On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives
New sharp inequalities of the Kolmogorov type are established, in particular, the following sharp inequality for $2 \pi$-periodic functions $x \in L^r_{\infty}(T):$ $$||x^{(k)}||_l \leq \left(\frac{\nu(x')}{2} \right)^{\left(1 - \frac1p \right)\alpha} \frac{||\varphi_{r-k}||_l}{||\varphi_r...
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| Date: | 2008 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2008
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3278 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509338288783360 |
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| author | Kofanov, V. A. Miropol'skii, V. E. Кофанов, В. А. Миропольский, В. Е. Кофанов, В. А. Миропольский, В. Е. |
| author_facet | Kofanov, V. A. Miropol'skii, V. E. Кофанов, В. А. Миропольский, В. Е. Кофанов, В. А. Миропольский, В. Е. |
| author_sort | Kofanov, V. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:49:49Z |
| description | New sharp inequalities of the Kolmogorov type are established, in particular, the following sharp inequality for $2 \pi$-periodic functions $x \in L^r_{\infty}(T):$
$$||x^{(k)}||_l \leq \left(\frac{\nu(x')}{2} \right)^{\left(1 - \frac1p \right)\alpha}
\frac{||\varphi_{r-k}||_l}{||\varphi_r||^{\alpha}_p} ||x||^{\alpha}_p \left|\left|x^{(r)}\right|\right|^{1-\alpha}_{\infty},$$
$k,\;r \in \mathbb{N},\quad k < r, \quad r \geq 3,\quad p \in [1, \infty],\quad \alpha = (r-k) / (r - 1 + 1/p), \quad \varphi_r$
is the perfect Euler spline of order $r,\quad \nu(x')$ is the number of sign changes of the derivative $x'$ on a period. |
| first_indexed | 2026-03-24T02:39:31Z |
| format | Article |
| fulltext |
UDK 517.5
V. A. Kofanov, V. E. Myropol\skyj (Dnepropetr. nac. un-t)
O TOÇNÁX NERAVENSTVAX TYPA KOLMOHOROVA,
UÇYTÁVAGWYX ÇYSLO PEREMEN
ZNAKA PROYZVODNÁX
New sharp inequalities of the Kolmogorov type are established, in particular, the following sharp
inequality for 2π-periodic functions x L Tr∈ ∞( ):
x k( )
1
≤
ν ϕ
ϕ
α
α
α α( ) – – ( ) –′( )
∞
x
x xp r k
r p
p
r
2
1
1
1 1
,
where k, r N∈ , k < r, r ≥ 3, p ∈ ∞[ ]1, , α = ( – )r k / ( – / )r p1 1+ , ϕr is the perfect Euler spline
of order r, ν( )′x is the number of sign changes of the derivative ′x on a period.
Otrymano novi toçni nerivnosti typu Kolmohorova, zokrema toçnu nerivnist\ dlq 2 π-periodyç-
nyx funkcij x L Tr∈ ∞( ):
x k( )
1
≤
ν ϕ
ϕ
α
α
α α( ) – – ( ) –′( )
∞
x
x xp r k
r p
p
r
2
1
1
1 1
,
de k, r N∈ , k < r, r ≥ 3, p ∈ ∞[ ]1, , α = ( – )r k / ( – / )r p1 1+ , ϕr — ideal\nyj splajn Ejlera
porqdku r, ν( )′x — çyslo zmin znaku ′x na periodi.
1. Vvedenye. Pust\ G — koneçn¥j otrezok I yly edynyçnaq okruΩnost\ T,
realyzovannaq kak otrezok 0 2, π[ ] s otoΩdestvlenn¥my koncamy. Budem ras-
smatryvat\ prostranstva L Gp( ), 1 ≤ p ≤ ∞, yzmerym¥x funkcyj x : G → R ta-
kyx, çto x L Gp ( ) < ∞, hde
x L Gp ( ) : =
G
p
p
t G
x t dt p
x t p
∫
≤ < ∞
= ∞
∈
( ) , ,
sup ( ) , .
/1
1esly
eslyvrai
Dlq s ∈ ∞[ ]1, y r N∈ oboznaçym çerez L Gs
r ( ) mnoΩestvo funkcyj x :
G → R takyx, çto x r( – )1 x x( )0 =( ) lokal\no absolgtno neprer¥vna y x r( ) ∈
∈ L Gs( ) . Symvolom ϕr t( ) , t R∈ , oboznaçym r-j 2π-peryodyçeskyj yntehral
so srednym znaçenyem na peryode, ravn¥m nulg ot funkcyy ϕ0( )t = sgnsin t , y
poloΩym g tr( ) : = 4 1– ϕr t– ( )1 .
V sluçae 2π-peryodyçeskyx funkcyj vmesto Lp 0 2, π[ ], x Lp 0 2, π[ ] y L Ts
r( )
budem pysat\ Lp , x p y L s
r
. PoloΩym W r
∞ : = x L Tr∈{ ∞( ): x r( )
∞
≤ }1 .
V nastoqwej stat\e budem yzuçat\ neravenstva dlq norm promeΩutoçn¥x
proyzvodn¥x funkcyj x Ls
r∈ vyda
x k
q
( ) ≤ C x xp
r
s
α α( ) –1
, (1)
a takΩe yx analohy, uçyt¥vagwye çyslo peremen znaka proyzvodn¥x.
Kak yzvestno [1], neravenstva typa Kolmohorova (1) dlq funkcyj x Ls
r∈ , k,
© V. A. KOFANOV, V. E. MYROPOL|SKYJ, 2008
1642 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
O TOÇNÁX NERAVENSTVAX TYPA KOLMOHOROVA, UÇYTÁVAGWYX … 1643
r N∈ , k < r, q, p, s ∈ ∞[ ]1, , α ∈( , )0 1 , ymegt mesto tohda y tol\ko tohda, kohda
α ≤ αcr : = min – ,
– / – /
/ – /
1
1 1
1 1
k
r
r k q s
r p s
+
+
. (2)
Osob¥j ynteres predstavlqgt neravenstva typa (1) s neuluçßaemoj kon-
stantoj C. Sredy neuluçßaem¥x neravenstv naybolee vaΩn¥ neravenstva (1) s
α = αcr , tak kak yz neuluçßaemoho neravenstva typa (1) s α = αcr , kak pravy-
lo, netrudno poluçyt\ neravenstvo s proyzvol\n¥m α < αcr y toçnoj konstan-
toj C.
Dlq summyruemoj 2π-peryodyçeskoj funkcyy symvolom ν( )x budem obo-
znaçat\ çyslo suwestvenn¥x peremen znaka x na peryode [2, s. 80]. V sylu
rezul\tata B. E. Kloca [1] neravenstva vyda (1) s α > αcr nevozmoΩn¥. Tem ne
menee A. A. Lyhun pokazal [3], çto esly neravenstvo vyda (1) vydoyzmenyt\ tak,
çtob¥ v nem b¥lo uçteno çyslo peremen znaka proyzvodn¥x funkcyy, to voz-
moΩn¥ neravenstva typa Kolmohorova s α > αcr . V sylu rezul\tata A. A. Ly-
huna dlq lgb¥x k, r N∈ , k < r, p ∈ ∞[ ]1, y x L r∈ 1 ymeet mesto neravenstvo
x k( )
1
≤
ν α
α
α α( ) – / – ( ) –′
( )x g
g
x x
p r k
r p
p
r
2
1 1
1
1
1
, (3)
hde α = (r – k) / (r – 1 + 1 / p). V [3] pryveden rqd pryloΩenyj neravenstva (3) v
teoryy approksymacyy.
V [4, 5] poluçen rqd neravenstv vyda
x k
q
( ) ≤ M x x x
i
m
i
p
r
s
i
=
∏ ( )( )
1
1
ν
α α α( ) ( ) –
, k, r N∈ , k < r,
hde αi ≥ 0 , α ∈ (0, 1) dlq funkcyj x L r∈ ∞ (v sluçae q = 1, p = s = ∞, m = r)
y dlq funkcyj x L r∈ +
1
1
(v sluçaqx q = 1, p ∈ ∞[ ]1, , s = ∞, r / 2 < k < r; q = 2,
p = s = ∞; q = 2, p ∈ 1, ∞[ ], s = ∞, m = r + 1).
V dannoj stat\e s pomow\g teorem¥ sravnenyq Σ-perestanovok Kornejçuka
Φ( , )x t dokazano neravenstvo
Φ ′ ⋅( )x q, ≤ 2
2
1 1
1 1 1 1– /
– / – ( ) –( )q
p r q
r p
p
rx x xν ϕ
ϕ
α
α
α α′
( )
∞
, x L r∈ ∞, (4)
hde r N∈ , r ≥ 3, q, p ∈ ∞[ ]1, , α = (r – 2 + 1 / q) / (r – 1 + 1 / p) (teoremaJ1). Yz teo-
rem¥J1 v¥vedeno sledugwee neravenstvo typa Kolmohorova, uçyt¥vagwee çys-
lo peremen znaka proyzvodn¥x:
x k( )
1
≤ ν ϕ
ϕ
α
α
α α( ) – / – ( ) ( – )′
( )
∞
x x x
p r k
r p
p
r
2
1 1
1 1
, k, r N∈ , k < r, r ≥ 3, (5)
hde α = (r – k) / (r – 1 + 1 / p) (teoremaJ2). Otmetym, çto v pravoj çasty neraven-
stva (5) v otlyçye ot analohyçnoho neravenstva yz rabot¥ [4] ne soderΩytsq
mnoΩytel\ ν x r( )+( )[ 1
/ 2
1] – α
. Ewe odno neravenstvo takoho typa poluçeno dlq
sluçaq q = 2, p ∈ ∞[ ]1, , s = ∞, r / 2 < k < r (teoremaJ3).
Otmetym, çto pry p = 1 neravenstvo (5) prynymaet vyd
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
1644 V. A. KOFANOV, V. E. MYROPOL|SKYJ
x k( )
1
≤
ϕ
ϕ
r k
r
k
r
k
r r
k
rx x
–
–
–
( )1
1
1
1
1
∞
. (6)
∏to neravenstvo b¥lo poluçeno v [6].
2. Vspomohatel\n¥e svedenyq. Dlq funkcyy x L a b∈ [ ]1 , y y > 0 polo-
Ωym
m x y( , ) : = mes t t a b x t y: , , ( )∈[ ] >{ }.
Budem oboznaçat\ çerez r x t( , ) perestanovku funkcyy x t( ) [6] (§ 6.1), t.Je.
r x t( , ) : = inf y m x y t: ( , ) ≤{ }, t b a∈[ ]0, – .
Yzvestno [7], çto
mes t t b a r x t y: , – , ( , )∈[ ] >{ }0 = m x y( , ) .
Dlq lgboj 2π-peryodyçeskoj funkcyy x L∈ 1 symvolom r x t( , ) budem
oboznaçat\ perestanovku suΩenyq x na 0 2, π[ ], a symvolom r tr( , ),ϕλ — pe-
restanovku suΩenyq ϕλ,r na 0 2, /π λ[ ]. Dlq x L∈ 1 poloΩym r x t( , ) = 0, es-
ly t ≥ 2π, y r tr( , ),ϕλ = 0 dlq t ≥ 2π / λ.
Pust\ D — mnoΩestvo vsex 2π-peryodyçeskyx funkcyj x yz L1, kotor¥e
ymegt odnostoronnye predel¥ v kaΩdoj toçke, a D1
— mnoΩestvo vsex 2π -pe-
ryodyçeskyx funkcyj x D∈ takyx, çto
0
2π
∫ x t dt( ) = 0. Dlq funkcyj x D∈ 1
Σ-perestanovka Kornejçuka Φ( , )x ⋅ opredelqetsq sledugwym obrazom [7,
s. 144]. Funkcyg g t( ), t R∈ , budem naz¥vat\ prostoj, esly ona opredelena na
otrezke a b,[ ], kotor¥j naz¥vaetsq osnovn¥m dlq funkcyy g t( ), y uravnenye
g t( ) = y ymeet rovno dva kornq dlq kaΩdoho y ∈ 0, ( )g L∞( )R . N. P. Kornejçuk
[7] dokazal, çto kaΩdaq funkcyq x D∈ 1
moΩet b¥t\ predstavlena v vyde
x t( ) =
k
kx t d∑ +( ) , t ∈ t t0 0 2, +[ ]π ,
hde
x t( )0 = min ( )
t
x t , d = x t( )0 ,
y x tk ( ) — prost¥e funkcyy, kotor¥e otlyçagtsq ot funkcyy x t( ) postoqn-
n¥my na kaΩdom yntervale monotonnosty funkcyy x . Dlq lgboj funkcyy
x D∈ 1
poloΩym
Φ( , )x t : =
k
kr x t d∑ +( , ) , t ∈ 0 2, π[ ],
y pust\ Φ( , )x t = 0 dlq t ≥ 2π. V [8, s. 14] b¥lo pokazano, çto Σ-perestanovka
moΩet b¥t\ opredelena dlq funkcyy x L∈ 1.
Çerez Φλ λϕ( , ),r ⋅ budem oboznaçat\ Σ-perestanovku ϕλ,r na a[ , a + 2π λ/ ],
hde a — nul\ ϕλ,r . Yzvestn¥ sledugwye svojstva Φ( , )x ⋅ [7, s. 144]:
Φ( , )x ⋅ 1 = x 1, (7)
2 0Φ( , )x – 2 min ( )
t
x t = ′x 1 = V
0
2π
x , (8)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
O TOÇNÁX NERAVENSTVAX TYPA KOLMOHOROVA, UÇYTÁVAGWYX … 1645
hde V0
2πx — varyacyq x na 0 2, π[ ].
Otmetym, çto dlq lgboj funkcyy x L∈ 1
1
[3]
0
t
x u du∫ Φ( , ) ≤
0
t x
r x u du
ν( )
( , )
′
∫ . (9)
Nam ponadobytsq sledugwaq teorema.
Teorema A [6]. Pust\ r N∈ , r ≥ 3. Esly funkcyq x W r∈ ∞ ymeet nuly y
λ v¥brano tak, çto
′x 1 ≤ λ ϕλ π λ, – , /r L1 0 21[ ]
, (10)
to poçty vsgdu na 0, /π λ[ ]
′Φ ( , )x t ≤ λ ϕλ λ′ ( )Φ , ,r t .
Bolee toho, esly
′x 1 = λ ϕλ π λ, – , /r L1 0 21[ ]
,
to dlq vsex t ∈ 0, /π λ[ ]
Φ( , )x t ≥ λ ϕλ λΦ ( , ),r t .
Esly Ωe λ v¥brat\ yz uslovyq
x 1 ≤ λ ϕλ π λ, , /r L1 0 2[ ]
,
to ymeet mesto (10) y v¥polneno neravenstvo
0
t
x u du∫ Φ( , ) ≤ λ ϕλ λ
0
t
r u du∫ Φ ( , ), , t > 0.
3. Toçn¥e neravenstva typa Kolmohorova.
Teorema 1. Pust\ r N∈ , r ≥ 3, x Lr∈ ∞. Tohda dlq lgb¥x p, q ∈ 1, ∞[ ]
v¥polnqetsq neravenstvo
Φ( , )′ ⋅x q ≤ 2
2
1 1
1 1 1 1– /
– / – ( ) –( )q
p r q
r p
p
rx x xν ϕ
ϕ
α
α
α α′
( )
∞
, (11)
hde α = (r – 2 + 1 / q) / (r – 1 + 1 / p).
Dokazatel\stvo. Zafyksyruem funkcyg x Lr∈ ∞. Bez potery obwnosty
moΩno sçytat\, çto x ymeet nuly. V sylu odnorodnosty neravenstva (11) mo-
Ωem predpoloΩyt\, çto
x r( )
∞
= 1. (12)
Tohda x W r∈ ∞ . V¥berem λ > 0, udovletvorqgwee uslovyg
′x 1 = λ ϕλ π λ, – ; /r L1 0 21[ ]
. (13)
Otsgda v sylu teorem¥ A sleduet, çto
Φ( , )x t ≥ λ ϕλ λΦ ( , ),r t , t ∈ 0, /π λ[ ],
y
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
1646 V. A. KOFANOV, V. E. MYROPOL|SKYJ
0
t
x u du∫ ′Φ( , ) ≤ λ ϕλ λ
0
1
t
r u du∫ Φ ( , ), – . (14)
Tohda dlq vsex t ∈ 0, /π λ[ ] v¥polneno neravenstvo
0
t
x u du∫ Φ( , ) ≥ λ ϕλ λ
0
t
r u du∫ Φ ( , ), . (15)
PokaΩem, çto dlq t ∈ 0, /π λ[ ]
λ ϕλ λ
0
t
r u du∫ Φ ( , ), = λ ϕ
λ
– ( , )r
t
rr u du
0
2
∫ . (16)
Oboznaçym çerez ϕλ, ( )r t suΩenye ϕλ, ( )r t na 0, /π λ[ ]. Sohlasno opredelenyg
Σ-perestanovky
Φλ λϕ( , ),r t = 2r trϕλ, ,( ) , t ∈
0; π
λ
. (17)
Qsno, çto r trϕλ, ,( ) = r trϕλ, , 2( ) . Poπtomu
0
t
r u du∫ Φλ λϕ( , ), = 2
0
t
rr u du∫ ( , ),ϕλ = 2 2
0
λ ϕ λ– ( ) ,r
t
rr u du∫ ⋅( )( ) =
= 2 2
0
λ ϕ λ– ,r
t
rr u du∫ ( ) = λ ϕ
λ
–( – ) ,r
t
rr u du1
0
2
∫ ( ) .
Otsgda sleduet (16). Yz (9), (15) y (16) poluçaem
0
t x
r x u du
ν( )
,
′
∫ ( ) ≥ λ ϕ
λ
– ,r
t
rr u du
0
2
∫ ( ) , t ∈
0; π
λ
. (18)
Polahaq m = ν( )′x , ξ = t m, neravenstvo (18) zapys¥vaem v vyde
0
ξ
∫ ( )r x u du, ≥ λ ϕ
λξ
–
/
,r
m
rr u du
0
2
∫ ( ) , ξ π
λ
∈
0,
m
. (19)
PokaΩem, çto
λ ϕ
λξ
–
/
,r
m
rr u du
0
2
∫ ( ) =
2 21
0
λ ϕ λ
ξ–( – )
( ) ,
r
rm
r
m
u du∫ ⋅
. (20)
Dejstvytel\no,
0
2λξ
ϕ
/
,
m
rr u du∫ ( ) =
0
2 2
ξ
ϕ λ λ∫
r
m
u
m
dur , =
2 2
0
λ ϕ λ
ξ
m
r
m
u dur∫ ⋅
( ) , ,
çto ravnosyl\no (20). Yz (19) y (20) sleduet, çto dlq lgboho ξ ∈ 0,
mπ
λ
v¥-
polnqetsq neravenstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
O TOÇNÁX NERAVENSTVAX TYPA KOLMOHOROVA, UÇYTÁVAGWYX … 1647
0
ξ
∫ ( )r x u du, ≥
2 21
0
λ ϕ λ
ξ–( – )
( ) ,
r
rm
r
m
u du∫ ⋅
. (21)
Poskol\ku r
m
urϕ λ2 ( ) ,⋅
= 0 dlq t ≥
mπ
λ
, neravenstvo (21) ymeet mesto dlq
vsex ξ > 0. Otsgda v sylu teorem¥ Xardy – Lyttlvuda (sm. predloΩenye 1.3.10
yz [9]) poluçaem
x p ≥ ψ π λr L mp 0, /[ ] , (22)
hde ψr t( ) = 2 1λ–( – )r
m
ϕ λ
r m
t2
, t ∈ 0,
mπ
λ
.
V¥çyslym normu funkcyy ψr v L
m
p 0,
π
λ
:
ψ π λr L mp 0, /[ ] =
2 21
0
1
λ ϕ λ
π λ−
∫
( – ) / /
r m
r
p
p
m m
t dt =
=
2
2
1
0
2 1
λ
λ
ϕ
π−
∫
( – )
/
( )
r
r
p
p
m
m u du = 2 1 1
1 1
m
p
r p
r p
– /
–( – )– /λ ϕ .
Tak kak m = ν( )′x , yz (22) ymeem
x p ≥ 2
1 1
1 1
ν
λ ϕ
( )
– /
–( – )– /
′
x
p
r p
r p . (23)
S druhoj storon¥, yz (14) v sylu teorem¥ Xardy – Lyttlvuda sleduet neravens-
tvo
Φ( , )′ ⋅x q ≤ λ ϕλ λ π λ
Φ ( , ), – , /r Lq
1 0
⋅
[ ]
, q ≥ 1. (24)
Obæedynqq (17) y oçevydnoe neravenstvo
ϕλ π λ, , /r Lq 0 2[ ]
= λ ϕ
– –r q
r q
1
,
poluçaem
Φλ λ π λ
ϕ , – , /
,r L
q
q
1 0
⋅( ) [ ]
= 2 1 0
q
r L
q
r
q
ϕλ π λ, – , /
, ⋅( ) [ ]
=
= 2 1
1 0 2
q
r L
q
q
–
, – , /
ϕλ π λ[ ]
= 2 1 1 1
1
q r q
r q
q– –( – ) –
–λ ϕ ,
y, sledovatel\no,
Φλ λ π λ
ϕ , – , /
,r Lq
1 0
⋅( ) [ ]
= 21 1 1 1
1
– / –( – )– /
–
q r q
r q
λ ϕ . (25)
Teper\ yz (23) – (25) poluçaem
Φ ′ ⋅( )x
x
q
p
,
α ≤
2
2
1 1 2 1
1
1 1 1 1 1 1
– / –( – / )
–
– / – – / / –( )
q r q
r q
p r p p
r px
λ ϕ
λ ν ϕ
α
+
′[ ]( )
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
1648 V. A. KOFANOV, V. E. MYROPOL|SKYJ
=
21 1 2 1 1 1 1– / – / –( – / )– ( – – / )
/ –
–
( )
α α α
α α α
λ
ν
ϕ
ϕ
+ +
′[ ]
p q r q r p
p
r q
r px
.
Uçyt¥vaq ravenstvo α = (r – 2 + 1 / q) / (r – 1 + 1 / p), ymeem
Φ ′ ⋅( )x
x
q
p
,
α ≤ 2
2
1 1
1 1 1– /
– / –( )q
p r q
r p
xν ϕ
ϕ
α
α
′
( )
ˆ.
Poslednee neravenstvo v sylu (12) ravnosyl\no (11).
Teorema dokazana.
Teorema 2. Pust\ k , r N∈ , k < r, r ≥ 3, p , q ∈ ∞[ ]1, . Tohda dlq lgboj
funkcyy x Lr∈ ∞ v¥polnqetsq neravenstvo
x k( )
1
≤
ν ϕ
ϕ
α
α
α α( ) – / – ( ) –′
( )
∞
x
x x
p r k
r p
p
r
2
1 1
1 1
, (26)
hde α = (r – k) / (r – 1 + 1 / p). Neravenstvo (26) qvlqetsq neuluçßaem¥m y ob-
rawaetsq v ravenstvo dlq funkcyj vyda x t( ) = a t bn rϕ , ( )+ , a, b R∈ , n N∈ .
Dokazatel\stvo. Zafyksyruem funkcyg x Lr∈ ∞. Bez potery obwnosty
moΩem sçytat\, çto funkcyq x ymeet nuly. Snaçala dokaΩem (26) pry k = 1.
Pry q = 1 yz teorem¥J1 v sylu (7) sleduet neravenstvo
′x 1 ≤
ν ϕ
ϕ
α
α
α α( ) – / – ( ) –′
( )
∞
x
x x
p r
r p
p
r
2
1 1 1 1 11
1
1 1
, (27)
hde α1 = (r – 1) / (r – 1 + 1 / p).
Pust\ teper\ k > 1. Dlq funkcyy ′ ∈ ∞x Lr –1
yz (6) sleduet neravenstvo
x k( )
1
≤
ϕ
ϕ
r k
r
k
r
k
r r
k
rx x
–
–
–
–
–
–
–
– ( )
–
–1
1 1
1
1
1
1
1
1
1
1
1′
∞
. (28)
Yspol\zuq (27) dlq ocenky ′x 1 v pravoj çasty (28), poluçaem
x k( )
1
≤
ϕ
ϕ
ν ϕ
ϕ
α
α
α αr k
r
k
r
p r
r p
p
r
k
r
r
k
rx
x x x
–
–
–
–
–
– – ( ) –
–
–
–
( )
–
–( )1
1 1
1
1
1
1 1
1 1 1
1
1
1 1
1
2
1
1 1′
∞ ∞
=
=
ν ϕ
ϕ
α
α
α α( ) – –
–
– –
–
–
–
–
–
– ( ) –
–
–
( – )
–
–
′
∞
+x
x xp
k
r r
r p
k
r
p
k
r r
k
r
k
r
2
1 1 1
1
1 1 1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1 ,
hde α1 = (r – 1) / (r – 1 + 1 / p).
Poskol\ku 1( – (k – 1) / (r – 1 1))α = (r – k) / (r – 1 + 1 / p), otsgda sleduet nera-
venstvo (26). Eho toçnost\ lehko proveryt\ s pomow\g oçevydnoho ravenstva
ϕn r p, = n r
r p
– ϕ .
Teorema dokazana.
Teorema 3. Pust\ k, r N∈ , r / 2 < k < r, r ≥ 3, p ∈ ∞[ ]1, . Tohda dlq lgboj
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
O TOÇNÁX NERAVENSTVAX TYPA KOLMOHOROVA, UÇYTÁVAGWYX … 1649
funkcyy x Lr∈ ∞
x k( )
2
≤
ν ϕ
ϕ
α
α
α α( ) – / – ( ) –′
( )
∞
x
x x
p r k
r p
p
r
2
1 1
2 1
, (29)
hde α = (r – k) / (r – 1 + 1 / p). Neravenstvo (29) obrawaetsq v ravenstvo dlq
funkcyj vyda x t( ) = a t bn rϕ , ( )+ , a, b R∈ , n N∈ .
Dokazatel\stvo. Yntehryruq po çastqm, ymeem
x k( )
2
2
=
0
2π
∫ x t x t dtk k( ) ( )( ) ( ) =
0
2
2
π
∫ x t x t dtr k r( ) ( – )( ) ( ) ≤ x xk r r( – ) ( )2
1 ∞
.
Ocenyvaq x k r( – )2
1
s pomow\g (26), poluçaem ocenku
x k( )
2
2
≤
≤
ν ϕ
ϕ
( )
( – )
– /
– ( – )
( – )/( – / )
– / ( )
– – /
– / ( )′
+
+
( − )
+
∞
+
+
∞
x
x x x
r k
r p p r k
r p
r k r p p
r k
r p r
k r p
r p r
2
2
1 1
1 1
2 1
2 1 1
2
1 1
2 1 1
1 1 ,
yz kotoroj sleduet (29) v sylu ravenstva ϕ2 1( – )r k = ϕr k– 2
2
. Toçnost\ nera-
venstva (29) oçevydna.
Teorema dokazana.
1. Kloc B. E. PryblyΩenye dyfferencyruem¥x funkcyj funkcyqmy bol\ßej hladkosty //
Mat. zametky. – 1977. – 21, # 1. – S. 21 – 32.
2. Kornejçuk N. P. Toçn¥e konstant¥ v teoryy pryblyΩenyq. – M.: Nauka, 1987. – 424 s.
3. Lyhun A. A. O neravenstvax meΩdu normamy proyzvodn¥x peryodyçeskyx funkcyj // Mat.
zametky. – 1983. – 33, # 3. – S. 385 – 391.
4. Babenko V. F., Kofanov V. A., Pyçuhov S. A. O toçn¥x neravenstvax typa Kolmohorova, uçy-
t¥vagwyx çyslo peremen znaka proyzvodn¥x // Dop. NAN Ukra]ny. – 1998. – # 8. – S. 12 –
16.
5. Babenko V. F., Kofanov V. A., Pyçuhov S. A. O nekotor¥x toçn¥x neravenstvax typa Kolmo-
horova, uçyt¥vagwyx çyslo peremen znaka proyzvodn¥x // Vestn. Dnepropetr. nac. un-ta.J–
2004. – # 11. – S. 3 – 8.
6. Kofanov V. A. Exact inequalities of Kolmogorov type and comparison of Korneichuk’s Σ-rearran-
gements // East J. Approxim. – 2003. – 9, # 1. – P. 67 – 94.
7. Kornejçuk N. P. ∏kstremal\n¥e zadaçy teoryy pryblyΩenyq. – Kyev: Nauk. dumka, 1976.
8. Lyhun A. A., Kapustqn V. E., Volkov G. Y. Specyal\n¥e vopros¥ teoryy pryblyΩenyq y op-
tymal\noho upravlenyq raspredelenn¥my systemamy. – Kyev: Vywa ßk., 1990.
9. Kornejçuk N. P., Babenko V. F., Lyhun A. A. ∏kstremal\n¥e svojstva polynomov y splajnov.
– Kyev: Nauk. dumka, 1992. – 304 s.
Poluçeno 15.10.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
|
| id | umjimathkievua-article-3278 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:39:31Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/15/0adc37e5007702cf8f7a460de1f0a515.pdf |
| spelling | umjimathkievua-article-32782020-03-18T19:49:49Z On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives O точных неравенствах типа Колмогорова, учитывающих число перемен знака производных Kofanov, V. A. Miropol'skii, V. E. Кофанов, В. А. Миропольский, В. Е. Кофанов, В. А. Миропольский, В. Е. New sharp inequalities of the Kolmogorov type are established, in particular, the following sharp inequality for $2 \pi$-periodic functions $x \in L^r_{\infty}(T):$ $$||x^{(k)}||_l \leq \left(\frac{\nu(x')}{2} \right)^{\left(1 - \frac1p \right)\alpha} \frac{||\varphi_{r-k}||_l}{||\varphi_r||^{\alpha}_p} ||x||^{\alpha}_p \left|\left|x^{(r)}\right|\right|^{1-\alpha}_{\infty},$$ $k,\;r \in \mathbb{N},\quad k < r, \quad r \geq 3,\quad p \in [1, \infty],\quad \alpha = (r-k) / (r - 1 + 1/p), \quad \varphi_r$ is the perfect Euler spline of order $r,\quad \nu(x')$ is the number of sign changes of the derivative $x'$ on a period. Отримано нові точні нерівності типу Колмогорова, зокрема точну нерівність для $2 \pi$-періодичних функцій $x \in L^r_{\infty}(T):$ $$||x^{(k)}||_l \leq \left(\frac{\nu(x')}{2} \right)^{\left(1 - \frac1p \right)\alpha} \frac{||\varphi_{r-k}||_l}{||\varphi_r||^{\alpha}_p} ||x||^{\alpha}_p \left|\left|x^{(r)}\right|\right|^{1-\alpha}_{\infty},$$ де $k,\;r \in \mathbb{N},\quad k < r, \quad r \geq 3,\quad p \in [1, \infty],\quad \alpha = (r-k) / (r - 1 + 1/p), \quad \varphi_r$ — ідеальний сплайн Ейлера порядку $r,\quad \nu(x')$ — число змін знаку $x'$ на періоді. Institute of Mathematics, NAS of Ukraine 2008-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3278 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 12 (2008); 1642–1649 Український математичний журнал; Том 60 № 12 (2008); 1642–1649 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3278/3302 https://umj.imath.kiev.ua/index.php/umj/article/view/3278/3303 Copyright (c) 2008 Kofanov V. A.; Miropol'skii V. E. |
| spellingShingle | Kofanov, V. A. Miropol'skii, V. E. Кофанов, В. А. Миропольский, В. Е. Кофанов, В. А. Миропольский, В. Е. On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives |
| title | On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives |
| title_alt | O точных неравенствах типа Колмогорова, учитывающих число перемен знака производных |
| title_full | On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives |
| title_fullStr | On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives |
| title_full_unstemmed | On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives |
| title_short | On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives |
| title_sort | on sharp kolmogorov-type inequalities taking into account the number of sign changes of derivatives |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3278 |
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