On exact Bernstein-type inequalities for splines
We establish new exact Bernstein-type and Kolmogorov-type inequalities. The main result of this work is the following exact inequality for periodic splines $s$ of order $r$ and defect 1 with nodes at the points $iπ/n, i ∈ Z, n ∈ N:$ $$\left\| {s^{(k)} } \right\|_q \leqslant n^{k + 1/p - 1/q} \frac{...
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| Datum: | 2006 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509647010529280 |
|---|---|
| author | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. |
| author_facet | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. |
| author_sort | Kofanov, V. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:57:10Z |
| description | We establish new exact Bernstein-type and Kolmogorov-type inequalities. The main result of this work is the following exact inequality for periodic splines $s$ of order $r$ and defect 1 with nodes at the points $iπ/n, i ∈ Z, n ∈ N:$
$$\left\| {s^{(k)} } \right\|_q \leqslant n^{k + 1/p - 1/q} \frac{{\left\| {\varphi _{r - k} } \right\|_q }}{{\left\| {\varphi _r } \right\|_p }}\left\| s \right\|_p ,$$
where $k, r ∈ N, k < r, p = 1$ or $p = 2, q > p$, and $ϕr$ is the perfect Euler spline of order $r$. |
| first_indexed | 2026-03-24T02:44:25Z |
| format | Article |
| fulltext |
UDK 517.5
V. A. Kofanov (Dnepropetrov. nac. un-t)
O TOÇNÁX NERAVENSTVAX TYPA BERNÍTEJNA
DLQ SPLAJNOV
New sharp inequalities of Bernstein and Kolmogorov type are established. The principal result of the
present paper is the following sharp inequality for periodic splines s of order r and defect 1 with
knots at the points iπ / n, i ∈ Z, n ∈ N:
s n sk k p q r k q
r p
pq
( ) + − −
≤ 1 1/ /
ϕ
ϕ
,
where k, r ∈ N, k < r, p = 1 or p = 2, q > p, and ϕr is the perfect Euler spline of order r.
Otrymano novi toçni nerivnosti typu Bernßtejna i Kolmohorova. Osnovnym rezul\tatom roboty
[ toçna nerivnist\ dlq periodyçnyx splajniv s porqdku r defektu 1 z vuzlamy v toçkax iπ / n,
i ∈ Z, n ∈ N:
s n sk k p q r k q
r p
pq
( ) + − −
≤ 1 1/ /
ϕ
ϕ
,
de k, r ∈ N, k < r, p = 1 abo p = 2, q > p, ϕr — ideal\nyj splajn Ejlera porqdku r.
1. Vvedenye. Budem rassmatryvat\ prostranstva Lp [ a, b ] yzmerym¥x funkcyj
x : [ a, b ] → R takyx, çto || x || Lp
[ a, b ] < ∞, hde
|| x || Lp
[ a, b ] : =
x t dt p
x t p
p
a
b p
t a b
( )
< < ∞
( ) = ∞
∫
∈[ ]
1
0
/
,
, ,
vrai sup , .
esly
esly
V sluçae 2π-peryodyçeskyx funkcyj vmesto Lp [ 0, 2π ] y x Lp[ π]0 2, budem
pysat\ Lp y || x || p .
Dlq r ∈ N çerez Lr
∞ oboznaçym mnoΩestvo 2π-peryodyçeskyx funkcyj x :
R → R , ymegwyx lokal\no absolgtno neprer¥vn¥e proyzvodn¥e do porqdka
r – 1 vklgçytel\no, pryçem x( r
) ∈ L∞ . PoloΩym W r
∞ : = x L xr r∈ ≤{ }∞
( )
∞
: 1 .
Symvolom Sn, r , n, r ∈ N, oboznaçym mnoΩestvo 2π-peryodyçeskyx polyno-
myal\n¥x splajnov porqdka r defekta 1 s uzlamy v toçkax i π / n, i ∈ Z.
Dannaq rabota posvqwena reßenyg zadaçy o toçnoj konstante v neravenst-
vax typa Bernßtejna dlq splajnov s ∈ Sn, r :
s Mn sk
q
k p q
p
( ) +( − )≤ +1 1/ /
, (1)
hde k, r, n ∈ N, k < r; q > 0, p > 0, u+ = max { u, 0 }.
Dlq r ∈ N çerez ϕr ( t ) oboznaçym r-j 2π-peryodyçeskyj yntehral s nule-
v¥m srednym znaçenyem na peryode ot funkcyy ϕ0 ( t ) = sgn sin t. Dlq λ > 0 po-
loΩym ϕλ, r ( t ) : = λ–
r
ϕr ( λ t ).
NyΩe pryveden¥ nekotor¥e toçn¥e neravenstva typa (1), neobxodym¥e v
dal\nejßem. V sluçae q ≤ p yzvestn¥ sledugwye toçn¥e neravenstva takoho
typa:
© V. A. KOFANOV, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10 1357
1358 V. A. KOFANOV
s n sk
p
k r k p
r p
p
( ) −
≤
ϕ
ϕ
, p ∈ { 1, 2, ∞ }, (2)
s n sk
q
k r k q
r p
p
( ) −
≤
ϕ
ϕ
, q ≥ 1, p = ∞, yly q = 1, p > 1. (3)
Neravenstvo (2) v sluçae p = ∞ ustanovleno V. M. Tyxomyrov¥m [1], v sluçae
p = 1 — G. M. Subbotyn¥m [2], a v sluçae p = 2 — V. F. Babenko y S. A. Pyçu-
hov¥m (sm., naprymer, teoremu 6.3.6 yz [3]). Neravenstva (3) dokazan¥ A. A. Ly-
hunom [4, 5].
V sluçae q > p otmetym toçnoe neravenstvo
s n sk k p r k
r p
p
( )
∞
+ − ∞≤ 1/
ϕ
ϕ
, k = 1, … , r, p ≥ 1, (4)
poluçennoe V. F. Babenko, V. A. Kofanov¥m y S. A. Pyçuhov¥m [6, 7].
Zametym, çto neravenstva (2) y (3) qvlqgtsq toçn¥my na klasse Sn, r (ony
obrawagtsq v ravenstvo dlq splajnov s ( t ) = a ϕn , r ( t ) ), a neravenstvo (4) toçnoe
na klasse S =
Sn rn ,∈N∪ (ono obrawaetsq v ravenstvo lyß\ dlq splajnov s ( t ) =
= a ϕr ( t ) ).
Osnovn¥m rezul\tatom nastoqwej rabot¥ qvlqetsq toçnoe neravenstvo typa
Bernßtejna
s n sk
q
k p q r k q
r p
p
( ) + − −
≤ 1 1/ /
ϕ
ϕ
(5)
dlq splajnov s ∈ Sn, r y proyzvol\noho q > p, hde p = 1 yly p = 2 (teorema 4).
Krome toho, v dannoj rabote poluçen¥ toçn¥e neravenstva typa Kolmohoro-
va y Bernßtejna v prostranstvax s lokal\n¥my „normamy” (teorem¥ 1 y 2), a
takΩe dokazana teorema sravnenyq perestanovok proyzvodn¥x splajnov s ∈ Sn, r
(teorema 3). Ymenno πty try teorem¥ sostavlqgt osnovu dokazatel\stva nera-
venstva (5).
2. Toçn¥e neravenstva typa Kolmohorova y Bernßtejna s lokal\n¥my
„normamy”. Dlq l-peryodyçeskoj funkcyy x ∈ Lp [ 0, l ] poloΩym
L ( x ) p : = sup {|| x || Lp
[ a, b ] : | x ( t ) | > 0 ∀t ∈ ( a, b ), a, b ∈ R}. (6)
Xarakterystyky takoho typa yzuçalys\ v rabote [8]. Funkcyonal L ( x ) p ne qv-
lqetsq normoj (on ne ymeet svojstva poluaddytyvnosty). Tem ne menee, v rabo-
te [8] pokazano, çto v prostranstvax s lokal\n¥my „normamy” L ( x ) p ostagtsq
spravedlyv¥my mnohye vaΩn¥e utverΩdenyq teoryy approksymacyy.
Dlq p > 0 symvolom E0( x ) Lp
[ a, b ] oboznaçym nayluçßee pryblyΩenye funk-
cyy x ∈ Lp [ a, b ] konstantamy v prostranstve Lp [ a, b ], t. e.
E0 ( x ) Lp
[ a, b ] : = inf { || x – c || Lp
[ a, b ] : c ∈ R }.
V sluçae 2π-peryodyçeskoj funkcyy x budem pysat\ E0 ( x ) p vmesto
E0 ( x ) Lp
[ 0, 2π ]. Çerez cp ( x ) oboznaçym konstantu nayluçßeho Lp-pryblyΩenyq
funkcyy x ∈ Lp .
Dlq x ∈ L1 [ a, b ] oboznaçym çerez r ( x, t ) perestanovku funkcyy | x | (sm.,
naprymer, [9], § 6.1). PoloΩym takΩe r ( x, t ) = 0 dlq t ≥ b – a. KaΩdoj l-pery-
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
O TOÇNÁX NERAVENSTVAX TYPA BERNÍTEJNA DLQ SPLAJNOV 1359
odyçeskoj funkcyy x postavym v sootvetstvye perestanovku r ( x, t ) suΩenyq
| x | na [ 0, l ].
Teorema 1. Pust\ k, r ∈ N, k < r, p > 0, q ≥ 1. Dlq lgboj funkcyy x ∈ Lr
∞
v¥polneno toçnoe neravenstvo
L ( x( k
)
) q ≤
L
E
x xr k q
r p
p
r( )
( )
− ( )
∞
−ϕ
ϕ α
α α
0
1
, (7)
hde α =
r k q
r p
− +
+
1
1
/
/
. Neravenstvo (7) obrawaetsq v ravenstvo dlq funkcyj
x ( t ) = a [ ϕr ( t + b ) – cp ( ϕr ) ], a, b ∈ R.
Dokazatel\stvo. Zafyksyruem proyzvol\nug funkcyg x ∈ Lr
∞ . Vsled-
stvye odnorodnosty neravenstva (7) moΩno predpoloΩyt\, çto
|| x( r
)
|| ∞ = 1. (8)
Tohda x ∈ W r
∞ . V¥berem λ > 0 yz uslovyq
|| x || p = E0 ( ϕλ, r ) Lp
[ 0, 2π / λ ] . (9)
Netrudno vydet\, çto E0 ( ϕλ, r ) Lp
[ 0, 2π / λ ] = λ–
r
–
1
/
p
E0 ( ϕ ) p
. Poπtomu yz (9) v sylu
neravenstva [10]
E0 ( x ) ∞ ≤
ϕ
ϕ
r
r p
r r p p
r r p r r r p
E
x x∞
( + )
( + ) ( )
∞
−( ( + ))
( )0
1
1 1 1
/ /
/ / / /
y uslovyq (8) sleduet ocenka
E0 ( x ) ∞ ≤ λ–
r
|| ϕr || ∞ = || ϕλ, r || ∞
. (10)
Prymenqq neravenstvo Kolmohorova [11]
|| x( i
)
|| ∞ ≤
ϕ
ϕ
r i
r
i r
i r r i r
E x x
− ∞
∞
− ∞
− ( )
∞( )1 0
1
/
/ /
, i = 1, … , r – 1,
yz (10) y (8) poluçaem sootnoßenye
|| x( i
)
|| ∞ ≤ || ϕλ, r – i || ∞
, i = 1, … , r – 1. (11)
Yz (11) y (10), v svog oçered\, sleduet neravenstvo
L ( x( k
)
) 1 ≤ L ( ϕλ, r – k ) 1 . (12)
Dejstvytel\no, pust\ [ a , b ] — proyzvol\n¥j promeΩutok, takoj, çto
| x( k
)
( t ) | > 0 dlq t ∈ ( a, b ), y c — nul\ funkcyy ϕλ, r – k . Tohda vsledstvye (11)
y (10)
x t dtk
a
b
( )( )∫ = | x( k
–
1
)
( b ) – x( k
–
1
)
( a ) | ≤ 2E0 ( x( k
–
1
)
) ∞ ≤ 2 || ϕλ, r – ( k – 1) || ∞ =
= | ϕλ, r – ( k – 1) ( c + π / λ ) – ϕλ, r – ( k – 1) ( c ) | = ϕλ
λ
,
/
r k
c
c
t dt−
+π
( )∫ = L ( ϕλ, r – k ) 1 ,
hde L ( x ) p opredeleno sohlasno (6). Poπtomu otsgda sleduet (12).
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1360 V. A. KOFANOV
PoloΩym σ : = [ a, b ] y oboznaçym çerez xσ suΩenye funkcyy x na σ. Pos-
kol\ku proyzvodnaq x( k
)
neprer¥vna, toçnaq verxnqq hran\ v opredelenyy (6)
velyçyn¥ L ( x( k
)
) p dostyhaetsq na takyx a y b, çto
x( k
)
( a ) = x( k
)
( b ) = 0. (13)
Budem sçytat\ uslovye (13) v¥polnenn¥m y dokaΩem neravenstvo
r x t dt r t dtk
r k( ) ≤ ( )( )
− +∫ ∫ ( )σ
ξ
λ
ξ
ϕ, ,,
0 0
, ξ > 0. (14)
Dlq dokazatel\stva (14) zametym, çto vsledstvye (11)
r x k( )( )
σ , 0 ≤ r r k( )( )− +ϕλ, , 0 .
PokaΩem, çto raznost\
∆ ( t ) : = r x t r tk
r k( ) − ( )( )
− +( )σ λϕ, ,,
menqet znak (s – na +) ne bolee odnoho raza.
Çtob¥ dokazat\ πtot fakt, zametym, çto v sylu (11) y (13) dlq lgboho y ∈
∈ 0, x k
σ
( )
∞( ) najdutsq toçky ti ∈ ( a, b ), i = 1, … , m, m ≥ 2, y dve toçky yj ∈
∈ ( c, c + 2π / λ ) takye, çto
y = x tk
iσ
( )( ) = ( ϕλ, r – k ) + ( yj ) ,
hde u+ : = max { u, 0 }. Pry πtom sohlasno teoreme sravnenyq Kolmohorova [11],
prymenennoj k funkcyy x k
σ
( )
(ee uslovyq v¥polnen¥ vsledstvye (8) y (11)), v¥-
polnqetsq neravenstvo
x t yk
i r k jσ λϕ
( + )
− +( ) ≤ ( ′ ) ( )1
, .
Poπtomu v sylu teorem¥ o proyzvodnoj perestanovky (sm., naprymer, [3], pred-
loΩenye 1.3.2), esly toçky θ1 y θ2 v¥bran¥ tak, çto
y = r x k( )( )
σ θ, 1 = r r k( )( )− +ϕ θλ, , 2 ,
to
′( ) = ( )
( ) ( + ) −
=
−
∑r x x tk k
i
i
m
σ σθ, 1
1 1
1
1
≤
≤ ( ′ ) ( )
= ′ ( )− +
−
=
−
− +∑ ( )ϕ ϕ θλ λ, , ,r k j
j
r ky r
1
1
2 1
2 .
Otsgda sleduet, çto raznost\ ∆ menqet znak (s – na +) ne bolee odnoho raza.
Rassmotrym yntehral
I ( ξ ) : = r x t r t dtk
r k( ) − ( )[ ]( )
− +( )∫ σ λ
ξ
ϕ, ,,
0
.
Qsno, çto I ( 0 ) = 0. Dalee, vsledstvye (12)
r x t dt x t dt L xk k k( ) = ( ) ≤ ( )( )
π
( )
π
( )∫ ∫σ σ,
0
2
0
2
1 ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
O TOÇNÁX NERAVENSTVAX TYPA BERNÍTEJNA DLQ SPLAJNOV 1361
≤ L r t dtr k r k( ) = ( )− − +
π
( )∫ϕ ϕλ λ
λ
, ,
/
,1
0
2
.
Poπtomu esly m¥ poloΩym M : = max { 2π, 2π / λ }, to I ( M ) ≤ 0. Krome toho, kak
m¥ vydely, I′ ( t ) = ∆ ( t ) menqet znak (s – na +) ne bolee odnoho raza. Takym
obrazom, I ( ξ ) ≤ 0 dlq vsex ξ ≥ 0, t. e. neravenstvo (14) dokazano.
Yz neravenstva (14) v sylu teorem¥ Xardy – Lytlvuda (sm., naprymer, [3],
predloΩenye 1.3.10) poluçaem
|| x( k
)
|| Lq
[ a, b ] ≤ || ( ϕλ, r – k ) + || Lq
[ 0, 2π / λ ]
dlq lgboho q ≥ 1 y dlq lgboho promeΩutka [ a, b ], dlq kotoroho | x( k
)
( t ) | > 0,
t ∈ ( a, b ). Otsgda sohlasno opredelenyg (6) sleduet
L ( x( k
)
) q ≤ L ( ϕλ, r – k ) q , q ≥ 1. (15)
Lehko vydet\, çto
L ( ϕλ, r ) q = λ–
r
–
1
/
q
L ( ϕr ) q , E0 ( ϕλ, r ) Lp
[ 0, 2π / λ ] = λ–
r
–
1
/
p
E0 ( ϕr ) p
. (16)
Poπtomu yz (9) y (15) s uçetom toho, çto α = ( r – k + 1 / q ) / ( r + 1 / p ), v¥vodym
L x
x
L
E
L
E
L
E
k
q
p
r k q
r L
r k q
r k q
r p
r p
r k q
r pp
( )
≤
( )
( )
=
( )
( )
=
( )
( )
( )
−
[ π ]
−( − )−
−
− −
−
[ ]α
λ
λ λ
α α α
ϕ
ϕ
λ ϕ
λ ϕ
ϕ
ϕ
,
, , /
/
/
0 0 2
1
1
0 0
.
∏to vsledstvye (8) zaverßaet dokazatel\stvo neravenstva (7).
Toçnost\ (7) oçevydna.
Teorema dokazana.
Poskol\ku L ( x ) ∞ = || x || ∞ dlq x ∈ L∞ , yz teorem¥ 1 v¥tekaet takoe sled-
stvye.
Sledstvye 1. V uslovyqx teorem¥ 1 dlq funkcyj x ∈ Lr
∞ v¥polnqetsq
toçnoe neravenstvo
|| x( k
)
|| ∞ ≤
ϕ
ϕ α
α αr k
r p
p
r
E
x x
− ∞ ( )
∞
−
( )0
1
, (17)
hde α = ( r – k ) / ( r + 1 / p ).
Neravenstvo (17) b¥lo dokazano v [6].
Zameçanye 1. Yspol\zuq kryteryj πlementa nayluçßeho Lp-pryblyΩenyq
(pry p ≥ 1), netrudno proveryt\ spravedlyvost\ ravenstva
E0 ( ϕr ) p = || ϕr || p . (18)
S druhoj storon¥, suwestvuet p0 < 1 / 2 takoe, çto ravenstvo (18) uΩe ne v¥-
polnqetsq dlq neçetn¥x r v sluçae p ∈ ( 0, p0 ) [12]. V to Ωe vremq πto raven-
stvo soxranqet sylu dlq neçetn¥x r v sluçae lgboho p ∈ ( 0, 1 ) y dlq lgboho
r v sluçae p ≥ 1 / 2 [12].
Zameçanye 2. V rabotax [13 – 15] poluçen¥ toçn¥e neravenstva typa Kol-
mohorova, kotor¥e ocenyvagt Lq-normu proyzvodnoj || x( k
)
|| q funkcyy x ∈ Lr
∞
çerez lokal\nug „normu”
||| x ||| p : = sup { E0 ( x ) Lp
[ a, b ] : x′ ( t ) ≠ 0 ∀t ∈ ( a, b ), a, b ∈ R } (19)
y L∞-normu proyzvodnoj || x( r
)
|| ∞
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1362 V. A. KOFANOV
Teorema 2. Pust\ k, r, n ∈ N, k < r, p > 0, q ≥ 1. Tohda dlq lgboho splajna
s ∈ Sn, r v¥polnqetsq neravenstvo
L ( s( k
)
) q ≤ n
L
E
sk p q r k q
r p
p
+ − −( )
( )
1 1
0
/ / ϕ
ϕ
. (20)
Neravenstvo (20) toçnoe na klasse S =
Sn rn ,∈N∪ y obrawaetsq v ravenstvo
dlq splajnov s ( t ) = a [ ϕr ( t ) – cp ( ϕr ) ], a ∈ R.
Dokazatel\stvo. Zafyksyruem proyzvol\n¥j splajn s ∈ Sn, r
. Qsno, çto
s ∈ Lr
∞ . Prymenym neravenstvo (7) k splajnu s:
L ( s( k
)
) q ≤
L
E
s sr k q
r p
p
r( )
( )
− ( )
∞
−ϕ
ϕ α
α α
0
1
,
hde α =
r k q
r p
− +
+
1
1
/
/
. Ocenym normu || s( r
)
|| ∞
. Vospol\zuemsq neravenstvom (4),
kotoroe soxranqet sylu y dlq p ∈ ( 0, 1 ), esly v nem ϕr p
zamenyt\ na
E0 ( ϕr ) p [7], t. e. ymeet mesto neravenstvo
|| s( k
)
|| ∞ ≤ n
E
sk p r k
r p
p
+ − ∞
( )
1
0
/
ϕ
ϕ
, k = 1, … , r, p > 0. (21)
Ocenyvaq normu || s( r
)
|| ∞ s pomow\g neravenstva (21), poluçaem
L ( s( k
)
) q ≤
L
E
s n
E
sr k q
r p
p
r p
r p
p
( )
( ) ( )
− + −ϕ
ϕ ϕα
α
α
0
1
0
1/
.
Otsgda sleduet (20), esly uçest\, çto
( r + 1 / p ) ( 1 – α ) = k
p q
+ −1 1
.
Toçnost\ neravenstva (20) oçevydna.
Teorema dokazana.
Sledstvye 2. Pust\ k, r, n ∈ N, k < r, p > 0, q ≥ 1. Esly splajn s ∈ Sn, r
udovletvorqet uslovyg
|| s || p = E0 ( ϕn , r ) Lp
[ 0, 2π / n ] ,
to
L ( s( k
)
) q ≤ L ( ϕn , r – k ) q ,
v çastnosty
|| s( k
)
|| ∞ ≤ || ϕn , r – k || ∞ .
Dokazatel\stvo. Vsledstvye (16) neravenstvo (20) moΩno perepysat\ v
vyde
L ( s( k
)
) q ≤
L
E
sn r k q
n r L n
p
p
( )
( )
−
[ π ]
ϕ
ϕ
,
, , /0 0 2
.
Teper\ utverΩdenye sledstvyq oçevydno.
Zameçanye 3. Neravenstva (4) y (21) pry k < r neposredstvenno sledugt yz
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
O TOÇNÁX NERAVENSTVAX TYPA BERNÍTEJNA DLQ SPLAJNOV 1363
(20). Yz neravenstva (21) pry k = r v uslovyqx sledstvyq 2 v¥tekaet takΩe
ocenka
|| s( r
)
|| ∞ ≤ || ϕn , 0 || ∞ = 1.
Zameçanye 4. V rabotax [13 – 15] poluçen¥ toçn¥e neravenstva typa Bern-
ßtejna, kotor¥e ocenyvagt Lq-normu || s( k
)
|| q proyzvodnoj splajna s ∈ Sn, r
çerez lokal\nug „normu” ||| s ||| p (sm. (19)).
3. Teorema sravnenyq perestanovok y toçnoe neravenstvo typa Bern-
ßtejna.
Teorema 3. Pust\ k, n, r ∈ N, k < r, p = 1 yly p = 2. Esly splajn s ∈ Sn, r
udovletvorqet uslovyg
|| s || p = || ϕn , r || Lp
[ 0, 2π / n ] , (22)
to dlq lgboho ξ > 0
r s t dt r t dtp k p
n r k( ) ≤ ( )( )
−∫ ∫, ,,
0 0
ξ ξ
ϕ . (23)
Dokazatel\stvo. Budem sledovat\ ydee N. P. Kornejçuka y A. A. Lyhuna,
yspol\zovannoj ymy pry dokazatel\stve teorem¥ sravnenyq proyzvodn¥x pere-
stanovok funkcyj x ∈ W r
∞ (sm., naprymer, [16], teorema 5.5.1).
Poskol\ku perestanovka r ( s( k
), t ) ynvaryantna otnosytel\no sdvyha arhu-
menta, a proyzvodnaq s( k
)
ymeet nuly, moΩno sçytat\, çto s( k
)
( 0 ) = 0.
Zafyksyruem proyzvol\n¥j splajn s ∈ Sn, r
, udovletvorqgwyj uslovyg
(22), y pust\
δ ( t ) : = r ( ϕn , r – k
, t ) – r ( s( k
), t ).
Dlq kaΩdoho ξ ∈ ( 0, 2π / n ), dlq kotoroho δ ( ξ ) = 0, poloΩym
z : = r ( s( k
), ξ ) = r ( ϕn , r – k
, ξ ).
Tohda
r s t dt s t dtp k k p
Ez
( ) = ( )( ) ( )∫ ∫,
0
ξ
, (24)
hde
Ez : = { t : | s( k
)
( t ) | > z, t ∈ ( 0, 2π ) }. (25)
Çerez ( ti , τi ), i = 1, … , m, oboznaçym sostavlqgwye ynterval¥ otkr¥toho
mnoΩestva Ez . Takym obrazom, Ez = ( )= ti ii
m
, τ
1∪ . Dlq prodolΩenyq dokaza-
tel\stva teorem¥ nam ponadobytsq ocenka sverxu çysla m πtyx yntervalov, so-
derΩawaqsq v sledugwej lemme.
Lemma 1. Pust\ toçka ξ ∈ ( 0, 2π / n ) takova, çto δ ( ξ ) = 0, pryçem dlq
nekotoroho ε > 0
δ ( t ) < 0, t ∈ ( ξ – ε, ξ ).
Tohda mnoΩestvo Ez soderΩyt ne bolee dvux sostavlqgwyx yntervalov.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1364 V. A. KOFANOV
Dokazatel\stvo lemm¥. PredpoloΩym protyvnoe, t. e. dopustym, çto v
uslovyqx lemm¥ suwestvuet m > 2 yntervalov ( ti , τi ) ⊂ Ez ⊂ ( 0, 2π ) takyx, çto
z = r ( ϕn , r – k
, ξ ) = r ( s( k
), ξ ) = | s( k
)
( ti ) | = | s( k
)
( τi ) |,
| s( k
)
( t ) | > z, t ∈ ( ti , τi ), i = 1, … , m.
V¥berem h > 0 stol\ mal¥m, çtob¥
Ez + h =
( ′ ′)
=
ti i
i
m
, τ
1
∪ , ( ′ ′)ti i, τ ⊂ ( ti , τi ),
y esly
r ( s( k
), ξ – γ ) = z + h,
to
δ ( t ) < 0, t ∈ ( ξ – γ, ξ ). (26)
Sohlasno opredelenyg (25)
s t sk
i
k
i
( ) ( )( ′) = ( ′)τ = z + h, i = 1, 2, … , m,
y, vsledstvye (26), esly
r ( ϕn , r – k
, ξ – γ0 ) = z + h,
to ξ – γ0 < ξ – γ, t. e.
γ < γ0
. (27)
Krome toho, v sylu svojstva ravnoyzmerymosty perestanovky
mes { t ∈ ( 0, 2π ) : | r ( s( k
), t ) | > y } = mes { t ∈ ( 0, 2π ) : | s( k
)
( t ) | > y }
dlq lgboho y > 0. Sledovatel\no,
γ = ′ − + ′ −( )
=
∑ t ti i i i
i
m
τ τ
1
.
Analohyçno, esly toçky α1
, α2 ∈ 0
2
, π
n
opredelen¥ uslovyqmy
ϕn , r – k ( α1 ) = z, ϕn , r – k ( α2 ) = z + h,
to
4 | α2 – α1 | = γ0
.
Yz uslovyq (22) v sylu sledstvyq 2, zameçanyq 3 y ravenstva
E0 ( ϕn , r ) Lp
[ 0, 2π / n ] = || ϕn , r || Lp
[ 0, 2π / n ] , p ≥ 1,
sleduet neravenstvo
|| s( k
)
|| ∞ ≤ || ϕn , r – k || ∞ , k = 1, … , r. (28)
Takym obrazom, prymenqq teoremu sravnenyq Kolmohorova (k splajnu s( k
)
), po-
luçaem
| α2 – α1 | ≤ ′ −t tk k , | α2 – α1 | ≤ ′ −τ τk k .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
O TOÇNÁX NERAVENSTVAX TYPA BERNÍTEJNA DLQ SPLAJNOV 1365
∏to pozvolqet vsledstvye predpoloΩenyq m > 2 zaklgçyt\, çto
γ = ′ − + ′ −( ) ≥ − = >
=
∑ t t m mk k k k
k
m
τ τ α α γ γ
1
2 1
0 02 2
4
4
4
= γ0
.
Poslednee nevozmoΩno vsledstvye (27).
Lemma dokazana.
ProdolΩenye dokazatel\stva teorem¥ 3. PreΩde vseho zametym, çto yz
uslovyq (22) y neravenstva (2), kotoroe s uçetom ravenstva || ϕn , r || Lp
[ 0, 2π / n ] =
= n–
r
–
1
/
p
|| ϕr || p moΩno perepysat\ v vyde
|| s( k
)
|| p ≤
ϕ
ϕ
n r k L n
n r L n
p
p
p
s
, , /
, , /
− [ π ]
[ π ]
0 2
0 2
,
sleduet ocenka
|| s( k
)
|| p ≤ || ϕn , r – k || Lp
[ 0, 2π / n ] . (29)
PredpoloΩym, çto utverΩdenye teorem¥ neverno. ∏to oznaçaet, çto
min , , , ,, ,
η
η ξ
ϕ ϕr t r s t dt r t r s tp
n r k
p k p
n r k
p k( ) − ( )[ ] = ( ) − ( )[ ]−
( )
−
( )∫ ∫
0 0
< 0. (30)
Poskol\ku sohlasno (29) dlq lgboho η ≥ 2π / n
r s t dt sp k k
p
p
n r k L n
p
p
( ) ≤ ≤( ) ( )
− [ π ]∫ , , , /
0
0 2
η
ϕ =
= r t dt r t dtp
n r k
n
p
n r k( ) = ( )−
π
−∫ ∫ϕ ϕ
η
,
/
,, ,
0
2
0
,
dlq toçky ξ, realyzugwej mynymum v (30), spravedlyvo vklgçenye ξ ∈ ( 0,
2π / n ). Pry πtom toçka ξ budet takoj, çto δ ( ξ ) = 0 y dlq nekotoroho ε > 0
v¥polnqetsq neravenstvo δ ( t ) < 0, t ∈ ( ξ – ε, ξ ), t. e. toçka ξ budet udovletvo-
rqt\ uslovyqm lemm¥ 1. Sohlasno πtoj lemme, esly ( aj , bj ), j = 1, … , l, — sos-
tavlqgwye ynterval¥ mnoΩestva E0 = { t : | s( k
)
( t ) | > 0, t ∈ ( 0, 2π ) }, takye, çto
( aj , bj ) ⊃ ( ti , τi ) dlq nekotoroho i ∈ { 1, … , m }, to
l ≤ m ≤ 2. (31)
Prymenqq k splajnu s( k
)
teoremu sravnenyq Kolmohorova (ee uslovyq v¥-
polnen¥ vsledstvye (28)), poluçaem
s t dt t dtk p
a b E
n r k
p
Lj j z z
( )
( )
−( ) ≥ ( )∫ ∫
, \
,
1
2
ϕ ,
hde Lz : = { t : | ϕn , r – k ( t ) | < z, t ∈ ( 0, 2π / n ) }. Qsno, çto
ϕ ϕ
ξ
n r k
p
L
p
n r k
n
t dt r t dt
z
, ,
/
,− −
π
( ) = ( )∫ ∫
2
.
Sledovatel\no,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1366 V. A. KOFANOV
s t dt r t dtk p
a b E
p
n r k
n
j j z
( )
( )
−
π
( ) ≥ ( )∫ ∫
, \
,
/
,1
2
2
ϕ
ξ
. (32)
Krome toho, sohlasno sledstvyg 2 teorem¥ 2 (dlq q = p )
s t dt L s Lk p
a
b
k
p
p
n r k p
p
j
j
( ) ( )
−( ) ≤ ( ) ≤ ( )∫ ϕ , =
= 1
2
1
2
0
2
0
2
ϕ ϕn r k
p
n
p
n r k
n
t dt r t dt,
/
,
/
,−
π
−
π
( ) = ( )∫ ∫ . (33)
Prymenqq (24) y (31) – (33), ymeem
r s t dt s t dt s t dtp k k p
E
k p
ti
m
z i
i
( ) = ( ) = ( )( ) ( ) ( )
=
∫ ∫ ∫∑,
0 1
ξ τ
=
= s t dt s t dtk p
a
b
j
l
k p
a b Ej
l
j
j
j j z
( )
=
( )
( )=
( ) − ( )∫∑ ∫∑
1 1 , \
≤
≤ l r t dt l r t dtp
n r k
n
p
n r k2 2
0
2 2
( ) − ( )−
π
−
π
∫ ∫ϕ ϕ
ξ
,
/
,, , =
= l r t dt r t dtp
n r k
p
n r k2
0 0
( ) ≤ ( )− −∫ ∫ϕ ϕ
ξ ξ
, ,, , ,
çto protyvoreçyt (30).
Teorema dokazana.
Teorema 4. Pust\ k, r, n ∈ N, k < r, p = 1 yly p = 2, q > p. Dlq lgboho
splajna s ∈ Sn, r v¥polnqetsq neravenstvo
|| s( k
)
|| q ≤ n sk p q r k q
r p
p
+ − −1 1/ /
ϕ
ϕ
. (34)
Neravenstvo (34) neuluçßaemo v tom sm¥sle, çto
sup sup
,
/ /
n s S
s
k
q
k p q
p
r k q
r pn r
s
n s∈ ∈
≠
( )
+ −
−
=
N
0
1 1
ϕ
ϕ
.
Dokazatel\stvo. Zafyksyruem lgboj splajn s ∈ Sn, r
, s ≠ 0. Vsledstvye
odnorodnosty neravenstva (34) moΩno sçytat\, çto
|| s || p = || ϕn , r || Lp
[ 0, 2π / n ] . (35)
Tohda sohlasno teoreme 3
r s t dt r s t dtk p p k( ) ( )( ) = ( )∫ ∫, ,
0 0
ξ ξ
≤
≤ r t dt r t dtp
n r k n r k
p( ) = ( )− −∫ ∫ϕ ϕ
ξ ξ
, ,, ,
0 0
, ξ > 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
O TOÇNÁX NERAVENSTVAX TYPA BERNÍTEJNA DLQ SPLAJNOV 1367
Otsgda sohlasno teoreme Xardy – Lytlvuda (sm., naprymer, [3], predloΩenye
1.3.10) sleduet, çto dlq lgboho m > 1
|| | s( k
)
| p || m ≤ || | ϕn , r – k | p || Lm
[ 0, 2π / n ] .
Poπtomu dlq lgboho q > p
|| s( k
)
|| q = s k p
q p
p
n r k
p
L n
p
q p
( )
− [ π ]
≤
/
/
, , /
/
/
1
0 2
1
ϕ = || ϕn , r – k || Lq
[ 0, 2π / n ] . (36)
Poskol\ku || ϕn , r || Lp
[ 0, 2π / n ] = n–
r
–
1
/
p
|| ϕr || p
, yz (35) y (36) v¥vodym
s
s
n
k
q
p
n r k L n
n r L n
k p q r k q
r p
q
p
( )
− [ π ]
[ π ]
+ − −
≤ =
ϕ
ϕ
ϕ
ϕ
, , /
, , /
/ /0 2
0 2
1 1
.
Tem sam¥m (34) dokazano.
Qsno, çto (34) obrawaetsq v ravenstvo dlq splajnov s ( t ) = a ϕr ( t ).
Teorema dokazana.
1. Tyxomyrov V. M. Popereçnyky mnoΩestv v funkcyonal\n¥x prostranstvax y teoryq nay-
luçßyx pryblyΩenyj // Uspexy mat. nauk. – 1960. – 15, # 3. – S. 81 – 120.
2. Subbotyn G. N. O kusoçno-polynomyal\noj ynterpolqcyy // Mat. zametky. – 1967. – 1,
#U1. – S. 24 – 29.
3. Kornejçuk N. P., Babenko V. F., Lyhun A. A. ∏kstremal\n¥e svojstva polynomov y splajnov.
– Kyev: Nauk. dumka, 1992. – 304 s.
4. Lyhun A. A. Toçn¥e neravenstva dlq splajn-funkcyj y nayluçßye kvadraturn¥e formu-
l¥ dlq nekotor¥x klassov funkcyj // Mat. zametky. – 1976. – 19, # 6. – S. 913 – 926.
5. Lyhun A. A. O neravenstvax meΩdu normamy proyzvodn¥x peryodyçeskyx funkcyj // Tam
Ωe. – 1983. – 33, # 3. – S. 385 – 391.
6. Babenko V. F., Kofanov V. A., Pichugov S. A. Inequalities for norms of intermediate derivatives of
periodic functions and their applications // East J. Approxim. – 1997. – 3, # 3. – P. 351 – 376.
7. Kornejçuk N. P., Babenko V. F., Kofanov V. A., Pyçuhov S. A. Neravenstva dlq proyzvodn¥x
y yx pryloΩenyq. – Kyev: Nauk. dumka, 2003. – 590 s.
8. Pinkus A., Shisha O. Variations on the Chebyshev and Lp-theories of best approximation // J.
Approxim. Theory. – 1982. – P. 148 – 168.
9. Kornejçuk N. P. Toçn¥e konstant¥ v teoryy pryblyΩenyj. – M.: Nauka, 1987. – 424 s.
10. Babenko V. F., Kofanov V. A., Pichugov S. A. Inequalities of Kolmogorov type and some their
applications in approximation theory // Rend. Circolo mat. Palermo. Ser. II, Suppl. – 1998. – 52. –
P. 223 – 237.
11. Kolmohorov A. N. O neravenstvax meΩdu verxnymy hranqmy posledovatel\n¥x proyzvod-
n¥x funkcyy na beskoneçnom yntervale // Yzbr. trud¥. Matematyka, mexanyka. – M.: Nauka,
1985. – S. 252 – 263.
12. Babenko V. F., Kofanov V. A., Pyçuhov S. A. Approksymacyq synusopodobn¥x funkcyj kon-
stantamy v prostranstvax Lp , p < 1 // Ukr. mat. Ωurn. – 2004. – 56, # 6. – S. 745 – 762.
13. Kofanov V. A. Some exact inequalities of Kolmogorov type // Mat. fyzyka, analyz, heometryq.
– 2002. – 9, # 3. – S. 412 – 419.
14. Kofanov V. A. O nekotor¥x neravenstvax typa Kolmohorova, uçyt¥vagwyx çyslo peremen
znaka // Ukr. mat. Ωurn. – 2003. – 55, # 4. – S. 456 – 469.
15. Kofanov V. A. O toçn¥x neravenstvax typa Kolmohorova y Bernßtejna // Teoryq nably-
Ωen\ ta harmoniçnyj analiz: Pr. Ukr. mat. konhresu. – 2001. – S. 84 – 99.
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Poluçeno 13.01.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
|
| id | umjimathkievua-article-3538 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:44:25Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/75/f71ca2eb0253eae0efe72ba2a384a175.pdf |
| spelling | umjimathkievua-article-35382020-03-18T19:57:10Z On exact Bernstein-type inequalities for splines О точных неравенствах типа Бернштейна для сплайнов Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. We establish new exact Bernstein-type and Kolmogorov-type inequalities. The main result of this work is the following exact inequality for periodic splines $s$ of order $r$ and defect 1 with nodes at the points $iπ/n, i ∈ Z, n ∈ N:$ $$\left\| {s^{(k)} } \right\|_q \leqslant n^{k + 1/p - 1/q} \frac{{\left\| {\varphi _{r - k} } \right\|_q }}{{\left\| {\varphi _r } \right\|_p }}\left\| s \right\|_p ,$$ where $k, r ∈ N, k < r, p = 1$ or $p = 2, q > p$, and $ϕr$ is the perfect Euler spline of order $r$. Отримано нові точні нерівності топу Бернштейна і Колмогорова. Основним результатом роботи є точна нерівність для періодичних сплайнів $s$ порядку $r$ дефекту 1 з вузлами в точках $iπ/n, i ∈ Z, n ∈ N:$ $$\left\| {s^{(k)} } \right\|_q \leqslant n^{k + 1/p - 1/q} \frac{{\left\| {\varphi _{r - k} } \right\|_q }}{{\left\| {\varphi _r } \right\|_p }}\left\| s \right\|_p ,$$ де $k, r ∈ N, k < r, p = 1$ або $p = 2, q > p$, $ϕr$ — ідеальний сплайн Ейлера порядку $r$. Institute of Mathematics, NAS of Ukraine 2006-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3538 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 10 (2006); 1357–1367 Український математичний журнал; Том 58 № 10 (2006); 1357–1367 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3538/3812 https://umj.imath.kiev.ua/index.php/umj/article/view/3538/3813 Copyright (c) 2006 Kofanov V. A. |
| spellingShingle | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. On exact Bernstein-type inequalities for splines |
| title | On exact Bernstein-type inequalities for splines |
| title_alt | О точных неравенствах типа Бернштейна для сплайнов |
| title_full | On exact Bernstein-type inequalities for splines |
| title_fullStr | On exact Bernstein-type inequalities for splines |
| title_full_unstemmed | On exact Bernstein-type inequalities for splines |
| title_short | On exact Bernstein-type inequalities for splines |
| title_sort | on exact bernstein-type inequalities for splines |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3538 |
| work_keys_str_mv | AT kofanovva onexactbernsteintypeinequalitiesforsplines AT kofanovva onexactbernsteintypeinequalitiesforsplines AT kofanovva onexactbernsteintypeinequalitiesforsplines AT kofanovva otočnyhneravenstvahtipabernštejnadlâsplajnov AT kofanovva otočnyhneravenstvahtipabernštejnadlâsplajnov AT kofanovva otočnyhneravenstvahtipabernštejnadlâsplajnov |