Shape-Preserving Smoothing of 3-Convex Splines of Degree 4
For every 3-convex piecewise-polynomial function s of degree ≤ 4 with n equidistant knots on [0, 1] we construct a 3-convex spline $s_1 (s_1 ∈ C (3))$ of degree ≤ 4 with the same knots that satisfies the inequality $$\left\| {S - S_1 } \right\|_{C_{[0,1]} } \leqslant c\omega _5 (s;1/n),$$ where $c$...
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| Datum: | 2005 |
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| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2005
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3596 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509713437818880 |
|---|---|
| author | Prymak, A. V. Примак, А. В. |
| author_facet | Prymak, A. V. Примак, А. В. |
| author_sort | Prymak, A. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:22Z |
| description | For every 3-convex piecewise-polynomial function s of degree ≤ 4 with n equidistant knots on [0, 1] we construct a 3-convex spline $s_1 (s_1 ∈ C (3))$ of degree ≤ 4 with the same knots that satisfies the inequality
$$\left\| {S - S_1 } \right\|_{C_{[0,1]} } \leqslant c\omega _5 (s;1/n),$$
where $c$ is an absolute constant and $ω_5$ is the modulus of smoothness of the fifth order. |
| first_indexed | 2026-03-24T02:45:29Z |
| format | Article |
| fulltext |
UDK 517.5
A. V. Prymak (Ky]v. nac. un-t im. T. Íevçenka)
ZHLADÛUVANNQ ZI ZBEREÛENNQM FORMY
3-OPUKLYX SPLAJNIV 4-HO STEPENQ
For each 3-convex piecewise polynomial function s of degree ≤ 4 with n equidistant knots on ·0, 1‚,
we construct a 3-convex spline s1 s C1
3∈( )( ) of degree ≤ 4 with the same knots, which satisfies the
inequality
s s c s n
C
− ≤1 5
0 1
1
[ , ]
( ; / )ω ,
where c is some absolute constant and ω5 is fifth order modulus of smoothness.
Dlq koΩno] 3-opuklo] kuskovo-polinomial\no] funkci] s stepenq ≤ 4 z n rivnoviddalenymy
vuzlamy na ·0, 1‚ pobudovano 3-opuklyj splajn s1 s C1
3∈( )( )
stepenq ≤ 4 z tymy Ω vuzlamy,
wo zadovol\nq[ nerivnist\
s s c s n
C
− ≤1 5
0 1
1
[ , ]
( ; / )ω ,
de c — absolgtna stala, a ω5 — modul\ hladkosti p’qtoho porqdku.
1. Vstup. Pry nablyΩenni dijsnoznaçno] funkci], vyznaçeno], napryklad, na
vidrizku [ 0, 1], inodi neobxidno zberehty deqki ]] vlastyvosti, taki qk znak, mo-
notonnist\, opuklist\ ta inßi. Formozberihagçe nablyΩennq alhebra]çnymy
mnohoçlenamy ta kuskovo-polinomial\nymy funkciqmy (splajnamy) rozvyva[t\-
sq vΩe majΩe 30 rokiv.
Neperervna funkciq f : [ 0, 1] → R nazyva[t\sq q-opuklog, qkwo
k
q
q k q
k
f x kh
=
−∑ −
+
0
1( ) ( ) ≥ 0, x ∈ [ 0, 1], h ∈ 0 1, −
x
q
.
Poznaçatymemo ce f ∈ ∆q
. Zrozumilo, wo 1- ta 2-opukli funkci] — vidpovidno
nespadni ta opukli donyzu funkci]. Rizni vlastyvosti q-opuklyx funkcij moΩ-
na znajty v ·1‚.
Zadaçi monotonnoho ta opukloho nablyΩennq kuskovo-polinomial\nymy
funkciqmy z fiksovanymy vuzlamy na skinçennomu intervali rozhlqdalys\,
napryklad, u robotax ·2;–;6‚. Formozberihagçe nablyΩennq vywyx porqdkiv,
tobto q-opukle nablyΩennq pry q ≥ 3, intensyvno doslidΩu[t\sq v ostanni
roky z dewo nespodivanymy rezul\tatamy. A same, analohy vstanovlenyx dlq
q = 1, 2 pozytyvnyx rezul\tativ ne magt\ miscq dlq q ≥ 4 (dyv. ·7‚), toçniße,
pry q ≥ 4 ne moΩna otrymaty Ωodno] ocinky typu DΩeksona z porqdkom
nablyΩennq navit\ n–3
qk dlq nablyΩennq mnohoçlenamy, tak i dlq nably-
Ωennq kuskovo-polinomial\nymy funkciqmy. Tomu zalyßavsq vypadok q = 3,
dlq qkoho newodavno dovedeno pozytyvni rezul\taty, analohiçni vstanovlenym
dlq q = 1, 2 (dyv. ·8;–;10‚).
Sformulg[mo ci rezul\taty dlq vypadku rivnoviddalenyx vuzliv. Nexaj
ωk ( f ; h) — modul\ hladkosti k-ho porqdku neperervno] na [ 0, 1] funkci] f z
krokom h. Todi qkwo nevid’[mni k, r, ( k, r ) ≠ ( 4, 0 ), taki, wo k + r ≥ 3 i abo k +
+ r ≤ 4 abo r ≥ 3, to dlq dovil\no] f ∈ ∆3 ∩ C
(r) [ 0, 1] isnu[ kuskovo-
polinomial\na funkciq s ∈ ∆3
stepenq ≤ k + r – 1 z n rivnoviddalenymy
vuzlamy taka, wo
f s C− [ , ]0 1 ≤ c k r n f
n
r
k
r( , ) ;( )−
ω 1 . (1)
© A. V. PRYMAK, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 277
278 A. V. PRYMAK
Ale wodo hladkosti takyx kuskovo-polinomial\nyx funkcij u statti ·10‚
dovedeno lyße moΩlyvist\ zhladΩuvannq 3-opuklyx kuskovo-polinomial\nyx
funkcij do kuskovo-polinomial\nyx funkcij z C
(2)
zi zbereΩennqm porqdku
nablyΩennq. Ce da[ moΩlyvist\ pobuduvaty kubiçni 3-opukli splajny naj-
menßoho defektu, wo dagt\ porqdok nablyΩennq n–4
(n — kil\kist\ vuzliv).
U cij roboti my navodymo konstrukcig 3-opukloho splajnu çetvertoho stepenq i
najmenßoho defektu, wo da[ ocinku dlq nablyΩennq porqdku n–5
,
poßyrggçy ocinku (1) dlq k + r = 5, r ≥ 3 i na splajny najmenßoho defektu
(dyv. naslidok 1). Toçniße, my dovodymo nastupnu teoremu, wo dozvolq[ zhla-
dyty dovil\nu 3-opuklu kuskovo-polinomial\nu funkcig stepenq ≤ 4 z rivno-
viddalenymy vuzlamy do splajnu najmenßoho defektu z tymy Ω vuzlamy, zberi-
hagçy ocinku na nablyΩennq.
Teorema 1. Dlq koΩno] 3-opuklo] kuskovo-polinomial\no] funkci] s stepe-
nq ≤ 4 z n rivnoviddalenymy vuzlamy na [ 0, 1] isnu[ 3-opuklyj splajn s1
(s1 ∈ C
(3)) stepenq ≤ 4 z tymy samymy vuzlamy, wo zadovol\nq[ nerivnist\
s s C− 1 0 1[ , ] ≤ c s
n
ω5
1;
, (2)
de c — absolgtna stala.
Vraxovugçy rezul\taty statej ·10, 6‚, otrymu[mo nastupni naslidky.
Naslidok 1. Dlq dovil\no] f ∈ ∆3 ∩ C
(3) [ 0, 1] isnu[ splajn s ∈ ∆3 çet-
vertoho stepenq minimal\noho defektu z n rivnoviddalenymy vuzlamy, wo za-
dovol\nq[ nerivnist\
f s C− [ , ]0 1 ≤ cn f
n
−
3
2
3 1ω ( ); ,
de c — absolgtna stala.
Naslidok 2. Dlq dovil\no] f ∈ ∆3 ∩ C
(1) [ 0, 1] isnu[ çyslo N ( f ) take, wo
dlq bud\-qkoho n ≥ N ( f ) isnu[ splajn s ∈ ∆3 çetvertoho stepenq minimal\-
noho defektu z n rivnoviddalenymy vuzlamy, wo zadovol\nq[ nerivnist\
f s C− [ , ]0 1 ≤ cn f
n
− ′
1
4
1ω ; , (3)
de c — absolgtna stala.
ZauvaΩymo, wo ocinka (3), vzahali kaΩuçy, ne vykonu[t\sq dlq vsix n.
U p. 2 my dovedemo deqki dopomiΩni lemy, a v p. 3 — teoremu 1.
2. DopomiΩni lemy. Poznaçymo ( )a k
+ : = max{ ; }0 a k( ) , k = 1, 2, … , ta
( )a +
0 : = 1, qkwo a ≥ 0, i ( )a +
0 : = 0, qkwo a < 0.
Nastupna lema [ modyfikaci[g lemy 3 z ·11‚ (dyv. takoΩ ·10‚, lema 11) dlq
rivnoviddalenyx vuzliv i B = 3. Vona dovodyt\sq analohiçno, tomu my ne navo-
dymo ]] dovedennq.
Lema 1. Nexaj x j = j l, j = 0, … , n, l = n–1. Dlq dovil\no] funkci]
g ( x ) =
i
n
i ix x
=
−
+∑ −
1
1
0α ( ) , x ∈ [ 0, 1] ,
de αi ≥ 0, isnu[ lamana
p ( x ) =
i
n
i il x x
=
−
−
+∑ −
1
1
1β ( )
taka, wo
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
ZHLADÛUVANNQ ZI ZBEREÛENNQM FORMY 3-OPUKLYX SPLAJNIV … 279
βi <
αi
3
, i = 1, … , n – 1,
i
| g ( x ) – p ( x ) | < 24
1 1
max
, ,i n
i
= … −
α , x ∈ [ 0, 1] .
Lema 2. Nexaj s — kuskovo-polinomial\na funkciq stepenq ≤ k – 1 na [ a,
b] z [dynym vuzlom rozbyttq c : = a b+
2
. Todi
s c s cr r( ) ( )( ) ( )+ − − ≤
c k r
b a
s b ar k
( , )
( )
( ; )
−
−ω , r = 0, 1, … .
Dovedennq. Za nerivnistg Uitni (dyv., napryklad, ·12‚) isnu[ mnohoçlen p
stepenq ≤ k – 1 takyj, wo
s p C a b− [ , ] ≤ 3ωk s b a( ; )− .
Za nerivnistg Markova
s c p cr r( ) ( )( ) ( )+ − ≤ 4 1 2( )
[ , ]
k
b a
s p
r
C a b
−
−
− ≤
≤ 3 4 1 2( ) ( ; )k
b a
s b a
r
k
−
−
−ω
i analohiçno
s c p cr r( ) ( )( ) ( )− − ≤ 3 4 1 2( ) ( ; )k
b a
s b a
r
k
−
−
−ω .
OtΩe,
s c s cr r( ) ( )( ) ( )+ − − ≤ s c p c p c s cr r r r( ) ( ) ( ) ( )( ) ( ) ( ) ( )+ − + − − ≤
≤ 6 4 1 2( ) ( ; )k
b a
s b a
r
k
−
−
−ω =
c k r
b a
s b ar k
( , )
( )
( ; )
−
−ω .
Lema 3. Nexaj x j = j l, j = 0, … , n , l = n–1, α
1, … , α
n – 1 — deqki dijsni
çysla. Isnugt\ β β1 1
* *, ,… −n taki, wo
β j
* ≥ – | α
j | , j = 1, … , n – 1, (4)
ta dlq x ∈ [ 0, 1] = [ x
0, x n ]
j
n
j
j
j
n
j
j jl x x l x x l x x
=
−
+
=
−
+ +∑ ∑
−
−
−
+
−
1
1 2
1
1 2 3 0
3 2 12
α β
( ) ( ) ( )* < cl
j n
i
3
1 1
max
, ,= … −
α . (5)
Dovedennq. Dlq zruçnosti poklademo α
–1 = α
0 = α
n = α
n + 1 : = 0. Spoçat-
ku pokaΩemo, wo dlq koΩno] poslidovnosti λ
j
, | λ
j | ≤ 1
3
, j = 1, … , n – 1,
isnugt\ β
j
, j = 0, … , n, taki, wo
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
280 A. V. PRYMAK
βj ≥ − α j , j = 0, … , n, (6)
i
j
n
j
j
j
n
j
j jl x x l x x l x x
=
−
+
=
+ +∑ ∑
−
−
−
+
−
1
1 2
0
2 3 0
3 2 12
α β
( ) ( ) ( )
=
=
j
n
j j j
j
n
j jl x x l x x r x
=
−
+
=
−
+∑ ∑− − − +
1
1
2
1
1
3 0λ α α( ) ( ) ( ), x ∈ [ 0, 1] , (7)
de r taka, wo
| r ( x ) | ≤ c l
j n
j0
3
1 1
max
, ,= … −
α , x ∈ [ 0, 1] . (8)
Dlq c\oho, pokladagçy λ
–1 = λ
0 = λ
n = λ
n + 1 : = 0, vyznaça[mo
βj : = − +
+
− +
−
+−
−
− +
+
+α
λ
α
α λ
α
α
j
j
j
j j
j
j1
2 6
1
2 6
1
1
1 1
1
1 , j = 0, … , n.
(9)
Zrozumilo, wo
1
2
1
1
∓ λ
αj
j
±
± ≥ 1
3 1α j± ≥
∓ 1
6 1α j± , j = 0, … , n,
tomu vykonu[t\sq (6). Bezposerednq perevirka pokazu[, wo pry j = 1, … , n – 1
funkciq rj, zadana spivvidnoßennqm
α
λ
α
α
j
j j
j
j j jl x x l x x l x x2
1
2 3
1
0
3
1
2 6 2 12
( ) ( ) ( )−
−
−
+
−
+
−
+ − + − +
+
+ α j
j jl x x l x x( ) ( )−
+
−
+ +
2 3 0
2 12
–
–
1
2 6 2 12
1
2 3
1
0+
−
−
+
−
+ + + +λ
α
αj
j
j j jl x x l x x( ) ( )
=
= λ α αj j j j j jl x x l x x r x2 3 0( ) ( ) ( )− − − ++ + , x ∈ [ 0, 1] , (10)
zadovol\nq[
rj ( x ) = 0, x ∉ x xj j− +[ ]1 1, , (11)
ta
| rj ( x ) | ≤ c l j1
3 α , x ∈ x xj j− +[ ]1 1, . (12)
Tomu (9) ta (10) zabezpeçugt\ (7) z
r ( x ) =
j
n
jr x
=
−
∑
1
1
( ), x ∈ [ 0, 1] .
Zavdqky (11) ta (12) otrymu[mo (8) z c0 = 2c1
. OtΩe, (7) dovedeno. Wob
zakinçyty dovedennq lemy, zastosu[mo lemu 1, qka zabezpeçu[ isnuvannq λ
j
,
| λ
j | ≤ 1
3
, j = 1, … , n, takyx, wo
λ α αj j
j
n
j j
j
n
jl x x l x x
=
−
+
=
−
+∑ ∑− − −
1
1
2
1
1
3 0( ) ( ) ≤ c l
j n
j0
3
1 1
max
, ,= … −
α , x ∈ [ 0, 1] .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
ZHLADÛUVANNQ ZI ZBEREÛENNQM FORMY 3-OPUKLYX SPLAJNIV … 281
Z ostann\o] nerivnosti ta z (7) i (8) dlq x ∈ [ 0, 1] ma[mo
j
n
j
j
j
n
j
j jl x x l x x l x x
=
−
+
=
+ +∑ ∑
−
−
−
+
−
1
1 2
0
2 3 0
3 2 12
α β
( ) ( ) ( )
< c l
j n
j0
3
1 1
max
, ,= … −
α .
Beruçy β1
* : = β1 + β0, β j
* : = βj, j = 2, … , n – 1, pryxodymo do (5).
3. Dovedennq teoremy 1. Zafiksu[mo n ≥ 2, l : = n–1, x j : = j l, j = –1, … , n +
+ 1. Vvedemo funkci] ϕ j , ψ j , j = 0, … , n. Poklademo
ϕj ( x ) : =
( ) , ( , ],
( ) , ( , ),
, [ , ] \ ( , ),
x x l x x x
x x l x x x
x x x
j j j
j j j
j j
− ∈
− − ∈
∈
−
−
−
+
−
+
− +
1
1
1
1
1
1
1 10 0 1
i
ψj ( x ) : =
( ) , ( , ],
( ) , ( , ),
, [ , ] \ ( , ).
x x l x x x
x x l x x x
x x x
j j j
j j j
j j
− ∈
− ∈
∈
−
−
−
+
−
+
− +
1
1
1
1
1
1
1 10 0 1
Nastupni dvi lemy perevirqgt\sq za dopomohog bezposeredn\oho obçyslennq.
Lema 4. KoΩna kuskovo-linijna funkciq q z vuzlamy x
j , j = 1, … , n – 1,
na [ 0, 1 ] , ma[ vyhlqd
q ( x ) =
j
n
i j
j
n
i jx u x
=
−
=
∑ ∑+
1
1
0
α ψ ϕ( ) ( ), x ∈ [ 0, 1] , x ≠ x k , k = 1, … , n – 1, (13)
de
αj =
q x q xj j( ) ( )− − +
2
, j = 1, … , n – 1, (14)
i
uj =
q x q xj j( ) ( )− + +
2
, j = 0, … , n.
Qkwo q nevid’[mna, to pry c\omu
uj =
q x q xj j( ) ( )− + +
2
≥ | α
j | ≥ 0, j = 0, … , n. (15)
Lema 5. Dlq funkcij rj k,
ϕ , j = 1, … , n – 1, k = 1, 2, 3, vyznaçenyx spivvidno-
ßennqmy
0
1 1
x
j t dt∫ ϕ ( ) = l x x r xj j( ) ( ),− ++
0
1
ϕ , x ∈ [ 0, 1] ,
0 0
2 2 1
1x t
j t dt dt∫ ∫ ϕ ( ) = l x x r xj j( ) ( ),− ++ 2
ϕ , x ∈ [ 0, 1] ,
0 0 0
3 3 2 1
1 2x t t
j t dt dt dt∫ ∫ ∫ ϕ ( ) = l x x l x x r xj j j2 12
2
3
0
3( ) ( ) ( ),− + − ++ +
ϕ , x ∈ [ 0, 1] ,
vykonu[t\sq
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
282 A. V. PRYMAK
r xj k, ( )ϕ = 0, x ∉ x xj j− +( )1 1, , j = 1, … , n – 1, k = 1, 2, 3,
ta
r xj k, ( )ϕ ≤ clk , x ∈ x xj j− +( )1 1, , j = 1, … , n – 1, k = 1, 2, 3.
TakoΩ dlq funkcij rj k,
ψ , j = 1, … , n – 1, k = 1, 2, 3, vyznaçenyx spivvidnoßen-
nqmy
0
1 1
x
j t dt∫ ψ ( ) = r xj, ( )1
ψ , x ∈ [ 0, 1] ,
0 0
2 2 1
1x t
j t dt dt∫ ∫ ψ ( ) = l x x r xj j
2
0
23
( ) ( ),− ++
ψ , x ∈ [ 0, 1] ,
0 0 0
3 3 2 1
1 2x t t
j t dt dt dt∫ ∫ ∫ ψ ( ) =
l
x x r xj j
2
33
( ) ( ),− ++
ψ , x ∈ [ 0, 1] ,
vykonu[t\sq
r xj k, ( )ψ = 0, x ∉ x xj j− +( )1 1, , j = 1, … , n – 1, k = 1, 2, 3,
ta
r xj k, ( )ψ ≤ clk , x ∈ x xj j− +( )1 1, , j = 1, … , n – 1, k = 1, 2, 3.
Dovedennq teoremy 1. Za teoremog 5 ·10‚ isnu[ 3-opukla kuskovo-polino-
mial\na funkciq s̃ ∈ C( )[ , ]2 0 1 stepenq ≤ 4 z rivnoviddalenymy vuzlamy, wo
zadovol\nq[ nerivnist\
s s C− ˜
[ , ]0 1 ≤ c s
n
ω5
1;
. (16)
Todi q ( x ) : = ˜ ( )( )s x3 , x ∈ [ 0, 1] , [ kuskovo-linijnog nevid’[mnog funkci[g, dlq
qko]
s ( x ) = p x q t dt dt dt
x t t
2
0 0 0
3 3 2 1
1 2
( ) ( )+ ∫ ∫ ∫ , x ∈ [ 0, 1] , (17)
de
p2 ( x ) = ˜( ) ˜ ( )
˜ ( )s s x s x0 0 0
2
2+ ′ + ′′ , x ∈ [ 0, 1] .
Teper, zastosovugçy lemu 4, otrymu[mo (13). Z (14) i lemy 2 vyplyva[, wo
l
j n j
3
1 1
max
, ,= … −
α ≤ c s
n
ω5
1˜;
. (18)
Zastosu[mo lemu 3 i poklademo
q1 ( x ) : = u x u x u xn n
j
n
j j j0 0
1
1
ϕ ϕ β ϕ( ) ( ) ( ) ( )*+ + +
=
−
∑ , x ∈ [ 0, 1] ,
de
s1 ( x ) : = p x q t dt dt dt
x t t
2
0 0 0
1 3 3 2 1
1 2
( ) ( )+ ∫ ∫ ∫ , x ∈ [ 0, 1] .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
ZHLADÛUVANNQ ZI ZBEREÛENNQM FORMY 3-OPUKLYX SPLAJNIV … 283
Z nerivnostej (4) i (15) ma[mo uj j+ β* ≥ 0, j = 1, … , n – 1, tomu q1 [ nevid’[m-
nog kuskovo-linijnog neperervnog funkci[g, zvidky vyplyva[, wo s1 [ 3-
opuklym splajnom çetvertoho stepenq z rivnoviddalenymy vuzlamy. Beruçy do
uvahy (5), (13), (17), (18) i lemu 5, otrymu[mo
s s C1 0 1− ˜
[ , ] ≤ c s
n
ω5
1˜;
.
Teper z (16) vyplyva[ (2).
1. Roberts A. W., Varbeg D. E. Convex functions. – New York: Acad. Press, 1973.
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P. 891 – 905.
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P. 69 – 82.
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algebraic polynomials // Constr. Approxim. – 1994. – 10. – P. 153 – 178.
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P. 357 – 368.
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functions on a finite interval // Isr. J. Math. – 2003. – 133. – P. 239 – 268.
8. Konovalov V. N., Leviatan D. Estimates on the approximation of 3-monotone function by 3-
monotone quadratic splines // East J. Approxim. – 2001. – 7. – P. 333 – 349.
9. Prymak A. V. Three-convex approximation by quadratic splines with arbitrary fixed knots // Ibid. –
2002. – 8, # 2. – P. 185 – 196.
10. Leviatan D., Prymak A. V. On 3-monotone approximation by piecewise polynomials // J.
Approxim. Theory. – 2005. – 133. – P. 97 – 121.
11. Bondarenko A. V. Jackson type inequality in 3-convex approximation // East J. Approxim. – 2002.
– 8, # 3 – P. 291 – 302.
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OderΩano 12.12.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
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| id | umjimathkievua-article-3596 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:45:29Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c8/02f1cff3bc0482af40ae07b61043fbc8.pdf |
| spelling | umjimathkievua-article-35962020-03-18T19:59:22Z Shape-Preserving Smoothing of 3-Convex Splines of Degree 4 Згладжування зі збереженням форми 3-опуклих сплайнів 4-го степеня Prymak, A. V. Примак, А. В. For every 3-convex piecewise-polynomial function s of degree ≤ 4 with n equidistant knots on [0, 1] we construct a 3-convex spline $s_1 (s_1 ∈ C (3))$ of degree ≤ 4 with the same knots that satisfies the inequality $$\left\| {S - S_1 } \right\|_{C_{[0,1]} } \leqslant c\omega _5 (s;1/n),$$ where $c$ is an absolute constant and $ω_5$ is the modulus of smoothness of the fifth order. Для кожної 3-опуклої кусково-поліноміальної функції s степеня ≤4 з п рівновіддалсиими вузлами на [0,1 ] побудовано 3-опуклий сплайн $s_1 (s_1 ∈ C (3))$ степеня ≤ 4 з тими ж вузлами, що задовольняє нерівність $$\left\| {S - S_1 } \right\|_{C_{[0,1]} } \leqslant c\omega _5 (s;1/n),$$ де $c$—абсолюгнастала, а $ω_5$ — модуль гладкості п'ятого порядку. Institute of Mathematics, NAS of Ukraine 2005-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3596 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 2 (2005); 277–283 Український математичний журнал; Том 57 № 2 (2005); 277–283 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3596/3923 https://umj.imath.kiev.ua/index.php/umj/article/view/3596/3924 Copyright (c) 2005 Prymak A. V. |
| spellingShingle | Prymak, A. V. Примак, А. В. Shape-Preserving Smoothing of 3-Convex Splines of Degree 4 |
| title | Shape-Preserving Smoothing of 3-Convex Splines of Degree 4 |
| title_alt | Згладжування зі збереженням форми 3-опуклих сплайнів 4-го степеня |
| title_full | Shape-Preserving Smoothing of 3-Convex Splines of Degree 4 |
| title_fullStr | Shape-Preserving Smoothing of 3-Convex Splines of Degree 4 |
| title_full_unstemmed | Shape-Preserving Smoothing of 3-Convex Splines of Degree 4 |
| title_short | Shape-Preserving Smoothing of 3-Convex Splines of Degree 4 |
| title_sort | shape-preserving smoothing of 3-convex splines of degree 4 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3596 |
| work_keys_str_mv | AT prymakav shapepreservingsmoothingof3convexsplinesofdegree4 AT primakav shapepreservingsmoothingof3convexsplinesofdegree4 AT prymakav zgladžuvannâzízberežennâmformi3opuklihsplajnív4gostepenâ AT primakav zgladžuvannâzízberežennâmformi3opuklihsplajnív4gostepenâ |