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.
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| author | Doronin, V. G. Ligun, A. A. Serdyuk, A. S. Shydlich, A. L. Доронин, В. Г. Лигун, А. А. Сердюк, А. С. Шидлич, А. Л. Доронин, В. Г. Лигун, А. А. Сердюк, А. С. Шидлич, А. Л. |
| author_facet | Doronin, V. G. Ligun, A. A. Serdyuk, A. S. Shydlich, A. L. Доронин, В. Г. Лигун, А. А. Сердюк, А. С. Шидлич, А. Л. Доронин, В. Г. Лигун, А. А. Сердюк, А. С. Шидлич, А. Л. |
| 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 | 2026-03-24T02:34:25Z |
| 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-3005 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:34:25Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b0/5ca1d4096ebae3e31a9ad4d29d7046b0.pdf |
| spelling | umjimathkievua-article-30052020-03-18T19:43:07Z 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. Serdyuk, A. S. Shydlich, A. L. Доронин, В. Г. Лигун, А. А. Сердюк, А. С. Шидлич, А. Л. Доронин, В. Г. Лигун, А. А. Сердюк, А. С. Шидлич, А. Л. 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/3005 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/3005/2759 https://umj.imath.kiev.ua/index.php/umj/article/view/3005/2760 Copyright (c) 2009 Doronin V. G.; Ligun A. A.; Serdyuk A. S.; Shydlich A. L. |
| spellingShingle | Doronin, V. G. Ligun, A. A. Serdyuk, A. S. Shydlich, A. L. Доронин, В. Г. Лигун, А. А. Сердюк, А. С. Шидлич, А. Л. Доронин, В. Г. Лигун, А. А. Сердюк, А. С. Шидлич, А. Л. 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/3005 |
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