Pointwise Estimates for the Coconvex Approximation of Differentiable Functions
We obtain pointwise estimates for the coconvex approximation of functions of the class $W^r,\; r > 3$.
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Institute of Mathematics, NAS of Ukraine
2005
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| author | Dzyubenko, H. A. Zalizko, V. D. Дзюбенко, Г. А. Залізко, В. Д. |
| author_facet | Dzyubenko, H. A. Zalizko, V. D. Дзюбенко, Г. А. Залізко, В. Д. |
| author_sort | Dzyubenko, H. A. |
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| datestamp_date | 2020-03-18T19:59:02Z |
| description | We obtain pointwise estimates for the coconvex approximation of functions of the class $W^r,\; r > 3$. |
| first_indexed | 2026-03-24T02:45:07Z |
| format | Article |
| fulltext |
UDK 517.5
H. A. Dzgbenko (MiΩnar. mat. centr NAN Ukra]ny, Ky]v),
V. D. Zalizko (Nac. ped. un-t, Ky]v)
POTOÇKOVI OCINKY KOOPUKLOHO
NABLYÛENNQ DYFERENCIJOVNYX FUNKCIJ
Pointwise estimates of the coconvex approximation of functions belonging to the class W
r
, r > 3, are
obtained.
Otrymano potoçkovi ocinky koopukloho nablyΩennq funkcij iz klasu W
r
, r > 3.
1. Vstup. Nexaj W
r
, r ∈ N, — mnoΩyna funkcij f ∈ C [ – 1, 1 ], qki magt\ ( r –
– 1 )-ßu absolgtno neperervnu poxidnu na I : = [ – 1, 1 ] i dlq qkyx pry majΩe
vsix x ∈ I vykonu[t\sq nerivnist\
| f (
r
)
( x ) | ≤ 1.
Poznaçymo çerez Y : = yi i
s{ } =1, s ∈ N, nabir z s fiksovanyx toçok yi :
ys + 1 : = –1 < ys < … < y1 < 1 = : y0.
Nexaj ∆
(
2
)
( Y ) — mnoΩyna neperervnyx na I funkcij, qki [ opuklymy dony-
zu na vidrizku [ yi + 1, yi ], qkwo i — parne, i opuklymy dohory na tomu Ω samomu
vidrizku, qkwo i — neparne. Funkci] z ∆
(
2
)
( Y ) nazyvagt\sq koopuklymy.
Nexaj
Π ( x ) : = Π ( x, Y ) : =
i
s
ix y
=
∏ −
1
( ), Π ( x, ∅ ) ≡ 1
(zauvaΩymo, wo qkwo f [ dviçi dyferencijovnog, to f ∈ ∆
(
2
)
( Y ) ⇔
⇔ f ′′ ( x ) Π ( x ) ≥ 0, x ∈ I ),
ρn ( x ) :=
1 2− x
n
+
1
2n
, n ∈ N, x ∈ I.
U cij roboti dovedeno nastupnu teoremu.
Teorema 1. Qkwo r > 3, s ≥ 2 i f ∈ W
r ∩ ∆
(
2
)
( Y ), to dlq koΩnoho natu-
ral\noho n > N ( Y, r ) isnu[ alhebra]çnyj mnohoçlen Pn stepenq ≤ n takyj,
wo
′′P x xn ( ) ( )Π ≥ 0, x ∈ I, (1)
| f ( x ) – Pn ( x ) | ≤ C ( Y, r )ρn
r x( ), x ∈ I, (2)
de N ( Y, r ) i C ( Y, r ) — stali, qki zaleΩat\ lyße vid min ( ), ,i s i iy y= … +−0 1 i r.
Dlq r = 1, 2, 3 teorema 1 takoΩ [ spravedlyvog. Dlq r = 1, 2 vona [ na-
slidkom rezul\tativ roboty [1], a dlq r = 3 — roboty [2]. Dlq s =1, r > 2
teorema 1, vzahali kaΩuçy, ne [ virnog (dyv. [1], teorema 2). Z [1, 2], teoremy 1 i
roboty [3] (dlq „malyx” n) vyplyva[ taka teorema.
Teorema 2. Qkwo r ∈ N, s ≥ 2 i f ∈ W
r ∩ ∆
(
2
)
( Y ), to dlq koΩnoho natu-
ral\noho n ≥ r – 1 isnu[ alhebra]çnyj mnohoçlen Pn stepenq ≤ n takyj, wo
Pn ∈ ∆
(
2
)
( Y ),
| f ( x ) – Pn ( x ) | ≤ C ( Y, r )ρn
r x( ), x ∈ I, (3)
de C ( Y, r ) — stala, qka zaleΩyt\ lyße vid min ( ), ,i s i iy y= … +−0 1 i r.
© H. A. DZGBENKO, V. D. ZALIZKO, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 47
48 H. A. DZGBENKO, V. D. ZALIZKO
Na cej ças vΩe otrymano toçni za porqdkom rivnomirni i potoçkovi ocinky qk
çysto opukloho ( s = 0, tobto bez toçok perehynu), tak i koopukloho (s ≥ 1) na-
blyΩennq nedyferencijovnyx ( r = 0 ) i „slabko” dyferencijovnyx ( r = 1, 2 )
funkcij (dyv., napryklad, [2, 4]). Navedemo çotyry rezul\taty wodo (ko)opuk-
loho nablyΩennq funkcij dlq vypadku r ≥ 3. Dlq s = 0 potoçkovu ocinku u
klasyçnij formi
| f ( x ) – Pn ( x ) | ≤ c ( r, k )ρn
r x( )ωk ( f (
r
); ρn ( x ) ), x ∈ I, n ≥ k + r – 1, (4)
de ωk ( ⋅ ) — k-j modul\ neperervnosti i c ( r, k ) — stala, bulo dovedeno v [5] ( r =
= 0, k = 3 ) i Maniq (dyv. [6, c. 148]) ( r > 1, k ∈ N ). U roboti [7] vstanovleno, wo
(4) ne vykonu[t\sq dlq r = 0, k ≥ 4 (navit\ z 1 / n zamist\ ρn ( x ) ). Dlq s ≥ 1
analohiçnyj nehatyvnyj rezul\tat dovedeno v [8]. Kopotun, Leviatan i Íevçuk
lgb’qzno povidomyly, wo nymy dovedeno rivnomirnyj analoh ocinky (4) ( s ≥ 1,
r ≥ 3, k ∈ N ), qkyj vklgça[ modul\ neperervnosti Ditzian – Totik.
Zahal\nu sxemu dovedennq teoremy 1 zapozyçeno v [9, 10]. Vona ©runtu[t\sq
na ide] DeVore [11] zobraΩennq poxidno] (tut f ′′ ( x ) ) sumog dvox funkcij: „ve-
lyko]” i „malo]”, na vykorystanni polinomial\nyx qder typu Dzqdyka [12, 6] i na
„monotonnomu” rozbytti odynyci DeVore i Yu [13]. Teoremu 1 bude dovedeno v
p.G2. U p. 3 u zruçnij dlq nas formi navedeno mirkuvannq z [3], qki dovodqt\ teo-
remu 2 dlq r – 1 ≤ n ≤ N ( Y, r ).
2. Oznaçennq i dopomiΩni tverdΩennq. 1. Nexaj toçky x j := x j, n :=
:= cos ( j π / n ), j = 0, … , n, skladagt\ çebyßovs\ke rozbyttq vidrizka I.
Poznaçymo
Ij : = I j, n : = [ x j , x j – 1 ], hj : = hj, n : = x j – 1 – x j , j = 1, n.
Bez special\nyx posylan\ budemo vykorystovuvaty nerivnosti
hj ± 1 ≤ 3 hj ,
ρn ( x ) < hj < 5 ρn ( x ), x ∈ Ij ,
ρn y2( ) < 4 ρn ( x ) x y xn− +( )ρ ( ) , x, y ∈ I,
2 x y xn− +( )ρ ( ) > | x – y | + ρn ( y ) >
x y xn− + ρ ( )
2
, x, y ∈ I .
Dlq fiksovanyx n ∈ N i Y = yi i
s{ } =1 poznaçymo
Oi : = Oi, n : = Oi, n ( Y ) : = ( x j + 2, x j – 3 ), qkwo yi ∈ [ x j , x j – 1 ),
de xn + 2 = xn + 1 : = – 1, x– 1 = x– 2 = x– 3 : = 1,
O : = O ( n, Y ) : =
i
s
iO
=1
∪ .
Budemo pysaty j ∈ H : = H ( n, Y ), qkwo Ij ∩ O = ∅, j = 1, n. Vyberemo çyslo
N ( Y ) tak, wob dlq koΩnoho n ≥ N ( Y ) bud\-qkyj interval ( yi + 1, yi ), i = 1 1, s − ,
mistyv prynajmni sim riznyx vidrizkiv Ij .
Çerez c budemo poznaçaty dodatni stali, qki moΩut\ zaleΩaty lyße vid r,
s i deqkoho fiksovanoho çysla b ∈ N, tobto c : = c ( r, s, b ). Ci stali, vzahali ka-
Ωuçy, [ riznymy, navit\ qkwo vony znaxodqt\sq v odnomu rqdku. Qkwo dali [
posylannq na znaçennq cyx stalyx, to budemo pysaty cν : = cν ( r, s, b ). Dotry-
mugçys\ [6], poklademo
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
POTOÇKOVI OCINKY KOOPUKLOHO NABLYÛENNQ DYFERENCIJOVNYX … 49
tj ( x ) : = tj, n ( x ) : =
cos arccos2
0 2
2n x
x x j−( )
+
sin arccos2
2
2n x
x x j−( )
, n ∈ N,
de
x j =
cos j
n
−( )/1 2 π
, x j
0 = cosβ j
0
, β j
0
=
j
n
j
n
j
n
j
n
−( ) ≤
−( ) >
/
/
1 4
2
3 4
2
π
π
, ,
, .
koly
koly
,
Toçky x j i x j
0
[ nulqmy vidpovidnyx çysel\nykiv i znaxodqt\sq stroho v sere-
dyni Ij , a tj — alhebra]çni mnohoçleny stepenq 4 n – 2 taki, wo
tj ( x ) ≤
c
x x hj j− +( )2 ≤ c tj ( x ), x ∈ I
t x
h
x Ij
j
j( ) ,≤ ∈
103
2 .
Naslidugçy [1, 4, 5], dlq koΩnoho j ∈ H rozhlqnemo çotyry mnohoçleny ste-
penq c n:
Tj ( x ) : = Tj, n ( x; b; Y ) : =
1
1
d
t u u du
j
x
j
b
−
∫ ( ) ( )Π ,
T xj ( ) : = T x b Yj n, ( ; ; ) : =
1
1
1
1
d
u x x u t u u du
j
x
j j j
b
−
−
+∫ − −( )( ) ( ) ( )Π ,
de
dj : = dj, n ( b; Y ) : =
−
∫
1
1
t u u duj
b( ) ( )Π ,
dj : = d b Yj n, ( ; ) : =
−
−
+∫ − −
1
1
1
1( )( ) ( ) ( )u x x u t u u duj j j
b Π ,
i
τj ( x ) : = τj, n ( x; b; Y ) : = α
−
+∫
1
1
x
jT u du( ) + ( ) ( )1
1
1−
−
−∫α
x
jT u du ,
τ j x( ) : = τ j n x b Y, ( ; ; ) : = β
−
+∫
1
1
x
jT u du( ) + ( ) ( )1
1
1−
−
−∫β
x
jT u du ,
de α i β ∈ [ 0, 1 ] vybrano z umovy
τj ( 1 ) = τ j ( )1 = 1 – xj ,
i T0 ( x ) ≡ T x0( ) : ≡ 0, Tn + 1 ( x ) ≡ T xn+1( ) : ≡ 1.
Nexaj
χ ( x; a ) : =
0
1
, ,
, ,
qkwo
qkwo
x a
x a
≤
>
a ∈ I,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
50 H. A. DZGBENKO, V. D. ZALIZKO
χ j ( x ) : = χ ( x; xj ),
( x – a ) + : = ( x – a ) χ ( x; a ),
Γj ( x ) : = Γj, n ( x ) : =
h
x x h
j
j j− +
.
ZauvaΩymo, wo
hj Γj ( x ) ≤ c ρn ( x ), x ∈ I. (5)
Lema 1 [1, 4, 5]. Qkwo j ∈H i b ≥ 6 ( s + 2 ), to
′′τ j jx x x( ) ( ) ( )Π Π ≥ 0, x ∈ I, (6)
′′τ ( ) ( ) ( )x x x jΠ Π ≤ 0, x ∈ I \ Ij , (7)
′ ±τ j ( )1 = ′ ±τ ( )1 = χ j ( ± 1 ),
τ j ( ± 1 ) = τ( )±1 = ( ± 1 – xj )+,
| ( x – xj )+ – τ j ( x ) | ≤ c1 hj ( Γj ( x ) )
2
b
–
s
–
2
, x ∈ I, (8)
| ( x – xj )+ – τ j x( ) | ≤ c1 hj ( Γj ( x ) )
2
b
–
s
–
2
, x ∈ I, (9)
| χ j ( x ) – ′τ j x( ) | ≤ c2 ( Γj ( x ) )
2
b
–
s
–
1
, x ∈ I,
| χ j ( x ) – ′τ j x( ) | ≤ c2 ( Γj ( x ) )
2
b
–
s
–
1
, x ∈ I,
′′τ j x( ) ≤ c
h
x
j
j
b s
3
21 Γ ( )( ) −
, x ∈ I, (10)
′′τ j x( ) ≤ c
h
x
j
j
b s
3
21 Γ ( )( ) −
, x ∈ I. (11)
Zokrema, qkwo j ≠ n, to
c
h
x
x
xj
j
b
j
4
21 Γ Π
Π
( )
( )
( )
≤ ′′τ j x( ) ≤ c
h
x
x
xj
j
b
j
5
21 Γ Π
Π
( )
( )
( )
, x ∈ I, (12)
c
h
x
x
xj
j
b
j
4
21 Γ Π
Π
( )
( )
( )
≤ ′′τ j x( ) ≤ c
h
x
x
xj
j
b
j
5
21 Γ Π
Π
( )
( )
( )
, x ∈ I \ Ij , (13)
′′τ j x( ) ≥ c
h
x
j
j
b s
6
2 21 Γ ( )( ) +
, x ∈ I \ O, (14)
′′τ j x( ) ≥ c
h
x
j
j
b s
6
2 21 Γ ( )( ) +
, x ∈ I \ ( O ∪ Ij ). (15)
Krim toho, qkwo n ≥ N ( Y ), to
′′τ j x( ) ≥ c
h
x
x y
x yj
j
b s i
j i
7
2 21 Γ ( )( ) −
−
+
, x ∈ Oi , i = 1, s , (16)
′′τ j x( ) ≥ c
h
x
x y
x yj
j
b s i
j i
7
2 21 Γ ( )( ) −
−
+
, x ∈ Oi , i = 1, s . (17)
ZauvaΩennq 1. Pry dovedenni lemy 1 bulo zastosovano, zokrema, nerivnosti
Π
Π
( )
( )
x
y
≤
x y
yn
s− +
ρ ( )
1 , x ∈ I, y ∈ I \ O,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
POTOÇKOVI OCINKY KOOPUKLOHO NABLYÛENNQ DYFERENCIJOVNYX … 51
γ j x2( ) < 16 Γj ( x ), Γj x2( ) < 400 γ j ( x ), x ∈ I,
de γ j ( x ) : = ρ ρn j nx x x x( ) ( )/ − +( ) . Dovedennq ocinok (12) – (15) spyragt\sq na
nerivnosti (6), (7) i totoΩnosti ′′τ j x( ) ≡ α ′+T xj 1( ) + ( ) ( )1 1− ′−α T xj , ′′τ j x( ) ≡
≡ β ′+T xj 1( ) + ( ) ( )1 1− ′−β T xj , a dovedennq ocinok (16) i (17) — krim toho, na
spivvidnoßennq Oi ∩ Oi – 1 = ∅, i = 2, s .
Dali zafiksu[mo n ≥ N ( Y ). Dlq dovil\noho intervalu E = x xj j1 2
,( ) ⊂ I, j1 >
> j2 , poznaçymo
*
E : = x xj j1 21+( ), ∩ I; | E | : = x j2
– x j1
.
Lema 2. Qkwo s ≥ 2 i funkciq g ∈ ∆
(
2
)
( Y ) ma[ „malen\ku” druhu poxidnu
| g ′′ ( x ) | ≤ ρn
r x−2( ), x ∈ I, r ≥ 3, (18)
to isnu[ mnohoçlen Gn ( x ) : = Gn ( x; g ) stepenq c n takyj, wo
′′G x xn( ) ( )Π ≥ 0, x ∈ I, (19)
i
| g ( x ) – Gn ( x ) | ≤ c ( Y, r )ρn
r x( ), x ∈ I, (20)
de c ( Y, r ) — stala, qka zaleΩyt\ lyße vid min ( ), ,i s i iy y= … +−0 1 i r.
Dovedennq. Nexaj L x( ) — neperervna lamana, qka interpolg[ g na I v
koΩnij toçci naboru x j Hj , ∈{ } ∪ Y (tobto, vzahali kaΩuçy, L( )±1 ≠ g ( ± 1 ) ).
Napevno L x( ) ∈ ∆
(
2
)
( Y ). „Pidpravymo” L tak, wob nova lamana L ∈ ∆
(
2
)
( Y ) i
toçky Y ne buly ]] vuzlamy. Nexaj
y yi i,( ) : = * Oi , i = 1, s ,
*
O : =
i
s
iO
=1
∪ *
;
l ( x; a, b ) — prqma, qka interpolg[ g ( x ) v a i b; i — takyj parnyj indeks i =
= 2, s , dlq qkoho
*Oi =
max *
i iO− parne
(qkwo takyx indeksiv dva, to nexaj i
— bil\ßyj z nyx); analohiçno, i : *Oi = max *
i iO− neparne
. Dlq koΩnoho i =
= 1, s poznaçymo
li : =
max ; , , ; , , ,
min ; , , ; , , ,
′( ) ′( ){ } −
′( ) ′( ){ } −
l x y y l x y y i
l x y y l x y y i
i i i i
i i i i
qkwo parne
qkwo neparne
∆i : =
y
y
i
i
i
l L t dt∫ − ′( )( )
(tobto ∆i ≥ 0, koly i — parne, i ∆i ≤ 0, koly i — neparne). Poklademo
L ′ ( x; A, B ) : =
′ ∈ −( )
∈ = ≠ ∨
∈
∈
L x x O
l x O i s i i i
A x O
B x O
i i
i
i
( ), , ,
, , , , ,
, ,
, ,
\ *
*
*
*
1 1
1
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
52 H. A. DZGBENKO, V. D. ZALIZKO
de çysla A ≥ li i B ≤ li vybrano z umovy
L ( 1; A, B ) : =
−
∫ ′
1
1
L t A B dt( ; , ) = L( )1 .
A same, nexaj ∆ : = L l li i1; ,( ) – L( )1 i
A = li , B = li –
∆
*Oi
, qkwo ∆ ≥ 0,
A = li –
∆
*Oi
, B = li , qkwo ∆ < 0.
OtΩe,
L ( x ) : = L ( x; A, B ) : =
−
∫ ′
1
x
L t A B dt( ; , ) ∈ ∆
(
2
)
( Y ),
i toçky Y ne [ vuzlamy L. Tomu
Π ( xj ) [ xj + 1 , xj , x j – 1 ; L ] ≥ 0, j ∈ H, (21)
[ xj + 1 , xj , x j – 1 ; L ] = 0, j ∉ H, (22)
de [ ⋅ ] — druha podilena riznycq L.
Ocinymo | g ( x ) – L ( x ) |, x ∈ I. Nerivnist\
| g ( x ) – L ( x; xj , x j – 1 ) | ≤
x
x t
j
g u dudt∫ ∫ ′′
Θ
( ) ≤ cρn
r x( ),
Θ ∈ Ij , x ∈ [ xj + 1 , x j – 2 ],
i analohiçna nerivnist\ dlq x ∈
*Oi , i = 1, s , pryvodqt\ do ocinky
g x L x( ) ( )− ≤ cρn
r x( ), x ∈ I.
Krim toho, dlq x ∈ −[ ]1, ys ∪ y1 1,[ ] L ( x ) – L x( ) ≡ 0; dlq reßty x vraxu[mo,
wo | ∆ | ≤ ( )max , ,s i s i− = …1 1 ∆ , i todi
L x L x( ) ( )− =
y
x
s
L t L t dt∫ ′ − ′( )( ) ( ) ≤
≤ 2 1
1
( ) max ( ) ( ) ( )
, ,
*
s l g t g t L t dt
i s
O
i
i
− − ′ + ′ − ′
= … ∫ ≤ c y
i s
n
r
imax ( )
, ,= …1
ρ ≤ c8 ρn
r x( ),
de c8 — dodatna stala, qka zaleΩyt\ lyße vid min ( ), ,i s i iy y= … +−0 1 i r. OtΩe,
| g ( x ) – L ( x ) | ≤ g x L x( ) ( )− + L x L x( ) ( )− ≤ c8 ρn
r x( ), x ∈ I. (23)
Zokrema,
| [ xj + 1 , xj , x j – 1 ; L ] | = | [ xj + 1 , xj , x j – 1 ; L – g + g ] | ≤
≤ c n
r
x
n x
j
j
8 2
ρ
ρ
( )
( )
+
1
2
′′g ( )Θ ≤ c n
r
x j8
2ρ −
( ), Θ ∈( xj + 1 , x j – 1 ), j = 1 1, n − . (24)
Zobrazymo L u vyhlqdi
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
POTOÇKOVI OCINKY KOOPUKLOHO NABLYÛENNQ DYFERENCIJOVNYX … 53
L ( x ) ≡ l ( x ) +
j
n
j j j j j jx x x L x x x x
=
−
+ − − + +∑ [ ] − −
1
1
1 1 1 1, , ; ( )( ) ≡
≡ l ( x ) +
j H
j j j j j jx x x L x x x x
∈
+ − − + +∑ [ ] − −1 1 1 1, , ; ( )( ) , (25)
de l ( x ) : = [ xn , xn – 1 ; L ] ( x + 1 ) + L ( – 1 ); pry c\omu my skorystalys\ (22).
Poklademo b1 = 6 ( s + 2 ) + r, τj ( x ) = τj, n ( x; b1 ; Y ),
Gn ( x ) : = l ( x ) +
j H
j j j j j jx x x L x x x
∈
+ − − +∑ [ ] −1 1 1 1, , ; ( ) ( )τ . (26)
Nerivnosti (6) i (21) harantugt\ vykonannq spivvidnoßen\
x x x L x x x xj j j j j j+ − − +[ ] − ′′1 1 1 1, , ; ( ) ( ) ( )τ Π =
=
1
2 1 1 1 1Π
Π Π Π
( )
( ) , , ; ( ) ( ) ( ) ( )
x
x x x x L x x x x x
j
j j j j j j j j+ − − +[ ]( ) − ′′( )τ ≥ 0,
x ∈ I, j ∈ H,
wo pryvodyt\ do (19). Ocinka (20) vyplyva[ z (5), (8) i (23) – (26). A same,
| g ( x ) – Gn ( x ) | ≤ | g ( x ) – L ( x ) | + | L ( x ) – Gn ( x ) | ≤ c8 ρn
r x( ) +
+
j H
j j j j j j jx x x L x x x x x
∈
+ − − + +∑ [ ] − − −( )1 1 1 1, , ; ( ) ( ) ( )τ ≤
≤ c8 ρn
r x( ) + cc x x x x
x
x x xj H
n
r
j j j n j
n j
j n j
b s
8
2
1 1
2 21
∈
−
− +
− −
∑ −
− +
ρ ρ
ρ
ρ
( )( ) ( )
( )
( )
≤
≤ c8 ρn
r x( ) + cc h x
j
n
j
r
j
r s
8
1
11
=
+∑ Γ ( ) ≤
≤ cc x x
h
x x x
n
r
n
j
n
j
j n
8
1
21ρ ρ
ρ
( ) ( )
( )
+
− +( )
=
∑ ≤ c c8ρn
r x( ), x ∈ I.
Lemu 2 dovedeno.
2. Nexaj β : = arccos x, x ∈ I; α : = arccos y, y ∈ I;
l : = 24 ( r – 1 ) s + 3 ( r – 1 ) + s + 3
i
D2 l + 1, n, l ( y, x ) : =
1
2
2 1
2 1
2
( )!
( ) ( ),l x
x y J t dt
l
l
l
n l
∂
∂ β α
β α+
+
−
+
− ∫ (27)
— polinomial\ne qdro typu Dzqdyka [6, c. 129], de
Jn, l ( t ) =
1 2
2
2 1
γ n l
l
nt
t,
( )
sin
sin
/
/
( )
( )
+
, γn, l =
−
+
∫ /
/
( )
( )
π
π
sin
sin
( )
nt
t
dt
l
2
2
2 1
— qdro typu DΩeksona.
Nexaj funkciq g = g ( x ) [ neperervnog na I i Lr – 1 ( x; g ) poznaça[ mnoho-
çlen LahranΩa stepenq ≤ r – 1, qkyj interpolg[ g u toçkax –1 + 2i / (r – 1),
i = 0 1, r − .
Lema 3 [6, c. 135]. Qkwo g ∈ W
r
, r ≥ 3 i g ′′ ( x ) = 0 dlq x ∈ F ⊂ I, t o
mnohoçlen
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54 H. A. DZGBENKO, V. D. ZALIZKO
Dn ( x; g ) : =
−
− +∫ −( )
1
1
1 2 1g y L y g D y x dyr l n l( ) ( ; ) ( , ), , + Lr – 1 ( x; g ) (28)
stepenq c n nablyΩa[ g ta ]] poxidni na I tak, wo
g x D x gp
n
p( ) ( )( ) ( ; )− ≤ c x
x
x I F xn
r p n
n
l r
9
2
ρ ρ
ρ
−
−
( ) +
( )
( )
, ( )\dist
, (29)
p = 0 ∨ 1 ∨ 2 ∨ 3, x ∈ I,
zokrema
g x D x gp
n
p( ) ( )( ) ( ; )− ≤ c xn
r p
9 ρ − ( ), x ∈ I. (30)
3. Dlq koΩnoho i = 1, s poznaçymo
Yi : = Y \ yi{ }; x xj ji i+ −( )3 3, : = *Oi ;
ji
* : = ji + 2, a u vypadku js + 2 = – 1 nexaj js
* : = js – 2;
τ∨i n x, ( ) : = τ
j n i
i
x b Y* ,
( ; ; )2 – τ
j n i
i
x b Y* ,
( ; ; )2 ,
de
b2 : = l – r + s – 1.
Nexaj
Kn ( x ) : =
1
1
, ,
, , , .
\qkwo
qkwo
x I O
x y
h
x O O i si
j
i
i
∈
− ∈ ⊂ =
Lema 4. Qkwo g ∈ W
r
, g ′′ ( x ) = 0, x ∈ F ⊂ I, i g ′′ ( yi ) = 0, i = 1, s , t o
mnohoçlen
Qn ( x; g ) : = Dn ( x; g ) –
j
s
n i
i n i
i n
D y g
y
x
=
∑ ′′
′′
∨
∨
1
( ; )
( )
( )
,
,
τ
τ (31)
stepenq c n zadovol\nq[ nerivnosti
| g ( x ) – Qn ( x; g ) | ≤ c xn
r
10 ρ ( ), x ∈ I, (32)
| g ′′ ( x ) – ′′Q x gn ( ; ) | ≤ c x
x
x I F xn
r n
n
l r
11
2
2
ρ ρ
ρ
−
−
( ) +
( )
( )
, ( )\dist
Kn ( x ), x ∈ I, (33)
zokrema
| g ′′ ( x ) – ′′Q x gn ( ; ) | ≤ c x K xn
r
n11
2ρ − ( ) ( ) , x ∈ I. (34)
Dovedennq. Z rivnosti g ′′ ( yi ) = 0 i ocinok (30), (5) – (9), (14), (15) vyplyva[
nerivnist\
V xi n, ( ) : =
′′
′′
∨
∨D y g
y
xn i
i n i
i n
( ; )
( )
( )
,
,
τ
τ ≤
≤ c y c
h
y c h xn
r
i
j
j i
b s
j j
b s
i
i i i
9
2
6
2 2 1
1
1
2 1
2
1
2
2 2
ρ − + −
−
− −( )
( )( ) ( ) ( )
*
* * *
( )
Γ Γ ≤
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POTOÇKOVI OCINKY KOOPUKLOHO NABLYÛENNQ DYFERENCIJOVNYX … 55
≤ ch x
j
r
j
b s
i i
* * ( )Γ( ) − −2 12
≤ c xn
rρ ( ), x ∈ I.
Tomu nerivnist\ (32) vykonu[t\sq. Analohiçno (29), (10) i (11) pryvodqt\ do
ocinky
′′V xi n, ( ) ≤
≤ c y
y
y I F y
c
h
c
h
xn
r
i
n i
i n i
l r
j j
j
b s
i i
i
9
2
2 1
3
2 11
2
1 2
ρ ρ
ρ
−
− −
− −
( ) +
( )( )
( )
, ( )
( )
\ * *
*
( )
dist
Γ ≤
≤ c x
y
y I F y
x
x y xn
r n i
i n i
l r
n
i n
b s r
ρ ρ
ρ
ρ
ρ
−
− − − − −( )
( ) +
− +
/
2
2 2 1 2 22
( )
( )
, ( )
( )
( )\
( ) ( )
dist
≤
≤ c x
x
x I F xn
r n
n
l r
ρ ρ
ρ
−
−
( ) +
2
2
( )
( )
, ( )\dist
= : c xn
rρ −2( )Ω , x ∈ I, (35)
z qko] z uraxuvannqm (29) otrymu[mo (33), koly x ∈ I \ O. Z (35) i nerivnosti Dzq-
dyka dlq modulq poxidno] alhebra]çnoho mnohoçlena [14] (abo dyv. [6, c. 120])
vyplyva[ ocinka
′′′V xi n, ( ) ≤ c xn
rρ −3( )Ω , x ∈ I.
Cq ocinka razom z umovog g ′′ ( yi ) = 0 i (29) harantugt\ vykonannq (33) dlq
x ∈ Oi ⊂ O, i = 1, s . Dijsno,
| g ′′ ( x ) – ′′Q x gn ( ; ) | =
y
x
n
i
s
i n
i
g u D u g V u du∫ ∑′′′ − ′′′ + ′′′
=
( ) ( ; ) ( ),
1
≤
≤ c
x y
h
h xi
j
j n
r
i
i
− −ρ 3( )Ω ≤ c x K xn
r
n11
2ρ − ( ) ( )Ω .
Lemu 4 dovedeno.
Lema 5. Qkwo mnoΩyna E ⊂ I \ O sklada[t\sq z qkyx-nebud\ vidrizkiv Ij ,
to mnohoçlen
Un ( x ) : = Un ( x; E ) : =
j I E
j
r
j n j n j
j
h x b Y x b Y x
:
, ,( ; ; ) ( ; ; ) ( )
⊂
−∑ −( )1
3 3τ τ Sign Π , (36)
b3 : = 6 ( s + 2 ) + r,
stepenq c n zadovol\nq[ nerivnosti
| Un ( x ) | ≤ c xn
r
12 ρ ( ), x ∈ I, (37)
′′U xn( ) ≤ c xn
r
13
2ρ − ( ) , x ∈ I, (38)
′′U xn( ) ≥ c x
x
x E x
K xn
r n
n
l r
n14
2
2 1
ρ ρ
ρ
−
− −
+
( )
( )
( , ) ( )
( )
dist
, x ∈ I \ E, (39)
′′U x xn( ) ( )Π ≥ 0, x ∈ I \ E. (40)
Dovedennq. Na pidstavi (5) z ocinok (8) i (9) vyplyva[ (37); z (10) i (11) —
(38); z (6) i (7) — (40); z (6), (7) i (14) – (17) — (39).
Lemu 5 dovedeno.
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56 H. A. DZGBENKO, V. D. ZALIZKO
Lema 6. Qkwo g ∈ W
r
i na vidrizku
Jj : =
ν
ν
=
+
0
20r
jI∪ , j = 1 20, n r− ,
sered usix Ij + ν znajdet\sq prynajmni 2 r – 1 vidrizkiv I j p+ν , 0 ≤ ν1 < ν2 < …
… < ν2 r – 1 ≤ 20 r, takyx, wo na koΩnomu z nyx [ xoça b odna toçka
x̃ j p+ν ∈ I j p+ν , p = 1 2 1, r − , v qkij
g x j p
˜ +( )ν ≤ ρ νn
r
jx
p
˜ +( ),
to dlq vsix x ∈ Jj vykonu[t\sq nerivnist\
| g ( x ) | ≤ c r xn
r
15( ) ( )ρ .
Lemu 6 dovodqt\, vykorystovugçy nerivnist\ Whitney [15]. Zaznaçymo, wo
| Jj | = mes Jj ≤ c xn16 ρ ( ), x ∈ Jj . (41)
3. Dovedennq teoremy 1. Nexaj f ∈ W
r ∩ ∆
(
2
)
( Y ). Zobrazymo funkcig
f ′′ ( x ) u vyhlqdi sumy „malen\ko]” g1 = g1 ( x ) i „velyko]” g2 = g2 ( x ) funkcij.
Poznaçymo
A : = max { c13 + c11 , 1 }. (42)
Oznaçennq 1. Nexaj j = 1, n. Budemo pysaty j ∈ V1 , qkwo
| f ′′ ( x ) | ≤ A c15 ( r – 2 )ρn
r x−2( ), x ∈ Ij ;
j ∈ V2 , qkwo j ∉ V1 , O ∩ ν ν=− +3
3∪ I j = ∅ i
| f ′′ ( x ) | ≥ Aρn
r x−2( ), x ∈ Ij ;
j ∈ V3 , qkwo j ∉ V1 ∪ V2. Poklademo
E1 : =
j V
jI
∈ 1
∪ ; E2 : =
j V
jI
∈ 2
∪ ; E3 : =
j V
jI
∈ 3
∪ .
MnoΩyna E3 (qkwo E3 ≠ ∅ ) sklada[t\sq z (skinçennoho çysla) vidrizkiv
a bν ν,[ ] = : lν , qki ne peretynagt\sq. KoΩen vidrizok lν zhidno z lemog 6 (dlq
f ′′ ∈ W
r–2
) ne moΩe skladatys\ iz bil\ß niΩ 20 ( r – 2 ) vidrizkiv Ij . (Inßymy
slovamy, qkwo j ∈ V3 , to miΩ indeksamy j, j + 1, … , j + 20 ( r – 2 ) znajdet\sq
prynajmni odyn, qkyj ne naleΩyt\ V3 .) Poznaçymo
E1, 3 : = E1 ∪
ν
ν
ν:l E
l
∩
∪
1 ≠ ∅
.
Budemo vvaΩaty E1, 3 ≠ I (abo, wo te same, E 2 ≠ ∅ ), inakße | f ′′ ( x ) | ≤
≤ Ac r xn
r
15
2 22( ) ( )− −ρ , x ∈ I, i teorema 1 vyplyva[ z lemy 2. Nexaj kµ — vidrizky,
qki ne peretynagt\sq i skladagt\ E1, 3 =
µ µ∪ k . Çerez k
pµ poznaçymo ti z nyx,
qki skladagt\sq z tr\ox i çotyr\ox vidrizkiv Ij (qkwo taki [). Poklademo
G1 : =
E k
p
p1 3, \ ∪ µ , G2 : = I G\ 1 ⊃ E2 .
Nexaj G2
* : = G n2
*( ) poznaça[ ob’[dnannq vsix Ij , j = 1, n, takyx, wo Ij ∩ G2 ≠
≠ ∅; G2
** : = G n2
**( ) — ob’[dnannq vsix Ij takyx, wo Ij ∩ G2
* ≠ ∅; analohiç-
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POTOÇKOVI OCINKY KOOPUKLOHO NABLYÛENNQ DYFERENCIJOVNYX … 57
no, G2
*** = G2
** *( ) = G2
* * *( )( ) . Oçevydno, G2 ⊂ G2
* ⊂ G2
** ⊂ G2
*** ⊂ I . Dlq
koΩnoho j = 1, n poznaçymo
Sj ( x ) : =
x
x
j
r
j
r
x
x
j
r
j
r
j j
j
y x x y dy y x x y dy∫ ∫− − − −
−
−
− −
−
−
−
−
( ) ( ) ( ) ( )2
1
2 2
1
2
1
1
.
Oznaçennq 2. Nexaj x ∈ Ij . Poklademo
g1 ( x ) : =
0
1
2
2
2 2 2
2 2 2
, ,
( ), ,
( ) ( ), ,
( ) ( ) , ,
*
**
** * *
** * *
\
\
\
qkwo
qkwo
qkwo i
qkwo i
I G
f x I I G
f x S x I G G x G
f x S x I G G x G
j
j
j j j
j j j
⊂
′′ ⊂
′′ ⊂ ∈
′′ −( ) ⊂ ∉
g2 ( x ) : = f ′′ ( x ) – g1 ( x ).
Vvedemo
f1 ( x ) : = f ( –1 ) + f ′ ( –1 ) ( x + 1 ) +
− −
∫ ∫
1 1
1
x t
g u dudt( ) , f2 ( x ) : =
− −
∫ ∫
1 1
2
x t
g u dudt( )
(tobto f ( x) = f x f x1 2( ) ( )+ ). Vidmitymo, wo
f1, f2 ∈ c W r
17 ∩ ∆( )( )2 Y (43)
i
′′f x1 ( ) ≤ c Ac r xn
r
17 15
2 22( ) ( )− −ρ , x ∈ I. (44)
Poznaçymo
A1 : = max ( ) ,
( )
c
c c c
c
r
l r r
17
2 17 11 16
2 1
14
3
1−
− − −
+
,
c c
c
c c
c
l r r l r r
17 11
2 1
14
3
17 11
2 1
14
2
14 66− − − − − −
, ,
n 1 : = [ A1 + 1 ] n,
de [ ⋅ ] — cila çastyna. ZauvaΩymo, wo
A
x
x
n
n
1
1
ρ
ρ
( )
( )
< 1, x ∈ I. (45)
Dovedemo, wo mnohoçlen
P xn1
( ) : = G x f Q x f U x En n n( ; ) ( ; ) ( ; )1 2 21
+ + , (46)
vyznaçenyj za formulamy (26), (31) i (36), [ ßukanym v teoremi 1. Nerivnist\ (2)
bezposeredn\o vyplyva[ z ocinok (44), (20), (32) i (37). Dovedemo (1), tobto
dovedemo nerivnist\
′′ + ′′ + ′′ − ′′ + ′′( )G x f x f x Q x f f x U x E xn n n( ; ) ( ) ( ) ( ; ) ( ) ( ; ) ( )1 2 2 2 21
Π Π = :
= : Ψ Ψ1 2( ) ( )x x+ ≥ 0, x ∈ I. (47)
Nerivnist\ Ψ1( )x ≥ 0, x ∈ I, [ naslidkom (43) i (19). Zhidno z (43) i (40) ocinka
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
58 H. A. DZGBENKO, V. D. ZALIZKO
′′ − ′′ − ′′ + ′′f x Q x f f x U x En n2 2 2 21
( ) ( ; ) ( ) ( ; ) ≥ 0, x ∈ I, (48)
pryvodyt\ do nerivnosti Ψ2( )x ≥ 0, x ∈ I. Dlq dovedennq (48) rozhlqnemo
çotyry vypadky. Pry c\omu budemo vraxovuvaty oznaçennq 1 i 2 ta (33), (38) –
(43), (45). Qkwo:
1) x ∈ E2 , tobto ′′f x2 ( ) = ′′f x( ) ≥ A xn
rρ −2( ) i K xn( ) = 1, to (48) vyplyva[
z nerivnosti
A x c c x c xn
r
n
r
n
rρ ρ ρ− − −− −2
17 11
2
13
2
1
( ) ( ) ( ) ≥ 0;
2) x ∈ G E2 2\ , tobto ′′f x2 ( ) = ′′f x( ), to z uraxuvannqm nerivnosti
ρn nx K x
1 1
( ) ( ) ≤ ρn nx K x( ) ( ), x ∈ I (49)
(qka [ pravyl\nog dlq bud\-qkoho n1 ≥ n) ma[mo
− +
+
− −
− −
c c x K x c x
x
x E x
K xn
r
n n
r n
n
l r
n17 11
2
14
2
2
2 1
1 1
ρ ρ ρ
ρ
( ) ( ) ( )
( )
dist ( , ) ( )
( ) ≥
≥ − +
+
−
− −
−c c x K x
c
c
x K xn
r
n l r n
r
n17 11
2 14
16
2 1
2
1 1 1
ρ ρ( ) ( )
( )
( ) ( ) ≥ 0,
tomu ocinka (48) tym bil\ße vykonu[t\sq:
3) x ∈ I G\ ***
2 , tobto ′′f x2 ( ) = 0, to z uraxuvannqm (49) i nerivnosti
14 dist , **x G2( ) > dist ( , ) ( )x E xn2 + ρ ma[mo
−
+
−
−
c c x
x
x G x
K xn
r n
n
l r
n17 11
2
2
2
1
1
1
1
ρ
ρ
ρ
( )
( )
dist ( , ) ( )
( )** +
+ c x
x
x E x
K xn
r n
n
l r
n14
2
2
2 1
ρ ρ
ρ
−
− −
+
( )
( )
dist ( , ) ( )
( ) ≥ 0,
zvidky vyplyva[ (48);
4) x ∈ G G2 2
*** \ , to x ∉ O i (48) [ naslidkom nerivnosti
− +−
−
− −c c x
c x
n
r n
r
l r17 11
2 14
2
2 11 66
ρ ρ
( )
( )
≥ 0.
Takym çynom, ocinku (48) dovedeno dlq vsix x ∈ I.
Teoremu 1 dovedeno.
4. Dovedennq teoremy 2 dlq „malyx” n. Nexaj n = r – 1. Rozhlqnemo dva
vypadky:
1. Nexaj s ≥ r – 2. Oskil\ky ′′f yi( ) = 0 dlq vsix i = 1 2, r − , to L x f( ; )′′ : =
: = L x f y yr( ; ; , , )′′ … −1 2 ≡ 0, de L — mnohoçlen LahranΩa stepenq ≤ r – 3, qkyj
interpolg[ ′′f x( ) v yi , i = 1 2, r − . Todi z nerivnosti
′′f x( ) = y y x f x yr
i
r
i1 2
1
2
, , , ;… ′′[ ] −−
=
−
∏ ≤ c ( r ) , f ∈ Wr
, r ≥ 3, (50)
de [ ⋅ ] — podilena riznycq funkci] f ′′ v y yr1 2, ,… − i x, vyplyva[, wo mnoho-
çlen P xr−1( ) : = f f x( ) ( )( )− + ′ − +1 1 1 [ ßukanym v teoremi 2 (c ( r ) ≤ C ( Y,
r )ρC Y
r x( )( ), x ∈ I).
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POTOÇKOVI OCINKY KOOPUKLOHO NABLYÛENNQ DYFERENCIJOVNYX … 59
2. Nexaj s < r – 2. Do toçok yi , i = 1, s , dodamo r – 2 – s rivnoviddalenyx
toçok yi , i = s r+ −1 2, , – 1 = ys+1 < ys+2 < … yr−2 < ys . ZauvaΩymo, wo
analohiçno (50) vykonu[t\sq nerivnist\
′′ − ′′f x L x f( ) ( ; ) ≤ 1
2 1
2
( )!r
x y
i
r
i−
−
=
−
∏ ≤ c r x18( ) ( )Π .
Poklademo
P xr−1( ) : = f f x L u f c r u du dt
x t
( ) ( )( ) ( ; ) ( ) ( )− + ′ − + + ′′ +( )
− −
∫ ∫1 1 1
1 1
18 Π
i zauvaΩymo, wo ′′−P x xr 1( ) ( )Π ≥ 0. Dlq r – 1 < n ≤ N ( Y, r ) poklademo P xn( ) : =
: = P xr−1( ) .
Teoremu 2 dovedeno.
1. Dzyubenko G. A., Gilewicz J., Shevchuk I. A. Coconvex pointwise approximation // Ukr. mat.
Ωurn. – 2002. – 54, # 9. – S. 1200 – 1212.
2. Dzgbenko H. A., Zalizko V. D. Koopukle nablyΩennq funkcij, qki magt\ bil\ße odni[] toç-
ky perehynu // Tam Ωe. – 2004. – 56, # 3. – S. 352 – 365.
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OderΩano 27.02.2004
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
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| id | umjimathkievua-article-3573 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:45:07Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b0/e0a5ee30ad6c64d7f4712dfbe4e72fb0.pdf |
| spelling | umjimathkievua-article-35732020-03-18T19:59:02Z Pointwise Estimates for the Coconvex Approximation of Differentiable Functions Поточкові оцінки коопуклого наближення диференційовних функцій Dzyubenko, H. A. Zalizko, V. D. Дзюбенко, Г. А. Залізко, В. Д. We obtain pointwise estimates for the coconvex approximation of functions of the class $W^r,\; r > 3$. Отримано поточкові оцінки коопуклого наближення функцій із класу $W^r,\; r > 3$. Institute of Mathematics, NAS of Ukraine 2005-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3573 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 1 (2005); 47–59 Український математичний журнал; Том 57 № 1 (2005); 47–59 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3573/3879 https://umj.imath.kiev.ua/index.php/umj/article/view/3573/3880 Copyright (c) 2005 Dzyubenko H. A.; Zalizko V. D. |
| spellingShingle | Dzyubenko, H. A. Zalizko, V. D. Дзюбенко, Г. А. Залізко, В. Д. Pointwise Estimates for the Coconvex Approximation of Differentiable Functions |
| title | Pointwise Estimates for the Coconvex Approximation of Differentiable Functions |
| title_alt | Поточкові оцінки коопуклого наближення диференційовних функцій |
| title_full | Pointwise Estimates for the Coconvex Approximation of Differentiable Functions |
| title_fullStr | Pointwise Estimates for the Coconvex Approximation of Differentiable Functions |
| title_full_unstemmed | Pointwise Estimates for the Coconvex Approximation of Differentiable Functions |
| title_short | Pointwise Estimates for the Coconvex Approximation of Differentiable Functions |
| title_sort | pointwise estimates for the coconvex approximation of differentiable functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3573 |
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