On one problem for comonotone approximation
For a comonotone approximation, we prove that an analog of the second Jackson inequality with generalized Ditzian - Totik modulus of smoothness $\omega^{\varphi}_{k, r}$ is invalid for $(k, r) = (2, 2)$ even if the constant depends on a function.
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| Дата: | 2005 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3697 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509827943366656 |
|---|---|
| author | Nesterenko, A. N. Petrova, T. O. Нестеренко, А. Н. Петрова, Т. О. Нестеренко, А. Н. Петрова, Т. О. |
| author_facet | Nesterenko, A. N. Petrova, T. O. Нестеренко, А. Н. Петрова, Т. О. Нестеренко, А. Н. Петрова, Т. О. |
| author_sort | Nesterenko, A. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:02:18Z |
| description | For a comonotone approximation, we prove that an analog of the second Jackson inequality with generalized Ditzian - Totik modulus of smoothness $\omega^{\varphi}_{k, r}$ is invalid for $(k, r) = (2, 2)$ even if the constant depends on a function.
|
| first_indexed | 2026-03-24T02:47:18Z |
| format | Article |
| fulltext |
UDK 517.518.8
O.�N.�Nesterenko, T.�O.�Petrova (Ky]v. nac. un-t im.T.�Íevçenka)
PRO ODNU ZADAÇU
DLQ KOMONOTONNOHO NABLYÛENNQ
For a comonotone approximation, we prove that an analog of the second Jackson inequality with
generalized Ditzian – Totik modulus of smoothness ωϕ
k r, is invalid for ( k , r ) = ( 2, 2 ) even if the
constant depends on a function.
Dovedeno, wo dlq komonotonnoho nablyΩennq analoh druho] nerivnosti DΩeksona z uzahal\ne-
nym modulem neperervnosti Diciana – Totika ωϕ
k r, pry ( k , r ) = ( 2, 2 ) [ xybnym navit\ zi stalog,
zaleΩnog vid funkci].
1. Nexaj C [ –1, 1 ] — prostir dijsnyx neperervnyx na [ –1, 1 ] funkcij iz rivno-
mirnog normog || ⋅ || , a Pn — sukupnist\ alhebra]çnyx mnohoçleniv stepenq
ne�vywe niΩ n – 1, n ∈ N . Poznaçymo çerez Ys , de s ∈ N , mnoΩynu vsix nabo-
riv toçok Ys : = { yj | j = 1, … , s }, dlq qkyx –1 < ys < … < y1 < 1, a çerez ∆1
( Ys )
sukupnist\ nespadnyx na [ y1 , 1] funkcij f ∈ C [ –1, 1 ] , wo zminggt\ naprqmok
monotonnosti v toçkax yj . Zokrema, qkwo f ∈ C1
[ –1, 1 ] ta
Π ( x ) : = ( )x yj
j
s
−
=
∏
1
,
to f ∈ ∆1
( Ys ) todi i til\ky todi, koly Π ( x ) f ′ ( x ) ≥ 0, x ∈ ( –1, 1 ) . Qkwo s = 0,
to vvaΩa[mo, wo Y0 : = ∅, ∆1
( Y0 ) — mnoΩyna nespadnyx na [ –1, 1] funkcij, a
Π ( x ) : = 1. Poznaçymo çerez
E f Yn s
( ) ,1 ( ) : = inf ( )f p p Yn n n s− ∈{ }P ∩ ∆1
velyçynu najkrawoho komonotonnoho nablyΩennq funkci] f ∈ ∆1(Ys ) ∩ C[–1, 1] .
Poklademo ϕ ( x ) : = 1 2− x , x ∈ [ –1, 1 ] , a takoΩ
ϕδ ( x ) : = 1
2
1
2
− −
+ −
x x x xδ ϕ δ ϕ( ) ( ) , x ± δ
2
ϕ ( x ) ∈ [ –1, 1 ] , δ > 0.
Dlq f ∈ C ( –1, 1 ) i r ∈ N ∪ { 0 } poznaçymo çerez
ωϕ
1, ( , )r f t : =
: = sup sup ( ) ( ) ( ) ( )
0
1
2
1
2
1
2
1
< ≤
+
− −
± <
h t
h
r x f x h x f x h x x h xϕ ϕ ϕ ϕ , t > 0,
ωϕ
2, ( , )r f t : =
: = sup sup ( ) ( ) ( ) ( ) ( )
0
2 2 1
< ≤
+( ) − + −( )( ) ± <{ }
h t
h
r x f x h x f x f x h x x h xϕ ϕ ϕ ϕ , t > 0,
uzahal\neni (pry r = 0 — zvyçajni) moduli neperervnosti Diciana – Totika funk-
ci] f perßoho i druhoho porqdkiv vidpovidno. Vyznaçymo pidprostir
Cr
ϕ : =
f C C x f xr
x
r r∈ − − ={ }| |→
( , ) [ , ] lim ( ) ( )( )1 1 1 1 0
1
∩ ϕ .
© O.�N.�NESTERENKO, T.�O.�PETROVA, 2005
1424 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
PRO ODNU ZADAÇU DLQ KOMONOTONNOHO NABLYÛENNQ 1425
U robotax [1 – 3], zokrema, dovedeno: qkwo Ys ∈ Ys , f ∈ Cr
ϕ ∩ ∆1
( Ys ) , to
E f Y f
n
O
n
n s r
r
r
( )
,
( )( , ) ,1
1
1 1≤
ωϕ , n → ∞ ,
a takoΩ pry r ≠ 2
E f Y f
n
O
n
n s r
r
r
( )
,
( )( , ) ,1
2
1 1≤
ωϕ , n → ∞ .
Pytannq pro spravedlyvist\ ostann\oho spivvidnoßennq u vypadku r = 2 zaly-
ßalos\ vidkrytym (dyv. [3]). Osnovnyj rezul\tat dano] roboty — teorema 1, v
qkij pokazano, wo ce spivvidnoßennq [ xybnym dlq r = 2. Pry c\omu my vyko-
rystaly metody robit [4, 5], de analohiçna zadaça rozv’qzana dlq opuklo] ta ko-
opuklo] aproksymaci].
Teorema 1. Nexaj s ∈ N ∪ {0}, Ys ∈ Ys . Todi isnu[ funkciq f ∈ Cϕ
2 ∩ ∆1
( Ys ) ,
dlq qko]
lim sup
( , )
,
( )
, /n
n sn E f Y
f n→∞ ′′( )
2 1
2 2 1ωϕ = + ∞ .
2. U podal\ßomu çerez c budemo poznaçaty dodatni stali, wo moΩut\ zale-
Ωaty lyße vid k , r ta Ys , pryçomu konstanty, wo poznaçagt\sq ci[g literog
u riznyx çastynax odni[] nerivnosti, moΩut\ buty, vzahali kaΩuçy, riznymy.
Magt\ misce nerivnosti [3; 6, s. 165 – 167]
ω ϕϕ
2 2
2
, ( , )f t c f≤ , t > 0, f ∈ C
2
[ –1, 1 ] , (1)
ω ϕϕ
2 2
2 2 4 4
,
( ) ( ),f t c t f( ) ≤ , t > 0, f ∈ C
4
[ –1, 1 ] . (2)
Pry b ∈ ( 0, 1 ) poklademo
gb ( x ) : = Π( ) lnx b
x b1 + +
, Gb ( x ) : = g u dub
x
( )
−
∫
1
, x ∈ [ –1, 1 ] .
Todi vykonugt\sq nastupni ocinky:
ωϕ
2 2
21 1
, , ln′′( ) ≤ +
G t c t
bb , t > 0, (3)
ϕ2 1 1′′ ≤ +
G c
bb ln , (4)
b
b
g x x x e
xbln ( ) ( ) ( ) ln1 1 3
1
2
≤ +
+
Π , x ∈ ( –1, 1 ] . (5)
Spravdi, poklademo
h1 ( x ) : = Π′ ( x ) ln b, h2 ( x ) : = Π′ ( x ) ln (1 + x + b) + Π( )x
x b
1
1+ +
, x ∈ [ –1, 1 ] .
Todi ′′G xb( ) = h1 ( x ) – h2 ( x ) . Poznaçyvßy çerez ω2 ( h1 , ⋅ ) (zvyçajnyj) druhyj
modul\ neperervnosti funkci] h1 i vykorystavßy joho vlastyvosti, otryma[mo
ωϕ
2 2 1, ( , )h t ≤ ω2 ( h1 , t ) ≤ ct h ct
b
2
1
2 1′′ ≤ ln .
Vraxuvavßy formulu (1), znajdemo
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
1426 O.�N.�NESTERENKO, T.�O.�PETROVA
ωϕ
2 2 2, ( , )h t ≤ c hϕ2
2 ≤
≤ c x x x x b
x
x b
x x
x
sup ( )( )( ) ln( ) ( ) ( )
[ , ]∈ −
′ − + + + + +
+ +
−
1 1
1 1 1
1
1
1Π Π ≤ c .
Zvidsy, vykorystovugçy nerivnist\ trykutnyka, oderΩu[mo ocinky�(3) i ( 4 ) .
Ocinku (5) vstanovleno v [4] (nerivnist\ (5.2)).
Poznaçymo çerez Pn
∗ mnoΩynu takyx mnohoçleniv pn stepenq ne vywe niΩ
n – 1, wo Π( ) ( )− ′ −1 1pn ≥ 0. Zrozumilo, wo Pn ∩ ∆1
( Ys ) ⊂ Pn
∗ . Nastupna ocinka
vstanovlg[t\sq doslivnym povtorennqm mirkuvan\ z dovedennq lemy 5.3 z [4].
Qkwo b ∈ 0 1
2,
n
, pn ∈ Pn
∗ , n ≥ s + 1, to
G p c
n n b nb n− ≥ −2 2 2
1 1ln . (6)
3. Dovedennq teoremy. Nexaj bn ∈
1 1
4 2n n
,
, n ≥ 2, take, wo b
b n
n
n
ln 1 1
2= .
Poklademo
fn ( x ) : = c
n
G xbn2 ( ) , x ∈ [ –1, 1 ] , n ≥ 2,
de 0 < c < 1 vybyra[t\sq nastil\ky malym, wob vykonuvalys\ nerivnosti (takyj
vybir c [ moΩlyvym na pidstavi ocinok (3) – (5))
ωϕ
2 2 2
1 1
, ,′′
≤f
n n
n , (7)
fn
j( ) < 1, j = 0, 1, ϕ2 ′′fn < 1. (8)
Krim c\oho, fn ∈ C∞
[ –1, 1 ] , fn ( –1 ) = ′ −fn( )1 = 0. Z oznaçennq bn otrymu[mo
nerivnosti
ln ln n ≤ ln ln n2 ≤ ln ln 1
bn
= ln 1
2n bn
,
zvidky vnaslidok (6) vyplyva[ isnuvannq stalo] c > 0 i nomera n1 ≥ 2 takyx, wo
dlq vsix n ≥ n1 i koΩnoho pn ∈ Pn
∗
f p c
n
n
n n− ≥ ln ln
4 . (9)
Poklademo D0 : = 1 i
Dσ : =
D
n n n
σ
σ σ
− =1
4
1
4 4
1 1… , σ ∈ N ,
de nσ vyznaçagt\sq za indukci[g takym çynom. Prypustymo, wo n1 , … , nσ –1
vΩe pobudovano. Poznaçymo
Fσ –1 ( x ) : = D f xj n
j
j−
=
−
∑ 1
1
1
( )
σ
.
Vyberemo nσ > nσ –1 nastil\ky velykym, wob vykonuvalys\ nerivnosti
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
PRO ODNU ZADAÇU DLQ KOMONOTONNOHO NABLYÛENNQ 1427
max , ( )σ σF −{ }1
4 < Dσ –1 ln ln ln nσ ,
(10)
Fσ−1
6( ) < Dσ –1 nσ .
Na pidstavi nerivnosti typu DΩeksona ta ostann\o] ocinky otrymu[mo
inf ( )′ − ∈{ } ≤ < =− − − − −
−F p p c
n
F
cD n
n
cDn n nσ
σ
σ
σ σ
σ
σσ σ σ1 1 1 1 5 1
6 1
5P . (11)
Poklademo
Φσ ( x ) : = D f xj n
j
j−
=
∞
∑ 1 ( )
σ
, x ∈ [ –1, 1 ] .
ZauvaΩymo, wo cej rqd moΩna dviçi poçlenno dyferencigvaty na ( –1, 1) ; ce
vyplyva[ z nerivnosti (8) ta ocinky
D D
n n
D
n
Dj
j jj
j
j
−
=
∞
−
−= +
∞
−
−
−
=
∞
∑ ∑ ∑= +
<
<1 1 4
1
4
1
1 4 11 1 1 2
σ
σ
σσ
σ
σ
σ
σ
σ…
. (12)
Pry c\omu
Φσ σ< −2 1D , ϕ σ σ
2
12′′ < −Φ D . (13)
Teper dlq velykyx σ ma[mo (poqsnennq dyv. nyΩçe)
ω ω ω ωϕ
σ
ϕ
σ
σ
ϕ
σ
σ
ϕ
σ
σ
σ2 2 1 2 2 1 2 2 1 2 2 1
1 1 1 1
, , , ,, , , ,′′
≤ ′′
+ ′′
+ ′′
− − +Φ Φ
n
F
n
D f
n nn ≤
≤ c
n
F
D
n
cD
σ
σ
σ
σ
σϕ2
4
1
4 1
2 2−
−+ +( ) ≤
≤ c
n
D n
D
n
c
D
n
cD
n
n
σ
σ σ
σ
σ
σ
σ
σ
σ
σ2 1
1
2
1
4
1
22−
− − −+ + ≤ln ln ln ln ln ln . (14)
U druhij nerivnosti dlq ocinky perßoho dodanka vykorystano formulu (2), dru-
hoho — formulu (7), a tret\oho — formuly (1) i (13). Tretq ta çetverta neriv-
nosti vyplyvagt\ z (10).
Dali, na pidstavi (11) isnu[ qnσ
— mnohoçlen stepenq ne�vywoho za nσ – 2,
dlq qkoho q Fnσ σ( ) ( )− = ′ −−1 11 = 0 ta ′ −−F qnσ σ1 ≤ 2cDσ . Tomu dlq mnohoçlena
Q x q u dun n
x
σ σ
( ) : ( )=
−
∫
1
vykonu[t\sq nerivnist\
F Q F u q u dun x n
x
σ σσ σ− ∈ − −
−
− = ′ −( )∫1 1 1 1
1
max ( ) ( )
[ , ]
≤ 4cDσ . (15)
Dlq pn nσ σ
∈ ∗P poklademo R
D
p Qn n n nσ σ σ σ
σ
:= −( ) ∈
−
∗1
1
P . Todi
Φ Φ1 1 1 1− = −( ) + −( ) +− − +p F Q D f Rn n n nσ σ σ σσ σ σ ,
zvidky, poslidovno vykorystovugçy (9), (15), (13) ta (10), dlq velykyx σ otry-
mu[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
1428 O.�N.�NESTERENKO, T.�O.�PETROVA
Φ Φ1 1 1 1− ≥ − − − −− − +p D f R F Qn n n nσ σ σ σσ σ σ ≥
≥ D c
n
n cD c D nσ
σ
σ σ σ σ− − ≥1 4 2
ln ln ln ln . (16)
Poklademo
′ = +
+
f x x x e
x
( ) : ( )( ) ln2 1 3
1
2
Π , f x f u du
x
( ) : ( )= ′
−
∫
1
, x ∈ ( –1, 1 ] , f ( )−1 : = 0.
Intehruvannqm çastynamy lehko perekonatys\, wo funkciq f ma[ vyhlqd
f x x x
x
x x( ) ( )( ) ln ( ) ( )= +
+
+ +Π Π1
2 2
21 2
1
1 , x ∈ ( –1, 1 ] ,
de Π1 , Π2 — deqki mnohoçleny stepenq ne vywe za s . U roboti [7] vstanovle-
no, wo dlq koΩnoho n ≥ 1 isnu[ takyj mnohoçlen Ωn ∈ Pn , wo
f c
n
n− ≤Ω 4 . (17)
Rozhlqdagçy mnohoçleny vydu
Ω ( x ) = Π Π1
2
4
21
2
21 1 1 1
1 1
1( )( )
( ) ( )
( ) ( )x x
n
T u
u
du
u
x xn
n
x
+ − − −
+
+
+ +∫ , x ∈ [–1, 1] ,
de Tn ( u ) = cos n arccos u — mnohoçlen Çebyßova, i mirkugçy analohiçno do [8],
lehko pokazaty, wo mnohoçlen Ωn ∈ Pn zi spivvidnoßennq (17) moΩna vybraty
tak, wob ′ −Ωn( )1 = 0.
ZauvaΩymo takoΩ, wo z formuly (2) vyplyva[ ocinka
ω ϕϕ
2 2 2
4 4
2
1
,
( ),′′
≤ ≤f
n
c
n
f c
n
. (18)
Poklademo
f ( x ) : = Φ1 ( x ) + f x( ), x ∈ [ –1, 1 ] ,
i pokaΩemo, wo funkciq f [ ßukanog. Vklgçennq f ∈ ∆1
( Ys ) [ naslidkom
(5)�i�(12).
Z nerivnostej (14), (18) i (10) vyplyva[, wo dlq velykyx σ
ωϕ
σ
2 2
1
, ,′′
≤f
n
≤ ′′
+ ′′
≤ + ≤− −ω ωϕ
σ
ϕ
σ
σ
σ
σ
σ
σ
σ
σ2 2 1 2 2
1
2 2
1
2
1 1
, ,, , ln ln ln ln ln lnΦ
n
f
n
cD
n
n c
n
cD
n
n . (19)
Oskil\ky prava çastyna ci[] nerivnosti prqmu[ do nulq pry σ → + ∞ , to (dyv.
[3]) f ∈ Cϕ
2 .
Vykorystovugçy te, wo dlq koΩnoho pn nσ σ
∈ ∗P spravdΩu[t\sq vklgçennq
pn n nσ σ σ
− ∈ ∗Ω P , z uraxuvannqm (16), (17) i (10) dlq velykyx σ ma[mo
f p p f c D n c
n
c D
n
nn n n n− ≥ − −( ) − − ≥ − ≥ −
σ σ σ σ σ σ
σ
σ
σ
σΦ Ω Ω1 4
1
42 4
ln ln ln ln . (20)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
PRO ODNU ZADAÇU DLQ KOMONOTONNOHO NABLYÛENNQ 1429
Oskil\ky E f Yn s
( ) ,1 ( ) ≥ inf f p pn n n− ∈{ }∗P , to z (19) i (20) vyplyva[, wo
n E f Y
f n
c
n
n
n sσ
ϕ
σ
σ
σ
σ
ω
2 1
2 2 1
( )
,
,
,
ln ln
ln ln ln/
( )
′′( )
≥ → + ∞ , σ → + ∞ .
Teoremu dovedeno.
ZauvaΩennq. Bezposerednim naslidkom teoremy [ analohiçnyj rezul\tat
dlq vypadku ( k , r ) = ( 3 , 1 ) (oznaçennq modulq neperervnosti ωϕ
3 1, ta joho vla-
styvosti, z qkyx i vyplyva[ cej rezul\tat, dyv. v [3 – 5]).
Avtory vyslovlggt\ wyru podqku profesoru I.�O.�Íevçuku za postanovku
zadaçi ta cinni zauvaΩennq.
1. Leviatan D. Pointwise estimates for convex polynomial approximation // Proc. Amer. Math. Soc. –
1986. – 98. – P. 471 – 474.
2. Kopotun K. A. Uniform estimates of monotone and convex approximation of smooth functions // J.
Approxim. Theory. – 1995. – 80. – P. 76 – 107.
3. Leviatan D., Shevchuk I. A. Some positive results and counterexamples in comonotone approxima-
tion // Ibid. – 1999. – 100. – P. 113 – 143.
4. Kopotun K. A., Leviatan D., Shevchuk I. A. Convex polynomial approximation in the uniform
norm: conclusion // Can. J. Math. – 2005. – 58. – P. 407 – 430.
5. Kopotun K. A., Leviatan D., Shevchuk I. A. Coconvex approximation in the uniform norm – the
final frontier // Acta math. hung. – 2005. – 108. – P. 305 – 333.
6. Íevçuk%Y.%A. PryblyΩenye mnohoçlenamy y sled¥ neprer¥vn¥x na otrezke funkcyj. –
Kyev: Nauk. dumka, 1992. – 225�s.
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toçkoj // Yzv. AN SSSR. Ser. mat. – 1946. – 10, #�5. – S.�429 – 456.
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OderΩano 08.06.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
|
| id | umjimathkievua-article-3697 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:47:18Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/97/8d802bbafa47bf746089c33dceda2b97.pdf |
| spelling | umjimathkievua-article-36972020-03-18T20:02:18Z On one problem for comonotone approximation Про одну задачу для комонотонного наближення Nesterenko, A. N. Petrova, T. O. Нестеренко, А. Н. Петрова, Т. О. Нестеренко, А. Н. Петрова, Т. О. For a comonotone approximation, we prove that an analog of the second Jackson inequality with generalized Ditzian - Totik modulus of smoothness $\omega^{\varphi}_{k, r}$ is invalid for $(k, r) = (2, 2)$ even if the constant depends on a function. Доведено, що для комонотонного наближення аналог другої нерівності Джексона з узагальненим модулем неперервності Діціана - Тотіка $\omega^{\varphi}_{k, r}$ при $(k, r) = (2, 2)$ є хибним навіть зі сталою, залежною від функції. Institute of Mathematics, NAS of Ukraine 2005-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3697 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 10 (2005); 1424–1429 Український математичний журнал; Том 57 № 10 (2005); 1424–1429 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3697/4119 https://umj.imath.kiev.ua/index.php/umj/article/view/3697/4120 Copyright (c) 2005 Nesterenko A. N.; Petrova T. O. |
| spellingShingle | Nesterenko, A. N. Petrova, T. O. Нестеренко, А. Н. Петрова, Т. О. Нестеренко, А. Н. Петрова, Т. О. On one problem for comonotone approximation |
| title | On one problem for comonotone approximation |
| title_alt | Про одну задачу для комонотонного наближення |
| title_full | On one problem for comonotone approximation |
| title_fullStr | On one problem for comonotone approximation |
| title_full_unstemmed | On one problem for comonotone approximation |
| title_short | On one problem for comonotone approximation |
| title_sort | on one problem for comonotone approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3697 |
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