Negative result in pointwise 3-convex polynomial approximation
Let $Δ^3$ be the set of functions three times continuously differentiable on $[−1, 1]$ and such that $f'''(x) ≥ 0,\; x ∈ [−1, 1]$. We prove that, for any $n ∈ ℕ$ and $r ≥ 5$, there exists a function $f ∈ C^r [−1, 1] ⋂ Δ^3 [−1, 1]$ such that $∥f (r)∥_{C[−1, 1]} ≤ 1$ and, fo...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509061132320768 |
|---|---|
| author | Bondarenko, A. V. Gilewicz, J. Бондаренко, А. В. Гилевич, Я. Я. Бондаренко, А. В. Гилевич, Я. Я. |
| author_facet | Bondarenko, A. V. Gilewicz, J. Бондаренко, А. В. Гилевич, Я. Я. Бондаренко, А. В. Гилевич, Я. Я. |
| author_sort | Bondarenko, A. V. |
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| datestamp_date | 2020-03-18T19:43:50Z |
| description | Let $Δ^3$ be the set of functions three times continuously differentiable on $[−1, 1]$ and such that $f'''(x) ≥ 0,\; x ∈ [−1, 1]$. We prove that, for any $n ∈ ℕ$ and $r ≥ 5$, there exists a function $f ∈ C^r [−1, 1] ⋂ Δ^3 [−1, 1]$ such that $∥f (r)∥_{C[−1, 1]} ≤ 1$ and, for an arbitrary algebraic polynomial $P ∈ Δ^3 [−1, 1]$, there exists $x$ such that
$$|f(x)−P(x)| ≥ C \sqrt{n}ρ^r_n(x),$$
where $C > 0$ is a constant that depends only on
$r, ρ_n(x) := \frac1{n^2} + \frac1n \sqrt{1−x^2}$. |
| first_indexed | 2026-03-24T02:35:06Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 517.5
A. V. Bondarenko (Ky]v. nac. un-t im. T. Íevçenka),
Q. Q. Hilevyç (Centr teor. fizyky, Lgmini, Marsel\, Franciq)
NEHATYVNYJ REZUL|TAT U POTOÇKOVOMU
3-OPUKLOMU NABLYÛENNI MNOHOÇLENAMY
Let ∆3 be a set of functions three times continuously differentiable on −[ ]1 1, and such that ′′′f x( ) ≥
≥ 0, x ∈ −[ ]1 1, . We prove that, for arbitrary n ∈N and r ≥ 5, there exists a function f ∈
∈ Cr −[ ]1 1, I ∆3 −[ ]1 1, such that f r
C
( )
,−[ ]1 1
≤ 1 and, for an arbitrary algebraic polynomial P ∈
∈ ∆3 −[ ]1 1, , there exists x such that
f x P x( ) – ( ) ≥ C n xn
rρ ( ) ,
where C > 0 is a constant depending only on r, ρn x( ) : =
1
2n
+
1
1 2
n
x− .
Pust\ ∆3 qvlqetsq mnoΩestvom tryΩd¥ neprer¥vno dyfferencyruem¥x funkcyj na −[ ]1 1,
takyx, çto ′′′f x( ) ≥ 0, x ∈ −[ ]1 1, . Dokazano, çto dlq proyzvol\n¥x n ∈N y r ≥ 5 suwestvuet
funkcyq f ∈ Cr −[ ]1 1, I ∆3 −[ ]1 1, takaq, çto f r
C
( )
,−[ ]1 1
≤ 1, y dlq proyzvol\noho alheb-
rayçeskoho polynoma P ∈ ∆3 −[ ]1 1, suwestvuet x takoe, çto
f x P x( ) – ( ) ≥ C n xn
rρ ( ) ,
hde C > 0 � postoqnnaq, zavysqwaq tol\ko ot r, ρn x( ) : =
1
2n
+
1
1 2
n
x− .
1. Vstup. Nexaj C a b,[ ] � prostir neperervnyx na a b,[ ] funkcij iz rivno-
mirnog normog ⋅ [ ]a b, : = ⋅ [ ]C a b, . Nexaj funkciq f naleΩyt\ prostoru So-
bol[va W r
∞ −[ ]1 1, , tobto prostoru funkcij, u qkyx poxidna f r( – )1 [ absolgtno
neperervnog na −[ ]1 1, ta f r( ) ∈ L∞ −[ ]1 1, . Sformulg[mo teper klasyçnu teo-
remu dlq potoçkovoho nablyΩennq alhebra]çnymy polinomamy [1].
Teorema. Nexaj f ∈ W r
∞ −[ ]1 1, , todi dlq dovil\noho n ≥ r – 1 isnu[ mnoho-
çlen pn ∈ Pn iz prostoru Pn mnohoçleniv stepenq ne vywoho za n takyj,
wo
f x p xn( ) – ( ) < C r x fn
r r
L
( ) ( ) ( )
,
ρ
∞ −[ ]1 1
, x ∈ −[ ]1 1, ,
de C r( ) � konstanta, qka zaleΩyt\ vid r, ρn x( ) : = 1
2n
+ 1
n
1 2– x .
Vynyka[ pytannq: çy spravdΩugt\sq analohy ci[] nerivnosti u vypadku for-
mozberihagçoho nablyΩennq?
Poznaçymo çerez ∆q a b,[ ] klas q-opuklyx funkcij na vidrizku a b,[ ]. Dlq
q = 1 ta q = 2 klasamy ∆q a b,[ ] [ klasy monotonnyx ta opuklyx na vidrizku
a b,[ ] funkcij vidpovidno. Dlq q ≥ 3 ∆q a b,[ ] � klas neperervnyx na a b,[ ]
© A. V. BONDARENKO, Q. Q. HILEVYÇ, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 563
564 A. V. BONDARENKO, Q. Q. HILEVYÇ
funkcij, qki magt\ opuklu poxidnu f q( – )2 na intervali ( , )a b . U cij roboti
bude rozhlqnuto okremyj vypadok tako] zadaçi: pry qkyx q, r ∈N dlq dovil\-
nyx f ∈ W r
∞ −[ ]1 1, I ∆q −[ ]1 1, ta n ≥ r – 1 isnu[ pn ∈ Pn I ∆q −[ ]1 1, take, wo
f x p xn( ) – ( ) < C r q x fn
r r
L
( , ) ( ) ( )
,
ρ
∞ −[ ]1 1
, x ∈ −[ ]1 1, , (1)
de C r q( , ) � konstanta, qka zaleΩyt\ til\ky vid r ta q.
Dlq q = 1 ta q = 2 nerivnist\ (1) bulo dovedeno pry vsix r ∈N (dyv., na-
pryklad, [1]). Z inßoho boku, dlq q ≥ 4, r ≥ 3 u roboti [2] dovedeno, wo neriv-
nist\ (1) ne ma[ miscq, ta pobudovano vidpovidni kontrpryklady. Dlq q ≥ 3 u
roboti [3] vstanovleno nerivnist\ (1) pry r = 1 i r = 2.
Osnovnym rezul\tatom dano] roboty [ pobudova kontrprykladiv do nerivnos-
ti (1) u vypadku q = 3 , r ≥ 5. Dlq c\oho vykorystano texniku, zaprovadΩenu v
[4]. My dovedemo nastupnu teoremu.
Teorema 1. Dlq dovil\noho n ∈N isnu[ funkciq f ∈ C5 1 1−[ ], I ∆3 1 1−[ ],
taka, wo f
C
( )
,
5
1 1−[ ]
≤ 1, ta dlq dovil\noho P ∈ Pn I ∆3 1 1−[ ], isnu[ x take,
wo
f x P x( ) – ( ) ≥ C n xnρ5 ( ),
de C > 0 � absolgtna stala.
Teorema 2. Dlq dovil\nyx n ∈N ta r ≥ 5 isnu[ funkciq f ∈ Cr −[ ]1 1, I
I ∆3 1 1−[ ], taka, wo f r
C
( )
,−[ ]1 1
≤ 1, ta dlq dovil\noho P ∈ Pn I ∆3 1 1−[ ],
isnu[ x take, wo
f x P x( ) – ( ) ≥ C n xn
rρ ( ),
de C > 0 � stala, wo zaleΩyt\ lyße vid r.
2. Dovedennq teorem. U podal\ßomu budemo pysaty ∆3 zamist\ ∆3 1 1−[ ], .
Poznaçymo In : = −[ 1, – 1 + 1
n
U 1
– 1
n
, 1
ta dlq funkci] f ∈ C −[ ]1 1, vy-
znaçymo f
nI : = maxx n∈I f x( ) . Dlq dovedennq teoremy 1 nam znadoblqt\sq
dekil\ka lem.
Lema 1. Nexaj f � opukla na a b,[ ] funkciq, x∗ ∈ a b,[ ], f a( ) ≤ 0,
f x( )∗ < 0. Todi pry vsix t ∈ a x, ∗[ ] vykonu[t\sq nerivnist\
a
t
f x dx∫ ( ) ≤ 1
2
2
f x
t a
b a
( )
( – )
–
∗ .
Dovedennq. Rozhlqnemo prqmu
l x( ) = f x
x a
x a
( )
–
–
∗
∗ , x a b∈[ ], .
Oskil\ky funkciq f [ opuklog, to l x( ) ≥ f (x) pry vsix x a x∈[ ]∗, , tomu
a
t
f x dx∫ ( ) ≤
a
t
l x dx∫ ( ) =
a
t
f x
x a
x a
dx∫ ∗
∗( )
–
–
=
= 1
2
2
f x
t a
x a
( )
( – )
–
∗
∗ ≤ 1
2
2
f x
t a
b a
( )
( – )
–
∗ .
Lemu 1 dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
NEHATYVNYJ REZUL|TAT U POTOÇKOVOMU 3-OPUKLOMU NABLYÛENNI … 565
Lema 2. Nexaj f � opukla na a b,[ ] funkciq, a < a1 < x∗ < b , f a( ) > 0,
f a( )1 = 0, f x( )∗ < 0. Todi pry vsix t a a∈[ ], 1 vykonu[t\sq nerivnist\
a
t
f x dx∫ ( ) ≥ − ∗1
2
2
f x
t a
b a
( )
( – )
–
.
Dovedennq. Rozhlqnemo prqmu
l x( ) = f x
x a
x a
( )
–
–
∗
∗
1
1
, x a b∈[ ], .
Oskil\ky funkciq f [ opuklog, to l x( ) ≤ f x( ) pry vsix x a a∈[ ], 1 . Zvidsy
otrymu[mo
a
t
f x dx∫ ( ) ≥
a
t
l x dx∫ ( ) ≥
a t a
a
l x dx
1
1
− −
∫
( )
( ) =
a t a
a
f x
x a
x a
dx
1
1
1
1− −
∗
∗∫
( )
( )
–
–
=
= − ∗
∗
1
2
2
1
f x
t a
x a
( )
( – )
–
≥ − ∗1
2
2
f x
t a
b a
( )
( – )
–
.
Lemu 2 dovedeno.
Poznaçymo teper x+ : = x( + x) / 2. Navedemo tverdΩennq, neobxidne nam u
podal\ßomu (dovedennq dyv. u [5]).
TverdΩennq. Dlq bud\-qkoho p n∈P ma[mo
x p x+ −[ ]– ( ) ,1 1 > 1
160n
> 1
28 n
. (2)
Dovedemo teper osnovnu lemu.
Lema 3. Nexaj n ∈N , n > 2, P n∈P I ∆3 , g x( ) : = P x( ) –
x +
2
2
, todi g In
≥
≥ 1
226 4n
.
Dovedennq. Nexaj ′′ −[ ]P 2 3 2 3/ , / > 2, todi, oskil\ky ′′P [ monotonnog,
ma[mo ′′P x( ) > 2 pry vsix x ∈ 1[ – 1/n , 1] abo ′′P x( ) < – 2 pry vsix x ∈ −[ 1, – 1 +
+ 1/n]. Qkwo ′′P x( ) > 2 pry vsix x ∈ 1[ – 1/n , 1], to ′′g x( ) > 1 pry vsix x ∈ 1[ –
– 1/n , 1]. Dali z formuly Tejlora z zalyßkovym çlenom u formi LahranΩa
vyplyva[, wo dlq deqkoho η ∈ 1[ – 1/n , 1]
g( )1 – 2 1 1
2
g
n
–
+ g
n
1 1–
= 1
4 2n
g′′( )η > 1
4 2n
,
a otΩe, g n1 1 1−[ ]/ , > 1
16 2n
. Analohiçno rozhlqda[t\sq vypadok ′′P x( ) < – 2 pry
vsix x ∈ −[ 1, – 1 + 1/n].
Nexaj ′′ −[ ]P 2 3 2 3/ , / ≤ 2. Prypustymo teper, wo lema ne [ pravyl\nog, ta
dovedemo isnuvannq x∗ ∈ −[ 1 + 1/n , 1 – 1/n] takoho, wo ′ ∗g x( ) < − 1
222 2n
. Dijs-
no, zhidno z (2) ′ −[ ]g 1 2 1 2/ , / ≥ 1
29 n
. Nexaj x1 ∈ −
1
2
1
2
, take, wo ′g x( )1 =
= ′ −[ ]g 1 2 1 2/ , / . Qkwo ′g x( )1 < 0, to lemu dovedeno. Qkwo Ω ′g x( )1 > 0 ta
′ ∗g x( ) > − 1
222 2n
pry vsix x∗ ∈ −[ 1 + 1/n , 1 – 1/n], to
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
566 A. V. BONDARENKO, Q. Q. HILEVYÇ
g
n
1 1–
– g
n
− +
1 1 =
− +
−
∫ ′
1 1
1 1
/
/
( )
n
n
g x dx >
x n
x n
g x dx
1
11
1
11
1 2
1 2
−
+
∫ ′
/
/
( ) – 2 1
222 2n
≥
≥
x n
x n
g x
1
11
1
11
1 2
1 2
1
−
+
∫ ′
/
/
( ) – ′′ −[ ]g x x dx2 3 2 3 1/ , / – – 2 1
222 2n
≥ 1
221 2n
.
Tomu u c\omu vypadku g In
≥ 1
222 2n
. OtΩe, isnu[ x∗ ∈ −[ 1 + 1/n , 1 – 1/n] ta-
ke, wo ′ ∗g x( ) < − 1
222 2n
. Bez obmeΩennq zahal\nosti mirkuvan\ moΩna vvaΩaty,
wo x∗ ∈ −[ 1 + 1/n , 0].
Nexaj ′ −g ( )1 ≤ 0. Todi na pidstavi lemy 1 dlq funkci] ′g na promiΩku
−[ 1, 0] otrymu[mo
g
n
− +
1 1 – g( )−1 =
−
− +
∫ ′
1
1 1/
( )
n
g x dx ≤ 1
2
1
2′ ∗g x
n
( ) ≤ − 1
223 4n
,
zvidky g In
≥ 1
224 4n
.
Nexaj ′ −g ( )1 > 0. Todi isnu[ a1 ∈ ( , )− ∗1 x take, wo ′g a( )1 = 0. Qkwo a1 ≤
≤ – 1 + 1
2n
, to zhidno z lemog 1 ma[mo
g
n
− +
1 1 – g a( )1 =
a
n
g x dx
1
1 1− +
∫ ′
/
( ) ≤ 1
2
1
4 2′ ∗g x
n
( ) ≤ − 1
225 4n
.
Zvidsy g In
≥ 1
226 4n
. Qkwo Ω a1 > – 1 + 1
2n
, to zhidno z lemog 2
g
n
− +
1 1
2
– g( )−1 =
−
− +
∫ ′
1
1 1 2/
( )
n
g x dx ≥ − ′ ∗1
2
1
4 2g x
n
( ) ≥ 1
225 4n
.
Zvidsy g In
≥ 1
226 4n
. Takym çynom, u dovil\nomu vypadku g In
≥ 1
226 4n
.
Lemu 3 dovedeno.
Dovedennq teoremy 1. Zafiksu[mo n ∈N . Poklademo h : = 1
214 n
,
s x( ) : =
0
1
2 2 3
2 2 3
, ,
–
–
, , ,
, ,
x h
h u du
h u du
x h h
x h
h
x
h
h
≤ −
( )
( )
∈ −[ ]
≥
−
−
∫
∫
S : =
− −
+∫ ∫
1
1
1
x
s t dt x dx( ) – , λ : =
1 1 8
2
− − S
i nareßti
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
NEHATYVNYJ REZUL|TAT U POTOÇKOVOMU 3-OPUKLOMU NABLYÛENNI … 567
g x( ) : = 1
1
1 1
−
−
− −
∫ ∫λ
λ
x t
s u dudt( ) .
Vraxovugçy, wo 2S = λ(1 – λ), ma[mo
g x( ) –
x +
2
2
=
0 1 1 1
1
1
1 1 1
2
2
, , ,
( )
( )
, , .
x
n
S x
x
n
∈ − − +
−
−
∈ −
λ
Z heometryçnyx mirkuvan\ oçevydno, wo S ≤ 3
2
2h , krim toho, z oznaçennq λ vy-
plyva[, wo 1
1 2( )− λ
≤ 2. Teper zhidno z lemog 3 dlq dovil\noho P n∈P I ∆3
otrymu[mo
g P In
− ≥
x
P
In
+ −
2
2
– g
x
In
− +
2
2
≥ 1
226 4n
– 3 1
2
1
28 2 2n n
= 1
228 4n
.
Lehko pokazaty, wo isnu[ absolgtna konstanta C taka, wo g( )
,
5
1 1−[ ]
≤ Cn3.
Poklademo f : =
g
Cn3 . Oçevydno, f ( )
,
5
1 1−[ ]
≤ 1. Krim toho, dlq dovil\noho
P n∈P I ∆3 ma[mo
f P In
− ≥ 1
228 7C n
.
Z inßoho boku, pry vsix x ∈ In ρn x5( ) < 6
7 5n , , zvidky v svog çerhu vyplyva[ teo-
rema 1.
Dovedennq teoremy 2 [ analohiçnym dovedenng teoremy 1, qkwo zamist\
s x( ) vzqty
s x0( ) : =
0
1
2 2 2
2 2 2
, ,
–
–
, , ,
, .
x h
h u du
h u du
x h h
x h
h
x r
h
h r
≤ −
( )
( )
∈ −[ ]
≥
−
−
−
−
∫
∫
1. Íevçuk Y. A. PryblyΩenye mnohoçlenamy y sled¥ neprer¥vn¥x na otrezke funkcyj. �
Kyev: Nauk. dumka, 1992. � 225 s.
2. Konovalov V. N., Leviatan D. Shape-preserving widths of Sobolev-type classes of s-monotone
functions on a finite interval // Isr. J. Math. – 2003. – 133. – P. 239 – 268.
3. Cao J., Gonska H. Pointwise estimates for higher order convexity preserving polynomial
approximation // J. Austral. Math. Soc. Ser. B. – 1994. – 36. – P. 213 – 233.
4. Bondarenko A. V., Prymak A. V. Otrycatel\n¥e rezul\tat¥ v formosoxranqgwem
pryblyΩenyy v¥sßyx porqdkov // Mat. zametky. � 2004. � 76, # 6. � S. 812 � 823.
5. Dzyadyk V. K., Shevchuk I. A. Theory of uniform approximation of functions by polynomials. –
Walter De Gruyter, 2008. – 280 p.
OderΩano 12.08.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
|
| id | umjimathkievua-article-3040 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:35:06Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f2/dc609e96d42d75bb4c7174b5795800f2.pdf |
| spelling | umjimathkievua-article-30402020-03-18T19:43:50Z Negative result in pointwise 3-convex polynomial approximation Негативний результат у поточковому 3-опуклому наближенні многочленами Bondarenko, A. V. Gilewicz, J. Бондаренко, А. В. Гилевич, Я. Я. Бондаренко, А. В. Гилевич, Я. Я. Let $Δ^3$ be the set of functions three times continuously differentiable on $[−1, 1]$ and such that $f'''(x) ≥ 0,\; x ∈ [−1, 1]$. We prove that, for any $n ∈ ℕ$ and $r ≥ 5$, there exists a function $f ∈ C^r [−1, 1] ⋂ Δ^3 [−1, 1]$ such that $∥f (r)∥_{C[−1, 1]} ≤ 1$ and, for an arbitrary algebraic polynomial $P ∈ Δ^3 [−1, 1]$, there exists $x$ such that $$|f(x)−P(x)| ≥ C \sqrt{n}ρ^r_n(x),$$ where $C > 0$ is a constant that depends only on $r, ρ_n(x) := \frac1{n^2} + \frac1n \sqrt{1−x^2}$. Пусть $Δ^3$ является множеством трижды непрерывно дифференцируемых функций на $[−1, 1]$ таких, что $f'''(x) ≥ 0,\; x ∈ [−1, 1]$. Доказано, что для произвольных $n ∈ ℕ$ и $r ≥ 5$ существует функция $f ∈ C^r [−1, 1] ⋂ Δ^3 [−1, 1]$ тaкaя, что $∥f (r)∥_{C[−1, 1]} ≤ 1$, и для произвольного алгебраического полинома $P ∈ Δ^3 [−1, 1]$ существует $x$ такое, что $$|f(x)−P(x)| ≥ C \sqrt{n}ρ^r_n(x),$$ где $C > 0$ — постоянная, зависящая только от $r, ρ_n(x) := \frac1{n^2} + \frac1n \sqrt{1−x^2}$. Institute of Mathematics, NAS of Ukraine 2009-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3040 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 4 (2009); 563-567 Український математичний журнал; Том 61 № 4 (2009); 563-567 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3040/2829 https://umj.imath.kiev.ua/index.php/umj/article/view/3040/2830 Copyright (c) 2009 Bondarenko A. V.; Gilewicz J. |
| spellingShingle | Bondarenko, A. V. Gilewicz, J. Бондаренко, А. В. Гилевич, Я. Я. Бондаренко, А. В. Гилевич, Я. Я. Negative result in pointwise 3-convex polynomial approximation |
| title | Negative result in pointwise 3-convex polynomial approximation |
| title_alt | Негативний результат у поточковому 3-опуклому наближенні многочленами |
| title_full | Negative result in pointwise 3-convex polynomial approximation |
| title_fullStr | Negative result in pointwise 3-convex polynomial approximation |
| title_full_unstemmed | Negative result in pointwise 3-convex polynomial approximation |
| title_short | Negative result in pointwise 3-convex polynomial approximation |
| title_sort | negative result in pointwise 3-convex polynomial approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3040 |
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