzuçalas\ vo mnohyx rabotax (sm., naprymer, [1 – 5] y pryve-
dennug v nyx byblyohrafyg).
Cel\g dannoj rabot¥ qvlqetsq rasprostranenye toçn¥x neravenstv typa
DΩeksona dlq nayluçßyx pryblyΩenyj tryhonometryçeskymy polynomamy pe-
ryodyçeskyx funkcyj v prostranstve L
2
, yssledovann¥x namy v [3, 4], na slu-
çaj approksymacyy cel¥my funkcyqmy πksponencyal\noho typa funkcyj na
vsej osy v prostranstve L
2
.
Spravedlyva sledugwaq teorema.
Teorema 1. Pry lgbom a > 1 dlq lgb¥x σ > 0 , r ≥ 0, p = 1, 2, … y lgboj
nenulevoj neotrycatel\noj summyruemoj funkcyy θ( )t , 0 < t < b < π, ymegt
mesto neravenstva
sup
( )
; / ( )( )f L
f
r
b
p
rr
A f
f t t dt∈
≠
∫ ( )2
2 2
0
2
const
σ
ω σ θ
σ ≤ a
a
y
r
r
p
y a
b r p
2
2 1
1
1
2
–
inf ( ; ), ,
–
≤ ≤
Φ θ , (6)
hde
Φb r p y, , ( ; )θ = y yt t dtr
b
p2
0
1∫ ( – cos ) ( )θ . (7)
Dokazatel\stvo. Yzvestno [1], çto dlq lgboj funkcyy f L∈
2
A fσ
2( ) =
ω σ
ω ω
≥
∫ F f d( ; ) 2 . (8)
V sylu πtoho s uçetom toho, çto vsledstvye vewestvennoznaçnosty f funkcyq
F f w( ; ) çetnaq, dlq lgboj funkcyy f Lr∈ 2 ymeem
A fσ
2( ) = 2 2
σ
ω ω
∞
∫ F f d( ; ) =
µ σ
σ
µ
µ
ω ω
=
∞
∑ ∫
+
0
2
1
2
a
a
F f d( ; ) =
=
µ σ
σ µ
µ
µ µ
µ
µ
ω
ω ω
σ
θ
σ ω
σ
ω
σ
θ
ω
=
∞
∑ ∫
∫
∫
+
( )
0
2
2
0
2 2
0
1
2
2 1
2 1a
a
p r
pb
p r r pb
F f a
t t dt
a
a a
t t dt
d( ; )
– cos ( )
– cos ( )
≤
≤
µ
σ
σ
µ
µ
µ
µ
ω ω ω
σ
ω θ
σ θ=
∞
≤ ≤
∑
∫∫
∫
+
( ) ( )0
2 2
0
2 2
1
2
0
1
2 2 1
2 1
a
a p r
pb
r r p
y a
r b p
F f
a
t d t dt
a y yt t dt
( ; ) – cos ( )
inf – cos ( )
≤
≤
µ
µ
µ
ω ω ω
σ
ω θ
σ θ=
∞
∞
≤ ≤
∑
∫∫
( )0
0
2 2
0
2 2
1
2 2 1
2
F f
a
t d t dt
a y
p r
pb
r r p
y a
b r p
( ; ) – cos ( )
inf ( ; ), ,Φ
. (9)
Yspol\zuq fundamental\n¥e svojstva transformacyj Fur\e, poluçaem
'
F fp r∆η ω( );( ) = ( ) – ( ; )i e F fr i p
ω ωηω 1( ) . (10)
'
Po teoreme Planßerelq, tak kak ∆η
p f r( ) ∈ L2, to F fp r∆η
( )( ) ∈ L2 y πty
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
94 A. A. LYHUN , V. H. DORONYN
funkcyy ymegt odynakov¥e norm¥. Poπtomu, prynymaq vo vnymanye (10), naxo-
dym
∆η
p rf ( ) ( )⋅
2
= 2 2 1
0
2 2
∞
∫ F f dp r p( ; ) ( – cos )ω ω ηω ω . (11)
Sledovatel\no, uçyt¥vaq opredelenye (2) modulq hladkosty, ustanavlyvaem,
çto
0
1 2 22 1
∞
+∫ p r pF f x d( ; ) ( – cos )ω ω ω ω ≤ ω p
rf x2 ( );( ) . (12)
Prymenqq πtu ocenku v sootnoßenyy (9), zaklgçaem, çto dlq lgboj funkcyy
f L r∈ 2
A fσ
2( ) ≤
µ
µ
µ
ω σ θ
σ θ=
∞
≤ ≤
∑ ∫ ( )
0
0
2
2 2
1
2
b
p
r
r r p
y a
b r p
f t a t dt
a y
( )
, ,
; / ( )
( ) inf ( ; )Φ
≤
≤ 0
2
2
1 0
22
1
b
p
r
r p
y a
b r p
r
f t t dt
y a
∫ ∑
( )
≤ ≤ =
∞ω σ θ
σ θ µ
µ( )
, ,
; / ( )
inf ( ; )Φ
=
= a
a
y f t t dt
r
r
p
y a
b r p r
b
p
r
2
2 1
1
2
0
2
1
2 1
–
inf ( ; ) ; / ( ), ,
–
( )
≤ ≤
( )∫Φ θ
σ
ω σ θ . (13)
Nakonec, perexodq v poluçennom neravenstve k supremumu po f Lr∈ 2, f ≠
≠ const, pryxodym k neravenstvu (6).
Teorema 1 dokazana.
Pust\ h > 0, αk ≥ 0 . Rassmotrym funkcyy
δh t( ) = 1 2 0 2/ ( / ); ( / )h t h t h< ≥{ }
y
θh t( ) =
k
n
k h kt
=
∑
1
α δ ξ( – ) ,
hde 0 < ξ1 < ξ2 < … < ξn , b ≥ ξn + h / 2.
Yz teorem¥ 1, polahaq v nej θ( )t = θh t( ), predel\n¥m perexodom po h → 0
poluçaem takoe sledstvye.
Sledstvye 1. Pry lgbom a > 1 dlq lgb¥x σ > 0, r ≥ 0, p = 1, 2, … , αk ≥
≥ 0, 0 < ξ1 < ξ2 < … < ξn ymegt mesto neravenstva
sup
( )
; /( )
f L
f
r
k
n
k p
r
k
r
A f
f∈
≠
=∑ ( )2
2 2
1
2
const
σ
α ω ξ σ
σ ≤
≤ a
a
y y
r
r
p
y a
r
k
n
k k
p
2
2 1
2
1
1
1
2 1
–
inf ( – cos )
–
≤ ≤ =
∑
α ξ . (14)
Zametym, çto predel\n¥j perexod v (6) y (14) po a → ∞ pozvolqet ustano-
vyt\ spravedlyvost\ sootvetstvenno sledugwyx neravenstv:
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
TOÇNÁE KONSTANTÁ V NERAVENSTVAX TYPA DÛEKSONA DLQ .… 95
sup
( )
; / ( )( )f L
f
r
b
p
rr
A f
f t t dt∈
≠
∫ ( )2
2 2
0
2
const
σ
ω σ θ
σ ≤ 2
1
2
1
p
y
r
b r py yinf ( ; ), ,
–
≥
Φ θ , (15)
sup
( )
; /( )
f L
f
r
k
n
k p
r
k
r
A f
f∈
≠
=∑ ( )2
2 2
1
2
const
σ
α ω ξ σ
σ ≤ 2 1
1
2
1
1
p
y
r
k
n
k k
py yinf ( – cos )
–
≥ =
∑
α ξ . (16)
Vproçem, πty rezul\tat¥ vpolne sohlasugtsq s rezul\tatamy [5] (ocenky
sverxu v sledstvyqx 1 y 2).
PoloΩym
cr p, = 4 2 2 2
– – / – /r p p( ) (17)
y
ξ = ξr p, = 2 2 1
π
arcsin –( / )–r p . (18)
Spravedlyva sledugwaq teorema.
Teorema 2. Pust\ r ≥ p takov¥, çto dlq kaΩdoj nesokratymoj droby
l L/ v¥polnqgtsq neravenstva
ξ – /l L ≥ 4 1L r p–( / )– . (19)
Tohda dlq lgboho σ > 0 pry lgbom δ ≥ (1 + ξ π) ymegt mesto neravenstva
ℵσ δ, , ( )r p ≤ cr p, . (20)
Dokazatel\stvo. V¥berem
a = ξ + r
pπ
πξtg( ) , (21)
α1 = ( )/1 2+ α , α2 = ( – )/1 2α , (22)
ξ1 = ( – )1 ξ π , ξ2 = ( )1 + ξ π. (23)
Tohda yz pravoho neravenstva v sootnoßenyy (14) sleduet, çto dlq lgboj
funkcyy f Lr∈ 2 y lgboho a > 1
σ σ
2 2r A f( ) ≤ a
a
f f
y y y
r
r
p
r
p
r
p
y a
r p p
2
2
1
2
1 2
2
2
1
2
1 1 2 21 2 1 1–
; / ; /
inf ( – cos ) ( – cos )
( ) ( )α ω ξ σ α ω ξ σ
α ξ α ξ
( ) + ( )
+{ }
≤ ≤
≤
≤ a
a
f f
y
r
r
p
r
p
r
p
y a
r p
2
2
1
2
2
2
1
1
1 1
2–
; ( – ) / ; ( ) /
inf ( )
( ) ( )
,
α ω ξ π σ α ω ξ π σ
θ
( ) + +( )
≤ ≤
≤
≤ a
a
f
y
r
r
p
r
p
y a
r p
2
2
2
1
1
1
2–
; ( ) /
inf ( )
( )
,
ω ξ π σ
θ
+( )
≤ ≤
, (24)
hde
θr p y, ( ) = y y yr p p2
1 21 1 1 1α ξ π α ξ π– cos( – ) – cos( )( ) + +( ){ }. (25)
Otsgda predel\n¥m perexodom po a → ∞ poluçaem, çto dlq lgboj funk-
cyy f Lr∈ 2
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
96 A. A. LYHUN , V. H. DORONYN
σ σ
2 2r A f( ) ≤
ω ξ π σ
θ
p
r
p
y
r p
f
y
2
1
1
2
( )
,
; ( ) /
inf ( )
+( )
≥
. (26)
Velyçyna
inf ( ),
y
r p y
≥1
θ
yssledovana v rabote [4], hde ustanovleno, çto
inf ( ),
y
r p y
≥1
θ = θr p, ( )1 = 2 1 2 2 2p r p p
– – ( / )–( ) . (27)
S uçetom πtoho yz sootnoßenyq (26) poluçaem, çto dlq lgboj funkcyy f Lr∈ 2
pry lgbom δ ≥ (1 + ξ π) ymegt mesto sootnoßenyq
σ σ
2 2r A f( ) ≤
2
2 2 1
1
2 2
2 2 2
2
r p
p r p p p
rf
+
+( )
+( )
( / )
( )
–
; ( ) /ω ξ π σ ≤ c fr p p
r
,
( ); /2 2ω δ σ( ).
(28)
Otsgda sleduet, çto v¥polnqetsq neravenstvo (20).
Teorema 2 dokazana.
Teorema 3. Dlq lgb¥x σ > 0, r ≥ 0, p = 1, 2, … , δ > 0 ymegt mesto nera-
venstva
ℵσ δ, , ( )r p ≥ sup
max ( – cos ), , ,β β β
δ
β
β1 2
1
2
1
2 22 1…
=
≤ =
∑
∑m
k
m
k
p
t k
m r
k
pk kt
. (29)
Dokazatel\stvo. V uslovyqx teorem¥ v¥berem lgboj vektor B = (β1,
β2, … , βm) y rassmotrym posledovatel\nost\ çetn¥x funkcyj f Ln B, ∈ 2 :
f xn B, ( ) =
k
n
k n
n
k
n
k n
k x x n
x k x n x n
x n
=
=
∑
∑
+ ≤ ≤
+ ≤ ≤ +
≥ +
1
1
0 2
2 2 1
0 2 1
β σ α π
ψ β σ α π π
π
cos ( ) , ,
( ) cos ( ) , ( ) ,
, ( ) ,
(30)
hde
ψn x( ) = H k y
n
dyr
x
n
k
m
k
r
( )
cos –
( )
2 1
1
2 1
2
+
=
∫ ∑ +
π
β π
,
konstanta Hr opredelena uslovyem ψn ( )2nπ = 1 y, nakonec, αn = 1 / n .
Po analohyy s [1], predvarytel\no postroyv F fn B( , ; ω) � transformacyg
Fur\e posledovatel\nosty f xn B, ( ) y prymenyv zatem formulu (8), poluçym
asymptotyçeskoe ravenstvo
A fn Bσ
2( ), = 2 1 1
1
2n o
k
m
kπ β
=
∑ +{ }( ) , n → ∞. (31)
Namerevaqs\ dalee operyrovat\ velyçynoj ω p n B
rf2
,
( )( ; δ σ/ ) , m¥, estestvenno,
snaçala (ßah za ßahom po p) stroym funkcyg ∆η
p f xn B
r
,
( ) ( ) y zatem ustanavly-
vaem, çto dlq kaΩdoho p = 1, 2, … pry n → ∞ ymeet mesto asymptotyçeskoe
ravenstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
TOÇNÁE KONSTANTÁ V NERAVENSTVAX TYPA DÛEKSONA DLQ .… 97
∆η
p
n B
rf ,
( ) ( )⋅
2
=
= 2 1 1 1 11 2
1
2 2p
n
r
k
m
r
k n
pn o k k o+
=
+ +{ } +( ) +
∑π σ α β σ α η( ) ( ) – cos ( ) ( ) . (32)
Otsgda sleduet, çto pry lgb¥x fyksyrovann¥x σ, B, r , p, η ravnomerno po
σ, 0 ≤ σ ≤ π, v¥polnqetsq sootnoßenye
∆η
p
n B
rf ,
( ) 2
= 2 1 1 11 2
1
2 2p r
k
m
r
k
pn o k k+
=
+{ } ( )∑π σ β ση( ) – cos , n → ∞. (33)
Teper\, uçyt¥vaq opredelenye (5) toçnoj konstant¥ v neravenstve typa DΩek-
sona, na osnovanyy sootnoßenyj (31) y (33) dlq lgboho vektora B = (β1, β2, …
… , βm) poluçaem
ℵσ δ, , ( )r p ≥
σ
ω δ σ
σ
2 2
2
r
n B
r
p n B
r
A f
f
,
( )
,
( ) ; /
( )
( ) =
σ π β
η δ σ
η
2
1
2
2
2 1 1r
k
m
k
p
n B
r
n o
f
=
≤
∑ +{ }( )
max
/
,
( )∆
=
=
σ π β
δ σ
2
1
2
2
2 1 1r
k
m
k
t
t
p
n B
r
n o
f
=
≤
∑ +{ }( )
max / ,
( )∆
=
=
σ π β
π σ β
δ
2
1
2
1 2
1
2 2
2 1 1
2 1 1 1
r
k
m
k
t
p r
k
m r
k
p
n o
n o k kt
=
≤
+
=
∑
∑
+{ }
+{ }
( )
max ( ) ( – cos )
=
= k
m
k
p
t k
m r
k
pk kt
o=
≤ =
∑
∑
+{ }1
2
1
2 22 1
1 1
β
β
δ
max ( – cos )
( ) , n → ∞. (34)
Perexodq k verxnej hrany po B = (β1, β2, … , βm) , poluçaem utverΩdenye teo-
rem¥ 3.
Teorema 4. Dlq lgb¥x σ > 0, r ≥ 0, p = 1, 2, … pry lgbom δ ≥ (1 – ξ)π
ymegt mesto neravenstva
ℵσ δ, , ( )r p ≥ cr p, . (35)
Dokazatel\stvo. V teoreme 3 poloΩym ( , )β β1 2 = (1, β). Tohda, na osnova-
nyy (30), dlq lgb¥x δ ≥ (1 – ξ)π ymegt mesto sootnoßenyq
ℵσ δ, , ( )r p ≥ sup
max ( – cos ) ( – cos )β
δ
β
β
1
2 1 2 1 2
2
2 2
+
+[ ]
≤
p
t
p r pt t
=
= sup
max ( )
,
β β
β
δ
1
2
2
1
+
∈[ ]
p
u u
uΨ
, (36)
hde u = cos t,
uδ =
cos , ( – )
– ,
δ ξ π δ π
δ π
1
1
≤ ≤
≥
y, nakonec,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
98 A. A. LYHUN , V. H. DORONYN
Ψβ( )u = ( – ) ( )1 1 2 12 2u up r p p+ +[ ]+ β . (37)
V rabote [4], v çastnosty, dokazano, çto pry β = βr p, ,
βr p,
–2 = 2 2 12 1( / ) –r p +( ), (38)
funkcyq Ψβr p
u
,
( ) ravna nulg v toçke u∗ ∈ uδ, 1[ ], hde
u∗ = cos( – )1 ξ π = 2 2 1– ( / )–r p – 1. (39)
Na osnovanyy πtoho poluçaem
max ( )
, ,u u r p
u
∈[ ]δ
β
1
Ψ = Ψβr p
u
,
( )∗ =
2 1
2 2 1
2 2
2 1 2 1
( / )
( / )
–
–
r p
r p r p
+
+ + +
( )
( ) . (40)
Nakonec, v sylu (36) y (40) ymeem
ℵσ δ, , ( )r p ≥
1
2
2
1
+
∈[ ]
β
δ
β
r p
p
u u r p
u
,
,
max ( )
,
Ψ
=
=
1
2
2+
∗
β
β
r p
p
r p
u
,
,
( )Ψ
= 1 2 2 1
2 2 1
2 2 1
2 1 1
2 1 2 1
2 2 1+ ( )( )[ ] ( )
( )
+
+ + +
+ +
( / ) –
( / )
( / )
–
–
–
r p
r p r p
p r p p =
= 2
2 1
2
2 2
r
r p p( / ) –+( )
= cr p,
2 .
Teorema 4 dokazana.
Sopostavlqq teoremu 2 s teoremoj 4, poluçaem sledugwee utverΩdenye.
Teorema 5. Pust\ v¥polnen¥ uslovyq teorem¥ 2. Tohda pry vsex δ ≥ (1 +
+ ξ)π ymegt mesto ravenstva
ℵσ δ, , ( )r p = cr p, .
1. Popov V. G. O nayluçßyx srednekvadratyçn¥x pryblyΩenyqx cel¥my funkcyqmy πkspo-
nencyal\noho typa // Yzv. vuzov. Matematyka. � 1972. � 121, # 6. � S. 65 � 73.
2. Lyhun A. A. Nekotor¥e neravenstva meΩdu nayluçßymy pryblyΩenyqmy y modulqmy ne-
prer¥vnosty v prostranstve L2 // Mat. zametky. � 1978. � 24, # 6. � S. 785 � 792.
3. Lyhun A. A. Toçn¥e neravenstva typa DΩeksona dlq peryodyçeskyx funkcyj v prostranst-
ve L2 // Tam Ωe. � 1988. – 43, # 6. � S. 757 � 769.
4. Doronin V., Ligun A. On the exact constants in Jackson’s type inequalities in the space L2 // East
J. Approxim. – 1995. – 1, # 2. – P. 189 – 196.
5. Doronyn V. H., Lyhun A. A. O toçn¥x neravenstvax typa DΩeksona dlq cel¥x funkcyj v
L2 // Visn. Dnipropetr. un-tu. Matematyka. � 2007. � # 8. � S. 89 � 93.
Poluçeno 18.02.08,
posle dorabotky � 07.07.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
|
| id | umjimathkievua-article-3004 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:34:24Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/92/f1ca9a83119937f1e9b15179c3596392.pdf |
| spelling | umjimathkievua-article-30042020-03-18T19:43:07Z Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis Точные константы в неравенствах типа Джексона для $L_2$-аппроксимации на прямой Doronin, V. G. Ligun, A. A. Доронин, В. Г. Лигун, А. А. Доронин, В. Г. Лигун, А. А. We investigate exact constants in Jackson-type inequalities in the space $L_2$ for the approximation of functions on an axis by the subspace of entire functions of exponential type. Проведено дослідження точних констант у нерівностях типу Джексона у просторі $L_2$ для наближення функцій на прямій підпростором цілих функцій експоненціального типу. Institute of Mathematics, NAS of Ukraine 2009-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3004 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 1 (2009); 92-98 Український математичний журнал; Том 61 № 1 (2009); 92-98 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3004/2757 https://umj.imath.kiev.ua/index.php/umj/article/view/3004/2758 Copyright (c) 2009 Doronin V. G.; Ligun A. A. |
| spellingShingle | Doronin, V. G. Ligun, A. A. Доронин, В. Г. Лигун, А. А. Доронин, В. Г. Лигун, А. А. Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis |
| title | Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis |
| title_alt | Точные константы в неравенствах типа Джексона для
$L_2$-аппроксимации на прямой |
| title_full | Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis |
| title_fullStr | Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis |
| title_full_unstemmed | Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis |
| title_short | Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis |
| title_sort | exact constants in jackson-type inequalities for $l_2$-approximation on an axis |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3004 |
| work_keys_str_mv | AT doroninvg exactconstantsinjacksontypeinequalitiesforl2approximationonanaxis AT ligunaa exactconstantsinjacksontypeinequalitiesforl2approximationonanaxis AT doroninvg exactconstantsinjacksontypeinequalitiesforl2approximationonanaxis AT ligunaa exactconstantsinjacksontypeinequalitiesforl2approximationonanaxis AT doroninvg exactconstantsinjacksontypeinequalitiesforl2approximationonanaxis AT ligunaa exactconstantsinjacksontypeinequalitiesforl2approximationonanaxis AT doroninvg točnyekonstantyvneravenstvahtipadžeksonadlâl2approksimaciinaprâmoj AT ligunaa točnyekonstantyvneravenstvahtipadžeksonadlâl2approksimaciinaprâmoj AT doroninvg točnyekonstantyvneravenstvahtipadžeksonadlâl2approksimaciinaprâmoj AT ligunaa točnyekonstantyvneravenstvahtipadžeksonadlâl2approksimaciinaprâmoj AT doroninvg točnyekonstantyvneravenstvahtipadžeksonadlâl2approksimaciinaprâmoj AT ligunaa točnyekonstantyvneravenstvahtipadžeksonadlâl2approksimaciinaprâmoj |