Inequalities of the Bernstein type for splines of defect 2
We obtain new exact inequalities of the Bernstein type for periodic polynomial splines of order r and defect 2.
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2009
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| author | Babenko, V. F. Parfinovych, N. V. Бабенко, В. Ф. Парфинович, Н. В. Бабенко, В. Ф. Парфинович, Н. В. |
| author_facet | Babenko, V. F. Parfinovych, N. V. Бабенко, В. Ф. Парфинович, Н. В. Бабенко, В. Ф. Парфинович, Н. В. |
| author_sort | Babenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:40Z |
| description | We obtain new exact inequalities of the Bernstein type for periodic polynomial splines of order r and defect 2. |
| first_indexed | 2026-03-24T02:35:46Z |
| format | Article |
| fulltext |
UDK 517.5
V. F. Babenko
(Dnepropetr. nac. un-t, Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck),
N. V. Parfynovyç (Dnepropetr. nac. un-t)
NERAVENSTVA TYPA BERNÍTEJNA
DLQ SPLAJNOV DEFEKTA 2
New exact inequalities of a Bernstein type for periodic polynomial splines of order r and defect 2 are
obtained.
Otrymano novi toçni nerivnosti typu Bernßtejna dlq periodyçnyx polinomial\nyx splajniv po-
rqdku r defektu 2.
Vo mnohyx sluçaqx dlq tryhonometryçeskyx polynomov y splajnov mynymal\-
noho defekta yzvestn¥ toçn¥e neravenstva typa Bernßtejna, kotor¥e yhragt
vaΩnug rol\ vo mnohyx voprosax teoryy pryblyΩenyq (obzor y yzloΩenye mno-
hyx yzvestn¥x toçn¥x neravenstv, a takΩe byblyohrafyg moΩno najty, napry-
mer, v [1, 2]). M¥ ustanovym nekotor¥e toçn¥e neravenstva typa Bernßtejna
dlq splajnov defekta 2.
Pust\ Lp , 1 ≤ p ≤ ∞, — prostranstva 2π-peryodyçeskyx funkcyj f : R →
→ R s sootvetstvugwymy normamy ⋅ Lp
= ⋅ p , C — prostranstvo neprer¥v-
n¥x 2π-peryodyçeskyx funkcyj.
Çerez S n r2
2
, , r = 1, 2, … , oboznaçym prostranstvo 2π-peryodyçeskyx poly-
nomyal\n¥x splajnov porqdka r defekta 2 s uzlamy v toçkax tk =
2k
n
π
,
k ∈Z , t.7e. mnoΩestvo 2π-peryodyçeskyx funkcyj s, udovletvorqgwyx sle-
dugwym uslovyqm:
1) s ymeet neprer¥vn¥e proyzvodn¥e do porqdka r – 2 vklgçytel\no;
2) dlq kaΩdoho k ∈Z najdetsq alhebrayçeskyj polynom p xk ( ) stepeny r
takoj, çto s x( ) = p xk ( ) dlq x ∈7 ( , )t tk k+1 .
Otmetym, çto s r( )−1
moΩet ymet\ razr¥v¥ (pervoho roda) v toçkax tk ,
k ∈Z , y v πtyx toçkax polahaem
s t s t s tr
k
r
k
r
k
( ) ( ) ( )( ) ( )− − −( ) = + + −
1 1 11
2
0 0 .
Nayluçßym pryblyΩenyem funkcyy f podprostranstvom konstant v prost-
ranstve Lp , 1 ≤ p ≤ ∞, naz¥vaetsq velyçyna
E f f cp
c p
( ) inf= −
∈R
.
Funkcyy Bernully opredelqgtsq sledugwym obrazom (sm. [3, s. 72]). B x1( ) —
2π-peryodyçeskaq funkcyq, kotoraq na 0 2, π[ ) zadaetsq tak:
B x1( ) =
π
π
π
−
∈
=
x
x
x
2
0 2
0 0
, ( , ),
, .
Esly r > 1, to B xr ( ) est\ (r – 1)-j 2π-peryodyçeskyj yntehral s nulev¥m
srednym znaçenyem na peryode ot B x1( ) .
© V. F. BABENKO, N. V. PARFYNOVYÇ, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7 995
996 V. F. BABENKO, N. V. PARFYNOVYÇ
PoloΩym ψn r x, ( ) = −
2π
n
B nx
r r ( ) (zametym, çto ψn r x, ( ) ∈ S n r2
2
, ).
DokaΩem sledugwye utverΩdenyq.
Teorema 1. Pust\ n , r ∈N , r ≥ 2. Dlq lgboho splajna s S n r∈ 2
2
, y lgbo-
ho j ∈Z ymegt mesto neravenstva
sup ( )( )
t t t
r
j j
s t
< < +1
≤
E s
E
t
n r t t t
n
j j
( )
( )
sup ( )
,
,
∞
∞ < < +
′
ψ
ψ
1
1 , (1)
sup ( )( )
t t t
r
j j
s t
< <
−
+1
1 ≤
E s
E
t
n r t t t
n
j j
( )
( )
sup ( )
,
,
∞
∞ < < +ψ
ψ
1
1 , (2)
ω s t tr
j j
( ), ( , )−
+( )1
1 ≤
E s
E
t t
n r
n j j
( )
( )
, ( , )
,
,
∞
∞
+( )
ψ
ω ψ 1 1 , (3)
hde ω( , )f M = sup ( ) ( )
,′ ′′∈
′ − ′′
t t M
f t f t — kolebanye funkcyy f na mnoΩestve
M ⊂ R .
Sledstvye 1. Pust\ n, r ∈N , r ≥ 2. Dlq lgboho s S n r∈ 2
2
,
V
0
2π
s r( )−
1 ≤
E s
E
t
n r
n
( )
( )
( )
,
,
∞
∞
[ ]
ψ
ψ
π
V
0
2
1 , (4)
s r( )−
∞
1 ≤
E s
E
t
n r
n
( )
( )
( )
,
,
∞
∞ ∞ψ
ψ 1 . (5)
Teorema 2. Pust\ n, r ∈N , r ≥ 2. Dlq lgboho s S n r∈ 2
2
, y lgboho j ∈Z
V
t
t
r
j
j
s
+
−
1
2( ) ≤
E s
E n r t
t
n
j
j
( )
( ),
,
∞
∞
+
[ ]
ψ
ψV
1
2 (6)
y, sledovatel\no,
V
0
2π
s r( )−
2 ≤
E s
E n r
n
( )
( ),
,
∞
∞
[ ]
ψ
ψ
π
V
2
0
2 . (7)
Teorema 3. Pust\ n , r ∈N , r ≥ 2. Dlq lgboho s S n r∈ 2
2
, 6 y lgboho p ∈
∈ [ ∞)1,
s r
p
( )−1 ≤
E s
E
t
n r
n p
( )
( )
( )
,
,
∞
∞ψ
ψ 1 . (8)
Zameçanye. Oçevydno, çto neravenstva (1) – (8) obrawagtsq v ravenstva
dlq funkcyy s n r= ψ , y, sledovatel\no, qvlqgtsq toçn¥my.
Dokazatel\stvo teorem¥ 1. Pust\ s S n r∈ 2
2
, . PoloΩym
ϕ
ψ
ψ( )
( )
( )
( )
,
,t
E s
E
t
n r
n r= ∞
∞
.
V sylu lynejnosty funkcyj ϕ( )r−1
y s r( )−1
na ( , )t tj j+1 , j = 0 1, n − , sootno-
ßenyq (1) y (3) budut sledovat\ yz sootnoßenyq (2). Poπtomu dlq dokaza-
tel\stva teorem¥ dostatoçno ustanovyt\ sootnoßenye (2), t.7e. dokazat\, çto
dlq lgboho j = 0 1, n −
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
NERAVENSTVA TYPA BERNÍTEJNA DLQ SPLAJNOV DEFEKTA 2 997
sup ( ) sup ( )( ) ( )
t t t
r
t t t
r
j j j j
s t t
< <
−
< <
−
+ +
≤
1 1
1 1ϕ . (9)
PredpoloΩym, çto najdetsq j0 takoe, çto na yntervale ( , )t tj j0 0 1+ sootnoße-
nye (9) ne v¥polnqetsq, t.7e. ymeet mesto po krajnej mere odno yz neravenstv
s t tr
j
r
j
( ) ( )( ) ( )− −+ >1 1
0 0
0 ϕ ,
s t tr
j
r
j
( ) ( )( ) ( )− −
+− >1 1
10 0
0 ϕ .
Pust\, dlq opredelennosty, s tr
j
( )( )− +1
0
0 > ϕ( )( )r
jt−1
0
. Tohda pry podxodq-
wem 0 < λ < 1 ymeem
λ ϕs t tr
j
r
j
( ) ( )( ) ( )− −+ =1 1
0 0
0 . (10)
PoloΩym δ ϕ( ) ( )t t= – α – λ βs t( ) −( ) , hde α y β — konstant¥ nayluçßeho
ravnomernoho pryblyΩenyq dlq funkcyj ϕ( )t y s t( ) sootvetstvenno.
Otmetym, çto δ( )r−1
na kaΩdom yz yntervalov ( , )t tj j+1 moΩet menqt\ znak
ne bolee odnoho raza, krome toho, peremena znaka u δ( )r−1
vozmoΩna pry pere-
xode arhumenta çerez toçku t j . V sylu (10) δ( )r−1
ne menqet znak na (t j0 ,
t j0 1+ ) y, znaçyt, δ( )r−1
ymeet na peryode ne bolee 2n – 1 peremen znaka. S
druhoj storon¥, tak kak E( )ϕ ∞ > E s( )λ ∞ , δ( )t ymeet po krajnej mere odnu
peremenu znaka meΩdu lgb¥my dvumq sosednymy toçkamy πkstremuma funkcyy
ϕ( )t – α. Znaçyt, δ( )t ymeet na peryode ne menee 2n peremen znaka. Tohda v
sylu teorem¥ Rollq u δ( )r−1
budet takΩe ne menee 2n peremen znaka. Polu-
çennoe protyvoreçye dokaz¥vaet, çto dlq lgboho j = 0 1, n −
sup ( ) sup ( )( ) ( )
t t t
r
t t t
r
j j j j
s t t
< <
−
< <
−
+ +
≤
1 1
1 1ϕ .
Takym obrazom, sootnoßenye (2) ustanovleno.
Teorema 1 dokazana.
V sylu lynejnosty funkcyj s r( )−1
y ψn,1 na kaΩdom yntervale ( , )t tj j+1
y
2π
n
-peryodyçnosty funkcyy ψn,1 yz sootnoßenyj (2) y (3) sledugt ut-
verΩdenyq (4) y (5).
Dokazatel\stvo teorem¥ 2. Pust\ snaçala promeΩutok t tj j, +[ ]1 takov,
çto s r( )−1
ymeet nul\ v ( , )t tj j+1 . Poskol\ku v sylu (5)
s r( )−
∞
1 ≤ ϕ( )r−
∞
1
y v sylu (1)
s tr( )( ) ≤ ϕ( )( )r t =
E s
E n r
( )
( ),
∞
∞ψ
, t ∈ ( , )t tj j+1 ,
dlq kaΩdoho x ≥ 0
mes t t t t xj j
r∈ ≥{ }+
−( , ) : ( )( )
1
1ϕ ≥ mes t t t s t xj j
r∈ ≥{ }+
−( , ) : ( )( )
1
1 . (11)
Yz (11) neposredstvenno sleduet, çto pry vsex p ∈ ∞[ )1,
s t dt t dtr p
t
t
r p
t
t
j
j
j
j
( ) ( )( ) ( )− −
+ +
∫ ∫≤1 1
1 1
ϕ (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
998 V. F. BABENKO, N. V. PARFYNOVYÇ
y, v çastnosty,
s t dt t dtr
t
t
r
t
t
j
j
j
j
( ) ( )( ) ( )− −
+ +
∫ ∫≤1 1
1 1
ϕ ,
tak çto
V V
t
t
r
t
t
r
j
j
j
j
s
+ +
− − ≤
1 1
2 2( ) ( )ϕ . (13)
Pust\ teper\ promeΩutok t tj j, +[ ]1 takov, çto s r( )−1
ne obrawaetsq v nul\
na yntervale ( , )t tj j+1 . V πtom sluçae s r
t tj j
( )
,
−
+
2
1
— monotonnaq funkcyq y,
sledovatel\no,
sup ( ) max ( ) , (( ) ( ) ( )
t t t
r r
j
r
j j
s t s t s
≤ ≤
− − −
+
=
1
2 2 2 tt j+{ }1) .
Pust\, dlq opredelennosty,
sup ( ) ( )( ) ( )
t t t
r r
j
j j
s t s t
≤ ≤
− −
+
=
1
2 2
.
Otmetym takΩe, çto
ϕ ϕ ϕ( ) ( ) ( )( ) ( )r r
j
r
jt t−
∞
− −
+= =2 2 2
1 .
Ustanovym neravenstvo
sup ( ) ( ) ( )( ) ( ) ( )
t t t
r r
j
r
j
j j
s t s t t
≤ ≤
− − −
+
= ≤
1
2 2 2ϕ , (14)
otkuda y budet sledovat\ sootnoßenye (13) dlq rassmatryvaemoho sluçaq.
PredpoloΩym, çto vopreky (14)
sup ( ) ( ) ( )( ) ( ) ( )
t t t
r r
j
r
j
j j
s t s t t
≤ ≤
− − −
+
= >
1
2 2 2ϕ .
Tohda najdetsq λ, 0 < λ < 1, takoe, çto λs tr
j
( )( )−2 = ϕ( )( )r
jt−2
y, sledova-
tel\no, λs tr
j
( )( )−
+
2
1 ≤ ϕ( )( )r
jt−
+
2
1 .
Pust\ δ( )t = ϕ( )t – α – λ βs t( ) −( ) , hde α y β — konstant¥ nayluçßeho
ravnomernoho pryblyΩenyq funkcyj ϕ( )t y s t( ) sootvetstvenno.
Poskol\ku δ( )r−2 ∈ S n2 2
2
, y δ( )( )r
jt−2 = 0, δ( )r−2
na promeΩutke t tj j, +[ ]1
moΩet ymet\ ne bolee trex peremen znaka. Tohda na peryode u δ( )r−2
budet ne
bolee 2n – 1 peremen znaka.
S druhoj storon¥, kak y pry dokazatel\stve teorem¥71, ubeΩdaemsq, çto
δ( )t ymeet na peryode ne menee 2n peremen znaka. No tohda v sylu teorem¥
Rollq δ( )( )r t−2
menqet znak na peryode ne menee 2n raz.
Poluçennoe protyvoreçye dokaz¥vaet neravenstvo (14). Yz (14) s uçetom mo-
notonnosty s r( )−2
na t tj j, +[ ]1 sleduet, çto
V V
t
t
r
t
t
r
j
j
j
j
s
+ +
− − ≤
1 1
2 2( ) ( )ϕ
takΩe dlq yntervalov, v kotor¥x s r( )−1
ne ymeet nulej.
Teorema 2 dokazana.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
NERAVENSTVA TYPA BERNÍTEJNA DLQ SPLAJNOV DEFEKTA 2 999
Dokazatel\stvo teorem¥ 3. Pry dokazatel\stve teorem¥72 ustanovleno,
çto esly s r( )−1
ymeet nul\ v ( , )t tj j+1 , to ymeet mesto neravenstvo (12). Poka-
Ωem, çto πto neravenstvo ymeet mesto y dlq takyx yntervalov, v kotor¥x s r( )−1
ne ymeet nulej. Dlq πtoho pokaΩem, çto dlq vsex x ∈ 0 2, /π n[ ] v¥polnqetsq
neravenstvo
r s t dt r t dtr
x
r
x
( ) ( ), ,− −( ) ≤ ( )∫ ∫1
0
1
0
ϕ , x ∈ 0 2, /π n[ ] , (15)
hde r f t( , ) — nevozrastagwaq perestanovka (sm., naprymer, [3], hl. 3) suΩenyq
funkcyy f na t tj j, +[ ]1 .
Pust\ ∆( )x = r t dtrx
ϕ( ),−( )∫ 1
0
– r s t dtrx ( ),−( )∫ 1
0
. V rassmatryvaemom sluçae
obe funkcyy r trϕ( ),−( )1
y r s tr( ),−( )1
lynejn¥ na 0 2, /π n[ ] , tak, çto yx raz-
nost\ lybo ne menqet znak na yntervale ( , / )0 2π n y( tohda v sylu (6) dlq
lgboho t ∈ ( , / )0 2π n budet r s tr( ),−( )1 ≤ r trϕ( ),−( ))1
, lybo menqet znak rovno
odyn raz v toçke x0 ∈ ( , / )0 2π n , pryçem s „+” na „–”.
V pervom sluçae neravenstvo (15) oçevydno. Vo vtorom sluçae, tak kak
′∆ ( )x = r xrϕ( ),−( )1 – r s xr( ),−( )1
, ∆( )x vozrastaet na yntervale ( , )0 0x y ub¥-
vaet na ( , / )x n0 2π . Krome toho, ∆( )0 = 0 y ∆( / )2π n ≥ 0 (poslednee neravenst-
vo ymeet mesto v sylu (6)). Takym obrazom, raznost\ ∆( )x neotrycatel\na na
0 2, /π n[ ] , çto πkvyvalentno (15).
Uçyt¥vaq (15) y predloΩenye 3.2.5 yz [3], vydym, çto (12) ymeet mesto takΩe
dlq yntervalov ( , )t tj j+1 , v kotor¥x s r( )−1
ne ymeet nulej.
Ytak, dlq lgboho p ∈ ∞[ )1, y lgboho j = 0 1, n −
s t dt t dtr p
t
t
r p
t
t
j
j
j
j
( ) ( )( ) ( )− −
+ +
∫ ∫≤1 1
1 1
ϕ ,
sledovatel\no,
s s t dtr
p
r p
t
t
j
n
p
j
j
( ) ( )
/
( )− −
=
−
=
+
∫∑1 1
0
1
1
1
≤≤
−
=
− +
∫∑ ϕ( )
/
( )r p
t
t
j
n
p
t dt
j
j
1
0
1
1
1
=
= ϕ ϕ
π
( )
/
( )( )r p
p
r
p
t dt− −∫
=1
0
2 1
1 .
Takym obrazom, yz posledneho neravenstva, s uçetom opredelenyq funkcyy ϕ,
poluçaem neravenstvo (8) dlq vsex p ∈ ∞[ )1, .
Teorema 3 dokazana.
1. Kornejçuk N. P., Babenko V. F., Lyhun A. A. ∏kstremal\n¥e svojstva polynomov y splajnov.
– Kyev: Nauk. dumka, 1992. – 304 s.
2. Babenko V. F., Kornejçuk N. P., Kofanov V. A., Pyçuhov S. A. Neravenstva dlq proyzvodn¥x
y yx pryloΩenyq. – Kyev: Nauk. dumka, 2003. – 591 s.
3. Kornejçuk N. P. Toçn¥e konstant¥ v teoryy pryblyΩenyq. – M.: Nauka, 1987. – 424 s.
Poluçeno 30.05.08,
posle dorabotky — 30.04.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
|
| id | umjimathkievua-article-3074 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:35:46Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/eb/831f87105bfcfea0c5ccefe8785668eb.pdf |
| spelling | umjimathkievua-article-30742020-03-18T19:44:40Z Inequalities of the Bernstein type for splines of defect 2 Неравенства типа Вернштейна для сплайнов дефекта 2 Babenko, V. F. Parfinovych, N. V. Бабенко, В. Ф. Парфинович, Н. В. Бабенко, В. Ф. Парфинович, Н. В. We obtain new exact inequalities of the Bernstein type for periodic polynomial splines of order r and defect 2. Отримано нoвi точш нeрiвнocтi типу Вернштейна для перюдичних пoлiнoмiaльниx сплайшв порядку r дефекту 2. Institute of Mathematics, NAS of Ukraine 2009-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3074 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 7 (2009); 995-999 Український математичний журнал; Том 61 № 7 (2009); 995-999 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3074/2897 https://umj.imath.kiev.ua/index.php/umj/article/view/3074/2898 Copyright (c) 2009 Babenko V. F.; Parfinovych N. V. |
| spellingShingle | Babenko, V. F. Parfinovych, N. V. Бабенко, В. Ф. Парфинович, Н. В. Бабенко, В. Ф. Парфинович, Н. В. Inequalities of the Bernstein type for splines of defect 2 |
| title | Inequalities of the Bernstein type for splines of defect 2 |
| title_alt | Неравенства типа Вернштейна для сплайнов дефекта 2 |
| title_full | Inequalities of the Bernstein type for splines of defect 2 |
| title_fullStr | Inequalities of the Bernstein type for splines of defect 2 |
| title_full_unstemmed | Inequalities of the Bernstein type for splines of defect 2 |
| title_short | Inequalities of the Bernstein type for splines of defect 2 |
| title_sort | inequalities of the bernstein type for splines of defect 2 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3074 |
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