Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials
We prove that $\max |p′(x)|$, where $p$ runs over the set of all algebraic polynomials of degree not higher than $n ≥ 3$ bounded in modulus by 1 on [−1, 1], is not lower than \( {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 -...
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| Дата: | 2009 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3053 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509075794558976 |
|---|---|
| author | Podvysotskaya, A. I. Подвысоцкая, А. И. Подвысоцкая, А. И. |
| author_facet | Podvysotskaya, A. I. Подвысоцкая, А. И. Подвысоцкая, А. И. |
| author_sort | Podvysotskaya, A. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:07Z |
| description | We prove that $\max |p′(x)|$, where $p$ runs over the set of all algebraic polynomials of degree not higher than $n ≥ 3$ bounded in modulus by 1 on [−1, 1], is not lower than \( {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} \) for all $x ∈ (−1, 1)$ such that \( \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} \). |
| first_indexed | 2026-03-24T02:35:20Z |
| 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. Y. Podv¥sockaq (Kyev nac. un-t ym. T. Íevçenko)
OCENKA SNYZU V NERAVENSTVE S. N. BERNÍTEJNA
DLQ PERVOJ PROYZVODNOJ
ALHEBRAYÇESKYX MNOHOÇLENOV
We prove that max ( )′p x , where p runs over all algebraic polynomials of degree not greater than
n ≥ 3 with modules bounded by 1 on −[ ]1 1, , is not less than (n – 1) / 1 2− x for all x ∈ (–1, 1)
such that x ∈ cos , cos
( )
/ 2 1
2 1
2 1
20
2 1 k
n
k
nk
n +
−
+
=
−[ ] π π∪ .
Dovedeno, wo max ( )′p x , de p probiha[ mnoΩynu vsix obmeΩenyx za modulem odynyceg na
−[ ]1 1, alhebra]çnyx polinomiv stepenq ne vywe n ≥ 3, ne menße (n – 1) / 1 2− x dlq vsix tyx
x ∈ (–1, 1), dlq qkyx x ∈ cos , cos
( )
/ 2 1
2 1
2 1
20
2 1 k
n
k
nk
n +
−
+
=
−[ ] π π∪ .
1. Vvedenye y osnovnoj rezul\tat. Pust\ Pn oboznaçaet mnoΩestvo vsex
alhebrayçeskyx mnohoçlenov s dejstvytel\n¥my koπffycyentamy stepeny ne
v¥ße n ∈N : = {1, 2, … }, C I( ) — klass vsex dejstvytel\noznaçn¥x neprer¥v-
n¥x funkcyj f na I : = −[ ]1 1, , f C I( ) : = max ( )x I f x∈ y
πn : =
p pn C I∈ ≤{ }P ( ) 1 , K p( ) : = x I p x∈ ={ }( ) 1 , p n∈π .
Esly p n∈π y card K(p) = n + 1, to suwestvugt takye znaky σ1, σ2 ∈ {+ 1,
– 1}, çto p x( ) = σ σ1 2T xn ( ) , hde card A oboznaçaet çyslo razlyçn¥x πlementov
nekotoroho koneçnoho mnoΩestva A ⊆ R1
, a T xn( ) : = cos n arccos x — mnoho-
çlen Çeb¥ßeva pervoho roda n-j stepeny.
Yzvestno [1, s. 99], çto vo mnohyx πkstremal\n¥x zadaçax na mnoΩestve πn
πkstremum dostyhaetsq na odnom yz mnohoçlenov p n∈π , dlq kotoroho n ≤
≤ card K(p) ≤ n + 1. Odnoj yz takyx zadaç qvlqetsq tesno svqzannaq s yzvestn¥m
neravenstvom Bernßtejna
′p x( ) ≤ n
x
p C I
1 2− ( ), x ∈ (– 1, 1), p n∈P , (1)
πkstremal\naq zadaça naxoΩdenyq
M xn( ) : = max ( )
p n
p x
∈
′
π
(2)
dlq proyzvol\noj fyksyrovannoj toçky x I∈ (sm. [2, s. 32], zadaça 1.1). ∏ta
zadaça b¥la reßena v 1892 h. V. A. Markov¥m [3] (sm. takΩe [2, s. 38]), kotor¥j
dokazal, çto M xn( ) = ′T xn( ) , esly x prynadleΩyt specyal\n¥m pod¥nterva-
lam α βj j,[ ]{ } yntervala I. Ym b¥lo dokazano takΩe, çto na ostavßyxsq
pod¥ntervalax I znaçenye M xn( ) dostyhaetsq na tak naz¥vaem¥x polynomax
Zolotareva, t.@e. na tex mnohoçlenax p n∈π \ πn−1, dlq kotor¥x card K(p) = n.
© A. Y. PODVÁSOCKAQ, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5 711
712 A. Y. PODVÁSOCKAQ
Neposredstvenno yz opredelenyq mnoΩestva πn y neravenstva Bernßtejna (1)
sledugt takye svojstva funkcyy M xn( ):
M xn( ) = M xn( )− , x I∈ ; 1 ≤ M xn( ) ≤ n
x1 2−
, x ∈ (– 1, 1). (3)
Yspol\zuq nov¥e svojstva mnohoçlenov Zolotareva, ustanovlenn¥e v [4], doka-
Ωem sledugwug ocenku snyzu funkcyy M xn( ) .
Teorema. Dlq proyzvol\noho n ≥ 3 ymeet mesto neravenstvo
M xn( ) ≥
n
x
−
−
1
1 2
, x ∈
k
n k
n
k
n=
[ ]− +
−
+
0
2 1 2 1
2 1
2 1
2
/
cos
( )
, cos∪ π π . (4)
2. Predvarytel\n¥e rezul\tat¥. Budem yspol\zovat\ oboznaçenyq dlq
mnohoçlenov Zolotareva, vvedenn¥e v [4, s. 154]. Dlq kaΩdoho natural\noho
n ≥ 2 y dejstvytel\noho çysla 1 ≤ b < ∞ suwestvuet edynstvenn¥j mnohoçlen
Zolotareva Z xn b, ( ) stepeny n so svojstvamy
′ −Z bn( ) = 0, Z bn( )− 1 = Z bn( )− 2 = 1,
(5)
b2 > b1 ≥ b ≥ 1, b1 = b1(b), b2 = b2(b),
dopuskagwyj analytyçeskoe predstavlenye (sm. [4], formula (11))
Z xn b, ( ) = cos ( )n xω( ) , x I∈ ,
(6)
ω( )x : = ω( , )x b : =
x
t b
t
dt
1
21∫ −
ρ( , )
, ρ( , )x b : =
( )
( ) ( )
b x
b b x b b x
+
+( ) +( )
2
1 2
.
Pry πtom Z xn, ( )1 = T xn
cos2
2
π
n
+ sin2
2
π
n
y v dopolnenye k formulam (6)
budem yspol\zovat\ oboznaçenye (sm. [4], formula (5))
Z xn b, ( ) : = T
x b
n
b
n
n
+
+
tg
tg
2
2
2
1
2
π
π , b ∈[ ]0 1, . (7)
V pryvedennom nyΩe dokazatel\stve teorem¥ budet yspol\zovano sledugwee
svojstvo funkcyy ρ( , )x b .
Lemma. Pust\ 1 < b < ∞ y funkcyq ρ( , )x b opredelena formuloj (6).
Tohda
ρ( , )x b ≥ 1 1 2
−
n
, x ∈[ ]0 1, . (8)
Dokazatel\stvo. MnoΩestvo πn qvlqetsq v¥pukl¥m y Zn b n, ∈π pry
lgbom b ≥ 0. Poπtomu dlq proyzvol\n¥x θ ∈[ ]0 1, , b ≥ 1 y ε < b
( ) ,1 − θ Zn b + θ εZn b, + = Zn b, + θ ε( ), ,Z Zn b n b+ − ∈ πn . (9)
PredpoloΩym, çto y I∈ y Z yn b, ( ) = σ ∈ {1, – 1}. Tohda yz (9) sleduet
σ εZ y Z yn b n b, ,( ) ( )+ −( ) ≤ 0, ε < b, b ≥ 1. (10)
V [5, s. 12 – 14] poluçeno v¥raΩenye mnohoçlenov Zn b, , b ≥ 1, çerez πllypty-
çeskye funkcyy, yz kotoroho sleduet, çto vse koπffycyent¥ mnohoçlena Zn b, ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
OCENKA SNYZU V NERAVENSTVE S. N. BERNÍTEJNA DLQ PERVOJ … 713
kak y funkcyy b1(b) y b 2 (b), qvlqgtsq neprer¥vno dyfferencyruem¥my
funkcyqmy parametra b. Poπtomu razdelyv neravenstvo (10) na ε y perexodq k
predelu pry ε → 0, poluçym
σ ∂
∂b
Z yn b, ( ) ≤ 0,
esly ε > 0, y
σ ∂
∂b
Z yn b, ( ) ≥ 0,
esly ε < 0, otkuda sleduet
y I∈ , Z yn b, ( ) = 1 ⇒ ∂
∂b
Z yn b, ( ) = 0. (11)
Sohlasno yzvestn¥m svojstvam mnohoçlena Zn b, pry b > 1 (sm. [4, s. 153])
Zn b, ( )±1 = 1 y Z yn b, ( ) = 1 dlq vsex (n – 2)-x kornej uravnenyq ′Z yn b, ( ) = 0,
kotor¥e raspoloΩen¥ v (– 1, 1), (n – 1)-j koren\ πtoho uravnenyq raven – b.
Poπtomu yz (11) sleduet, çto dlq kaΩdoho b > 1 suwestvuet takaq nenulevaq
dejstvytel\naq postoqnnaq λ( )b , çto
∂
∂b
Z xn b, ( ) = λ( ) ( ),b
x
x b
Z xn b
1 2−
+
′ , b > 1. (12)
Dlq naxoΩdenyq λ( )b podstavym predstavlenye (6) v (12):
−
−∫∂
∂
ρ
b
t b
t
dt
x
1
21
( , )
= λ
ρ
( )
( , )
b
x
x b
x b
x
1
1
2
2
−
+ −
= λ( )
( ) ( )
b
x
x b x b
1 2
1 2
−
+ +
,
y posle dyfferencyrovanyq po x budem ymet\
1
1 2− x b
x b∂
∂
ρ( , ) = λ ∂
∂
( )
( ) ( )
b
x
x
x b x b
1 2
1 2
−
+ +
. (13)
Poskol\ku ρ( , )x b = (x + b) / ( ) ( )x b x b+ +1 2 , oboznaçyv ′bk : = ′b bk ( ) , k =
= 1 2, , poluçym
1
1 2− x b
x b∂
∂
ρ( , ) =
x b
x x b x b x b
b
x b
b
x b
+
− + + +
− ′
+
− ′
+
2 1
2
2
1 2
1
1
2
2( ) ( ) ( )
,
∂
∂x
x
x b x b
1 2
1 2
−
+ +( ) ( )
= 1
2
1 1
1
1
1
1 12
1 2 1 2
−
+ + +
−
−
−
+
−
+
x
x b x b x x x b x b( ) ( )
y yz (13) najdem
( )x b
x b
b
x b
b
x b
+
+
− ′
+
− ′
+
2 1
1
2
2
=
= λ( ) ( )b x
x x x b x b
1
1
1
1
1
1 12
1 2
−
+
+
−
−
+
−
+
.
Otsgda s uçetom ravenstv
( )x
x x x b x b
2
1 2
1 1
1
1
1
1 1−
+
+
−
−
+
−
+
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
714 A. Y. PODVÁSOCKAQ
= ( )
( ) ( ) ( ) ( )
x
b
x x b
b
x x b
2 1
1
2
2
1
1
1
1
1
− −
+ +
+ +
− +
=
( )( )b x
x b
1
1
1 1− −
+
+
+
( )( )b x
x b
2
2
1 1+ +
+
= b1 + b2 –
( )( )1 11 1
1
+ −
+
b b
x b
+
( ) ( )b b
x b
2 2
2
1 1+ −
+
=
= b1 + b2 –
b
x b
1
2
1
1−
+
–
b
x b
2
2
2
1−
+
poluçym
2 – ′b1 – ′b2 +
b b
x b
b1
1
1
−
+
′ +
b b
x b
b2
2
2
−
+
′ = λ( )b
b
x b
b
x b
b b1
2
1
2
2
2
1 2
1 1−
+
+ −
+
− −
.
Takym obrazom, dolΩn¥ ymet\ mesto ravenstva
′b1 + ′b2 – 2 = λ( ) ( )b b b1 2+ ,
′ −b b b1 1( ) = λ( )b b1
2 1−( ),
′ −b b b2 2( ) = λ( )b b2
2 1−( ).
Podstavlqq v pervoe ravenstvo znaçenyq proyzvodn¥x yz vtoroho y tret\eho,
poluçaem
λ( ) ( )b b b1 2+ = λ( )b
b
b b
1
2
1
1−
−
+ λ( )b
b
b b
2
2
2
1−
−
– 2,
otkuda s uçetom neravenstv b2 > b1 > b > 1 (sm. (5)) naxodym
2
λ( )b
=
b
b b
1
2
1
1−
−
– b1 +
b
b b
2
2
2
1−
−
– b2 =
bb
b b
1
1
1−
−
+
bb
b b
2
2
1−
−
=
=
( )( ) ( )( )
( )( )
bb b b bb b b
b b b b
1 2 2 1
1 2
1 1− − + − −
− −
> 0,
t.@e.
λ( )b =
2
1 1
1 2
1 2 2 1
( )( )
( )( ) ( )( )
b b b b
bb b b bb b b
− −
− − + − −
> 0 ∀ >b 1. (14)
Teper\ zametym, çto znak pravoj çasty (13) pry x ∈[ ]0 1, sovpadaet so znakom
v¥raΩenyq
1
1x +
– 1
1 − x
– 1
1x b+
– 1
2x b+
= −
−
2
1 2
x
x
– 1
1x b+
– 1
2x b+
< 0,
poπtomu s uçetom (13) y (14) ymeem
∂
∂
ρ
b
x b( , ) < 0 ∀ >b 1, x ∈[ ]0 1, . (15)
No v [3, s. 65] dokazano, çto lim ,b n bZ→+∞ = Tn−1, a πto v sylu formul (6) ozna-
çaet, çto lim ( , )b x b→+∞ ρ = ( )n − 1 2
/ n2
pry kaΩdom x ∈ −[ ]1 1, . Poπtomu na os-
novanyy (15) poluçaem ρ( , )x b ≥ ( )n − 1 2
/ n2
dlq proyzvol\n¥x b > 1 y x ∈ [0,
1], çto y trebovalos\ dokazat\.
3. Dokazatel\stvo teorem¥. Pust\ n ≥ 2, 0 ≤ k ≤ n/2[ ] – 1, αk n( ) : =
: = cos
2 1
2
k
n
+
y βn : = tg2
2
π
n
. Poskol\ku T nn kα ( )( ) = 0, po opredelenyg (7)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
OCENKA SNYZU V NERAVENSTVE S. N. BERNÍTEJNA DLQ PERVOJ … 715
mnohoçlen Zn b, na promeΩutke cos
( )
2 1
2 1
k
n
+
−
π, cos
2 1
2
k
n
+
π ymeet rovno odyn
nul\ αk n( ) @( )1 + b nβ – b nβ , kotor¥j ub¥vaet ot αk n( ) do αk n( ) @( )1 + βn – βn ,
kohda b vozrastaet ot 0 do 1. Poπtomu dlq kaΩdoho x ∈ αk n( )[ ( )1 + βn – βn ,
αk n( )] suwestvuet takoe b ∈[ ]0 1, , çto koren\ αk n( ) @( )1 + b nβ – b nβ mnoho-
çlena Zn b, sovpadaet s x. No yz (7) sleduet, çto
′Z xn b, ( ) = n
x b n1
1
12− + β
,
otkuda ymeem
1 2− ′x
n
Z xn b, ( ) ≥ 1
1 + βn
= cos2
2
π
n
≥ 1 – π2
24n
≥ 1 – 1
n
,
çto dokaz¥vaet neravenstvo (4) dlq vsex x ∈ αk n( )[ ( )1 + βn – βn , αk n( )].
Rassmotrym sluçaj, kohda v (4) x ∈ αk n( )−[ 1 , αk n( ) ( )1 + βn – βn) .
Dlq proyzvol\noho b ≥ 1 oboznaçym nuly polynoma Zn b, çerez αr
n n( ), 0 ≤
≤ r ≤ n – 1, pryçem (sm. [4, s. 155]) αn
b n−1( ) < αn
b n−2( ) < …. < α1
b n( ) < α0
b n( ) < 1.
V [3, s. 65] dokazano, çto kohda b vozrastaet ot 1 do + ∞, αk
b n( ) ub¥vaet ot
αk n( ) (1 + βn) – βn do αk n( )− 1 , pryçem lim ,b n bZ→+∞ = Tn−1. Poπtomu dlq
proyzvol\noho x ∈ αk n( )−( 1 , αk n( ) ( )1 + βn – βn) suwestvuet takoe b > 1,
çto αk
b n( ) = x, a v πtoj toçke yz formul¥ (6) sleduet ravenstvo
′Z xn b, ( ) = n
x
x
1 2−
ρ( ) ,
yz kotoroho vsledstvye (8) v¥tekaet neravenstvo (4). Esly Ωe x = αk n( )− 1 , to
neravenstvo (4) neposredstvenno sleduet yz ravenstva ′−Tn 1 αk n(( – 1)) = (n –
– 1) / 1 1 2− −αk n( ) . Çetnost\ funkcyy M xn( ) (sm. (3)) zaverßaet dokaza-
tel\stvo teorem¥.
1. Tyxomyrov V. M. Nekotor¥e vopros¥ teoryy approksymacyy. – M.: Yzd-vo Mosk. un-ta,
1976. – 303 s.
2. Bojanov B. D. Markov-type inequalities for polynomials and splines // Approxim. Theory X:
Abstract and Classical Anaysis. – Vanderbilt Univ. Press, 2002. – P. 31 – 90.
3. Markov V. A. O funkcyqxæ, naymenæ uklonqgwyxsq otæ nulq væ dannom promeΩutkæ. –
Sankt Peterburhæ, 1892.
4. Avvakumova L. S. Comparison of integral functionals depending on the second derivative of Che-
byshev and Zolotarev polynomials // East J. Approxim. – 1999. – 5, # 2. – P. 151 – 182.
5. Zolotarev E. Y. Polnoe sobranye soçynenyj. T. 2. – L.: Yzd-vo AN SSSR, 1932. – T. 2. –
364 s.
Poluçeno 29.09.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
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| id | umjimathkievua-article-3053 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:35:20Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/21/22b3f2392a9db37b846f6dacdf529121.pdf |
| spelling | umjimathkievua-article-30532020-03-18T19:44:07Z Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials Оценка снизу в неравенстве С. Н. Вернштейна для первой производной алгебраических многочленов Podvysotskaya, A. I. Подвысоцкая, А. И. Подвысоцкая, А. И. We prove that $\max |p′(x)|$, where $p$ runs over the set of all algebraic polynomials of degree not higher than $n ≥ 3$ bounded in modulus by 1 on [−1, 1], is not lower than \( {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} \) for all $x ∈ (−1, 1)$ such that \( \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} \). Доведено, що max $\max |p′(x)|$, де $p$ пробігає множину всіх обмежених за модулем одиницею на [-1,1] алгебраїчних поліномів степеня не вище $n ≥ 3$, не менше \( {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} \) для всіх тих $x ∈ (−1, 1)$ для яких \( \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} \). Institute of Mathematics, NAS of Ukraine 2009-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3053 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 5 (2009); 711-715 Український математичний журнал; Том 61 № 5 (2009); 711-715 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3053/2855 https://umj.imath.kiev.ua/index.php/umj/article/view/3053/2856 Copyright (c) 2009 Podvysotskaya A. I. |
| spellingShingle | Podvysotskaya, A. I. Подвысоцкая, А. И. Подвысоцкая, А. И. Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials |
| title | Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials |
| title_alt | Оценка снизу в неравенстве С. Н. Вернштейна для первой производной алгебраических многочленов |
| title_full | Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials |
| title_fullStr | Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials |
| title_full_unstemmed | Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials |
| title_short | Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials |
| title_sort | lower bound in the bernstein inequality for the first derivative of algebraic polynomials |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3053 |
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