Kolmogorov and linear widths of classes of s-monotone integrable functions
Let $s \in \mathbb{N}$ and let $\Delta^s_+$ be the set of functions $x \mapsto \mathbb{R}$ on a finite interval $I$ such that the divided differences $[x; t_0, ... , t_s ]$ of order $s$ of these functions are nonnegative for all collections of $s + 1$ distinct points $t_0,..., t_s \in I$. For the...
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509847074635776 |
|---|---|
| author | Konovalov, V. N. Коновалов, В. Н. Коновалов, В. Н. |
| author_facet | Konovalov, V. N. Коновалов, В. Н. Коновалов, В. Н. |
| author_sort | Konovalov, V. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:02:57Z |
| description | Let $s \in \mathbb{N}$ and let $\Delta^s_+$ be the set of functions $x \mapsto \mathbb{R}$ on a finite interval $I$ such that the divided differences
$[x; t_0, ... , t_s ]$ of order $s$ of these functions are nonnegative for all collections of $s + 1$ distinct points $t_0,..., t_s \in I$.
For the classes $\Delta^s_+ B_p := \Delta^s_+ \bigcap B_p$ , where $B_p$ is the unit ball in $L_p$, we obtain orders of the Kolmogorov and linear widths in the spaces $L_q$ for $1 \leq q < p \leq \infty$.
|
| first_indexed | 2026-03-24T02:47:36Z |
| format | Article |
| fulltext |
UDK 517.5
V. N. Konovalov (Yn-t matematyky NAN Ukrayn¥, Kyev)
KOLMOHOROVSKYE Y LYNEJNÁE POPEREÇNYKY
KLASSOV s -MONOTONNÁX YNTEHRYRUEMÁX FUNKCYJ
Let s ∈ N and let ∆+
s be the set of functions x : I � R on a finite interval I such that the divided
differences [ ; , , ]x t ts0 … of order s of these functions are nonnegative for all collections of s + 1
distinct points t ts I0, ,… ∈ . For the classes ∆ ∆+ +=s
p
s
pB B: ∩ , where Bp is the unit ball in Lp , we
obtain orders of the Kolmogorov and linear widths in the spaces Lq for 1 ≤ q < p ≤ ∞.
Nexaj s ∈ N i ∆+
s
— mnoΩyna funkcij x : I � R na skinçennomu intervali I takyx, wo
podileni riznyci [ ; , , ]x t ts0 … porqdku s cyx funkcij [ nevid’[mnymy dlq vsix naboriv z s + 1
riznyx toçok t ts I0, ,… ∈ . Dlq klasiv ∆ ∆+ +=s
p
s
pB B: ∩ , de Bp — odynyçna kulq v Lp ,
znajdeno porqdky u prostorax Lq pry 1 ≤ q < p ≤ ∞ kolmohorovs\kyx i linijnyx popereç-
nykiv.
1. Vvedenye. Formulyrovky osnovn¥x rezul\tatov. Pust\ X — vewestven-
noe lynejnoe prostranstvo x s normoj x X , a W — proyzvol\noe nepustoe
mnoΩestvo yz X . Kolmohorovskym n -popereçnykom v prostranstve X
mnoΩestva W naz¥vaetsq velyçyna
d Wn X( )kol : = inf sup inf
M x W y M
Xn n
x y
∈ ∈
− ,
hde perv¥j ynfymum beretsq po vsem affynn¥m mnohoobrazyqm M
n
razmerno-
sty ≤ n yz X. Lynejn¥m n-popereçnykom v prostranstve X mnoΩestva W
naz¥vaetsq velyçyna
d Wn X( )lin : = inf inf sup
M A x W
Xn
x Ax
∈
− ,
hde perv¥j ynfymum beretsq po vsem affynn¥m mnohoobrazyqm M
n
razmerno-
sty ≤ n yz X, a vtoroj — po vsem affynn¥m neprer¥vn¥m otobraΩenyqm
A W Mn: ( )aff � affynnoj oboloçky aff ( )W mnoΩestva W v mnohoobrazye
M
n. Vxodqwye v opredelenyq kolmohorovskoho y lynejnoho popereçnykov ve-
lyçyn¥
E X Mn
X( , ) : = sup inf
x W y M
Xn
x y
∈ ∈
−
y
E X Mn
X( , )lin : = inf sup
A x W
Xx Ax
∈
−
naz¥vagtsq sootvetstvenno nayluçßym y nayluçßym lynejn¥m pryblyΩenyem
v X mnoΩestva W (fyksyrovann¥m) affynn¥m mnohoobrazyem M
n. Oçevyd-
no, çto
E X Mn
X( , ) ≤ E X Mn
X( , )lin y d Wn X( )kol ≤ d Wn X( )lin.
Pry s ∈N ∪ { }0 funkcyg x : I � R budem naz¥vat\ s-monotonnoj na ko-
neçnom yntervale I ⊂ R , esly dlq vsex naborov yz s + 1 razlyçn¥x toçek
t0 , … , ts ∈ I sootvetstvugwye razdelenn¥e raznosty [ x; t0 , … , ts ] porqdka s
© V. N. KONOVALOV, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12 1633
1634 V. N. KONOVALOV
πtoj funkcyy qvlqgtsq neotrycatel\n¥my. Oçevydno, çto s-monotonn¥e
funkcyy pry s = 0, 1, 2 — πto sootvetstvenno neotrycatel\n¥e, neub¥vagwye
y v¥pukl¥e funkcyy na yntervale I. Takym obrazom, parametr s xarakteryzu-
et formu funkcyj.
Osnovn¥e svojstva s-monotonn¥x funkcyj opysan¥ v [1 – 3]. Otmetym, çto
te svojstva s-monotonn¥x funkcyj, kotor¥e budut yspol\zovat\sq v dannoj
rabote, pryveden¥ takΩe v [4].
Klass vsex s -monotonn¥x na I funkcyj budem oboznaçat\ çerez ∆+
s I( ).
Krome toho, esly na I opredelen nekotor¥j klass funkcyj W ( I ), to polahaem
∆ ∆+ +=s sW I I W I( ) : ( ) ( )∩ . Çerez Lp ( I ) , 1 ≤ p ≤ ∞ , oboznaçym, kak ob¥çno, ly-
nejnoe prostranstvo vsex yzmerym¥x po Lebehu funkcyj x : I � R s koneçnoj
normoj x L Ip( ) . Edynyçn¥j ßar prostranstva Lp ( I ) oboznaçym çerez Bp ( I ) .
Dlq yntervala I = ( – 1, 1 ) eho oboznaçenye budem ynohda opuskat\, t. e. W : =
: = W ( I ) .
Netrudno proveryt\, çto ∆+ ⊄s
p qB L pry 1 ≤ p < q ≤ ∞ . Sleduet takΩe
otmetyt\, çto, nesmotrq na nalyçye pry s > 1 u funkcyj x Bs
p∈ +∆ oprede-
lenn¥x dyfferencyal\n¥x svojstv, nel\zq, voobwe hovorq, harantyrovat\ pry
1 ≤ p < ∞ ohranyçennost\ norm proyzvodn¥x x k( )
porqdka k ≥ 1 v kakom-ly-
bo yz prostranstv Lq , 1 ≤ q ≤ ∞ . Lyß\ dlq funkcyj x Bs∈ + ∞∆ , hde s > 1,
moΩno utverΩdat\, çto ′ < ∞x L1
.
Cel\ dannoj rabot¥ — opysat\, v termynax kolmohorovskyx y lynejn¥x
popereçnykov, vlyqnye form¥ funkcyj, xarakteryzuemoj parametrom s, na
porqdky pryblyΩenyq πtyx funkcyj affynn¥my mnohoobrazyqmy koneçnoj
razmernosty.
PreΩde çem sformulyrovat\ poluçenn¥e rezul\tat¥, uslovymsq ewe o
nekotor¥x oboznaçenyqx. Çerez I budem oboznaçat\ dlyn¥ promeΩutkov I,
a çerez c : = c ( α, β, … , γ ) — razlyçn¥e poloΩytel\n¥e „postoqnn¥e”, zavysq-
wye ot parametrov α, β , … , γ . Esly zadan¥ dve posledovatel\nosty { }an y
{ }bn , n ≥ 1, poloΩytel\n¥x çysel an y bn , to πty posledovatel\nosty udov-
letvorqgt sootnoßenyg an � bn , n ≥ 1, tohda y tol\ko tohda, kohda suwest-
vugt ne zavysqwye ot n çysla c1 > 0 y c2 > 0 takye, çto c1 ≤ an / bn ≤ c2 ,
n ≥ 1.
Osnovn¥m rezul\tatom dannoj rabot¥ qvlqetsq sledugwaq teorema, v koto-
roj predpolahaetsq, çto klass¥ ∆+
s
pB zadan¥ na yntervale I : = ( – 1, 1 ) .
Teorema,1.1. Esly s ∈ N, s > 1, y 1 ≤ q < p ≤ ∞ , yly s = 1 y 1 ≤ q <
< p ≤ 2, yly s = 1 y 1 ≤ q ≤ 2 < p ≤ ∞ , to
d Bn
s
p Lq
( )∆+
kol � d Bn
s
p Lq
( )∆+
lin � n s q− + ′min{ / , / }1 1 2 , n ≥ 1, (1.1)
hde 1 1/ /q q+ ′ = 1. Esly s = 1 y 2 < q < p ≤ ∞ , to suwestvugt c c q1 1= ( )
y c c q2 2= ( ) takye, çto
c n1
1 2− / ≤ d Bn p Lq
( )∆+
1 kol ≤ d Bn p Lq
( )∆+
1 lin ≤ c n n2
1 2 3 21− +/ /(ln( )) , n ≥ 1. (1.2)
Zameçanye,1.1. V rabotax [4, 5] dlq klassov ∆+
s
pB yssledovano povedenye
formosoxranqgwyx popereçnykov d B Ln
s
p
s
q Lq
( ),∆ ∆+ +
kol
typa Kolmohorova (op-
redelenye sm. v [4, 5]). B¥lo ustanovleno, çto pry s = 1, 2 y 1 ≤ q < p ≤ ∞
formosoxranqgwye popereçnyky ymegt porqdky
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
KOLMOHOROVSKYE Y LYNEJNÁE POPEREÇNYKY … 1635
d B Ln
s
p
s
q Lq
( ),∆ ∆+ +
kol � n s q− + ′1/ , n ≥ 1. (1.3)
Esly Ωe s ≥ 3 y 1 ≤ q < p ≤ ∞ , to
d B Ln
s
p
s
q Lq
( ),∆ ∆+ +
kol � n−2, n ≥ 1. (1.4)
Sravnyvaq ocenky (1.3) y (1.4) s ocenkamy (1.1) y (1.2), vydym, çto dlq klassov
∆+
s
pB porqdky kolmohorovskyx y lynejn¥x popereçnykov mohut suwestvenno
otlyçat\sq ot porqdkov formosoxranqgwyx popereçnykov typa Kolmohorova.
Esly Ωe s ∈ N y 1 ≤ q = p ≤ ∞ , yly s = 0 y 1 ≤ q ≤ p ≤ ∞ , to ny kol-
mohorovskye, ny lynejn¥e, ny formosoxranqgwye popereçnyky typa Kolmoho-
rova klassov ∆+
s
pB v Lq ne stremqtsq k nulg pry n → ∞ . ∏to ustanovleno v
[4, 5].
Otmetym takΩe, çto v [6 – 8] yssledovalos\ povedenye v prostranstvax Lq ,
1 ≤ q ≤ ∞ , kolmohorovskyx, lynejn¥x y formosoxranqgwyx popereçnykov
typa Kolmohorova klassov ∆+
s
p
rW , 0 ≤ s ≤ r + 1, sostoqwyx yz s-monotonn¥x
funkcyj, prynadleΩawyx klassam Soboleva Wp
r , hde r ∈ N y 1 ≤ p ≤ ∞ .
2. Vspomohatel\n¥e utverΩdenyq. Sformulyruem v vyde lemm utverΩ-
denyq, kotor¥e budut yspol\zovat\sq pry dokazatel\stve teorem¥I1.1.
Esly s ∈ N y k ∈ Z, to polahaem
( )k s : =
s k
s
k
k
− +
−
≥
≤
2
1
1
0 0
, ,
, .
(2.1)
Lemma,2.1. Pust\ s , n ∈ N , a = ( a1 , … , an ) ∈ R
n
— fyksyrovann¥j vek-
tor s neotrycatel\n¥my koordynatamy ai , b = ( b1 , … , bn ) ∈ R
n
— fyksy-
rovann¥j vektor s poloΩytel\n¥my koordynatamy bi , ω = ( ω1 , … , ωn ) ∈ R
n
y 1 ≤ p ≤ ∞ . Pust\ takΩe
f an( ; )ω : = ai i
i
n
ω
=
∑
1
, ω ∈ R
n, (2.2)
y
Ωs p n b, , ( ) : = ω ωi
i
n
i
j
i
s j
p p
i n b i j≥ ≤ ≤ − +
≤
= =
∑ ∑0 1 1 1
1 1
1
, , ( )
/
. (2.3)
Tohda v¥polnqetsq neravenstvo
max ( ; )
, , ( )ω
ω
∈Ωs p n b nf a ≤
i
n
k
s
k
i k i
p p
s
k
a b
= =
+
−
′ ′
∑ ∑ −
1 0
1
1
1( )
/
,
hde ai := 0, i = n + 1, … , n + s y 1 1 1/ /p p+ ′ = .
Pry s = 1 lemmaI2.1 dokazana v [6] (lemmaI2), a pry s = 2 — v [5] (lem-
maI11). Dokazatel\stvo lemm¥I2.1 pry vsex s ∈ N ymeetsq v [4] (lemmaI4).
V sledugwej lemme y dalee dlq funkcyj x : I � R , ymegwyx v toçke t yz
yntervala I koneçn¥e odnostoronnye proyzvodn¥e x tk
−
( )( ) y x tk
+
( )( ) porqdka
k ∈ N , polahaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1636 V. N. KONOVALOV
x tk( )( ) : =
1
2
x t x tk k
− ++( )( ) ( )( ) ( ) .
Qsno, çto esly v kakoj-lybo toçke yntervala I suwestvuet ob¥çnaq proyzvod-
naq porqdka k ∈ N , to ona sovpadaet s tak opredelqemoj obobwennoj proyz-
vodnoj. Pry k = 0 polahaem x t( )( )0 : = x ( t ) , t ∈ I .
Lemma,2.2. Pust\ I : = ( – 1, 1 ) , s ∈ N, 1 ≤ p ≤ ∞ , x L Is
p∈ +∆ ( ) , a takΩe
πs t x( ; ; )0 : =
k
s k
kx
k
t
=
−
∑
0
1 0( )( )
!
, t ∈ I ,
y
˜( )x t : = x t t xs( ) ( ; ; )− π 0 , t ∈ I .
Tohda suwestvuet c = c ( s, p ) takoe, çto
˜
( )x L Ip
≤ c x L Ip( ).
∏ta lemma dokazana v [4] (lemmaI3).
Lemma,2.3. Pust\ I — proyzvol\n¥j koneç¥j ynterval, s ∈N ∪ { }0 y
1 ≤ q ≤ ∞ . Tohda suwestvuet c = c ( s, q ) takoe, çto dlq vsex alhebrayçeskyx
mnohoçlenov πs porqdka ≤ s v¥polnqetsq neravenstvo
πs L I∞ ( ) ≤ c I q
s L Iq
−1/
( )π .
UtverΩdenye lemm¥I2.3 — çastn¥j sluçaj teorem¥I2.7 yz [9] (hl.I4, § 2).
Esly n ∈ N y 1 ≤ p ≤ ∞ , to çerez lp
n
oboznaçym, kak ob¥çno, prostranstvo
vektorov x x xn
n= … ∈( , , )1 R s normoj
x lp
n : =
i
n
i
p
p
x
=
∑
1
1/
, 1 ≤ p ≤ ∞ ,
hde x xii
n
i n i
∞
=
∞
= …∑( ) =
1
1
1
/
, ,: max , çerez bp
n
— edynyçn¥j ßar v prostran-
stveII lp
n
.
Lemma,2.4. Esly 2 < q < ∞ , to pry vsex n, m ∈ N takyx, çto m < n ,
ymeet mesto neravenstvo
d bm
n
lq
n( )1
lin ≤ cn mq1 1 2/ /− ,
hde c = c ( q ) .
LemmaI2.4 — neposredstvennoe sledstvye teorem¥I2 yz [10].
Pust\ n ∈ N y Ξn i
i
n
:= { } =
ξ
1
— proyzvol\naq systema vektorov v vewestven-
nom lynejnom prostranstve X. Esly 1 ≤ p ≤ ∞ , to mnoΩestvo
Sp
n+( )Ξ : = ξ ξ: , ( , , ) , , ,= = … ∈ ≥ ≤ ≤ ≤
=
∑a a a a a i n ai
i
i
n
n
n
i lp
n
1
1 0 1 1R
budem naz¥vat\ (neotrycatel\n¥m) p-sektorom po systeme Ξ
n
v X .
Lemma,2.5. Pust\ m , n ∈ N takov¥, çto m < n , y 1 ≤ q ≤ ∞ . Esly
E en i
i
n
:= { } =1
— standartnaq systema edynyçn¥x vektorov e1 1 0 0= …( , , , ), …
… , en = …( , , , )0 0 1 v R
n, to
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
KOLMOHOROVSKYE Y LYNEJNÁE POPEREÇNYKY … 1637
d S Em
n
lq
n( ( ))1
+ kol ≥ max ( ) ( ) , ( ( ) )/ / ( / / )/2 1 11 1 1 2 1 2 1n n m m n nq q− − −− − +{ } + ,
hde a a+ =: max{ , }0 .
LemmaI2.5 qvlqetsq çastn¥m sluçaem lemm¥I1 yz [6].
3. Dokazatel\stvo ocenok sverxu v teoreme,1.1. Pry kaΩdom n ∈ N y
β ≥ 1 razob\em ynterval I : = ( – 1, 1 ) toçkamy
t n iβ, , : =
1 0 1
1 1
− − = …
− + + = − … −
(( ) ) , , , , ,
(( ) ) , , , ,
/
/
n i n i n
n i n i n
β
β
(3.1)
na 2n promeΩutkov
I n iβ, , : =
[ ), , , ,
( ], , , .
, , , ,
, , , ,
t t i n
t t i n
n i n i
n i n i
β β
β β
−
+
= …
= − … −
1
1
1
1
(3.2)
Esly s = 1, to dlq kaΩdoj funkcyy x : I � R pry kaΩdom τ ∈ I pola-
haem
π τ1( ; ; )t x : = x ( τ ) , t ∈ I . (3.3)
Esly Ωe s > 1, to dlq kaΩdoj funkcyy x : I � R, ymegwej koneçn¥e odno-
storonnye proyzvodn¥e x s
–
( )( )−1 τ y x s
+
−( )( )1 τ v toçke τ ∈ I , polahaem
π τs t x( ; ; ) : =
x
k
t
k
k
k
s ( )( )
!
( )
τ τ−
=
−
∑
0
1
, t ∈ I , (3.4)
hde x s( )( )−1 τ , voobwe hovorq, obobwenn¥e proyzvodn¥e porqdka s – 1. Oçevyd-
no, çto pry fyksyrovannom τ ∈ I funkcyy π τs t x( ; ; ), t ∈ I , qvlqgtsq obob-
wenn¥my mnohoçlenamy Tejlora porqdka ≤ s (t. e. stepeny ≤ s – 1 ) po t,
postroenn¥my dlq funkcyy x otnosytel\no toçky τ.
Dlq kaΩdoj funkcyy x L Is
p∈ +∆ ( ) y zadann¥x razbyenyj yntervala I na
promeΩutky I n iβ, , vyda (3.2) polahaem
π βs n it x I( ; ; ), , : =
π
π
β
β
s n i
s n i
t x t t I i n
t x t t I i n
( ; ; ), , , , ,
( ; ; ), , , , ,
, ,
, ,
−
+
∈ = …
∈ = − … −
1
1
1
1
(3.5)
hde mnohoçlen¥ π βs n ix t( ; ; ), ,⋅ −1 opredelen¥ sohlasno (3.1), (3.3) y (3.4). Teper\
opredelym na I kusoçno-polynomyal\n¥e splajn¥
σβ, , ( ; ; )s n t x I : = π βs n it x t( ; ; ), , , t ∈ I n iβ, , , i = ± 1, … , ± n, (3.6)
porqdka ≤ s s uzlamy v toçkax t n iβ, , .
Ocenym uklonenye splajnov σβ, , ( ; ; )s n x I⋅ ot funkcyj x L Is
p∈ +∆ ( ) v met-
ryke L Iq( ) pry uslovyy, çto 1 ≤ q < p ≤ ∞ . Vnaçale budem rassmatryvat\
funkcyy ˜ ( )x L Is
p∈ +∆ takye, çto
˜ ( )( )x k 0 = 0, k = 0, … , s – 1. (3.7)
Lehko ubedyt\sq, çto v πtom sluçae pry vsex k = 0, … , s – 1 budut v¥polnqt\-
sq neravenstva ˜ ( )( )x tk ≥ 0 , t I∈ =+ : [ , )0 1 . Esly Ωe t I∈ = −− : ( , ]1 0 , to
( ) ˜ ( )( )− ≥−1 0s k kx t , k = 0, … , s – 1. Otmetym takΩe, çto vse proyzvodn¥e ˜( )x k ,
k = 0, … , s – 1, qvlqgtsq neub¥vagwymy funkcyqmy na promeΩutke I+ , a na
promeΩutke I– proysxodyt çeredovanye monotonnosty proyzvodn¥x ˜( )x k . Da-
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1638 V. N. KONOVALOV
lee budem rassmatryvat\ takΩe promeΩutok I+ , poskol\ku dlq promeΩutka I–
rassuΩdenyq analohyçn¥.
Otpravlqqs\ ot splajnov σβ, , ( ; ˜; )s n x I⋅ , opredelenn¥x dlq funkcyy x̃ so-
hlasno (3.1) – (3.6), polahaem
σβ, , ( ; ˜; )s n t x I+ : = σβ, , ( ; ˜; )s n t x I , t ∈ I+ , (3.8)
y
σβ, , ( ; ˜; )s n t x I− : = σβ, , ( ; ˜; )s n t x I , t ∈ I– . (3.9)
V dal\nejßem çyslo β budet zavyset\ lyß\ ot parametrov s, p y q. Po-
πtomu ynohda budem opuskat\ yndeks β, çtob¥ neskol\ko uprostyt\ obozna-
çenyq.
Esly n = 1, to yz (3.1) – (3.8) sleduet
σs t x I, ( ; ˜; )1 + =
˜ ( )
!
( )
( )
,
,
x t
k
t t
k
k
k
s
1 0
1 0
0
1
−
=
−
∑ =
˜ ( )
!
( )x
k
t
k
k
k
s 0
0
1
=
−
∑ = 0, t ∈ I+
.
No tohda oçevydno, çto ˜( ) ( ; ˜; ),x t t x Is− +σ 1 = ˜( )x t , t ∈ I+
. Yspol\zuq neravenst-
vo Hel\dera, poluçaem
˜( ) ( ; ˜; ), ( )
x x Is L Iq
⋅ − ⋅ +
+
σ 1 ≤ ˜
( )x L Ip +
. (3.10)
Dalee budem rassmatryvat\ sluçaj n > 1. Esly s = 1, to
˜( ) ( ; ˜; ),x t t x In− +σ1 = ˜( ) ˜( ),x t x tn i− −1 , t ∈ In, i
, i = 1, … , n. (3.11)
Esly Ωe s > 1, to dlq t ∈ In, i
, 1 ≤ i ≤ n , yz (3.1) – (3.8) y formul¥ Tejlora
sledugt ravenstva
˜( ) ( ; ˜; ),x t t x Is n− +σ =
1
2
1
1 1
1
2
( )!
˜ ( ) ˜ ( ) ( )
,
( ) ( )
,s
x x t t d
t
t
s s
n i
s
n i
−
−( ) −
−
∫ − −
−
−τ τ τ . (3.12)
Polahaq
ω ˜ ;( )
,x Is
n i
−( )1 : = ˜ ( ) ˜ ( )( )
,
( )
,x t x ts
n i
s
n i
− −
−−1 1
1 , i = 1, … , n – 1,
zameçaem, çto proyzvodnaq ˜( )x s−1
qvlqetsq neotrycatel\noj y neub¥vagwej
funkcyej na promeΩutke I+
. Poπtomu yz (3.11) y (3.12) lehko poluçaem ocenky
˜( ) ( ; ˜; ), ( ),
x x Is n L Iq n i
⋅ − ⋅ +σ ≤
I
s
x I
n i
s q
s
n i
,
/
( )
,( )!
˜ ;( )
− ′
−
−
1
1
1
ω , i = 1, … , n – 1, (3.13)
hde 1 1/ /q q+ ′ = 1 y s ≥ 1.
Esly Ωe i = n, to, uçyt¥vaq, çto ˜ ( )( )
,x tk
n n−1 ≥ 0, k = 0, … , s – 1, ymeem
0 ≤ ˜( ) ( ; ˜; ),x t t x Is n− +σ ≤ ˜( )x t , t ∈ In, n
.
Snova yspol\zuq neravenstvo Hel\dera, poluçaem
˜( ) ( ; ˜; ), ( ),
x x Is n L Iq n n
⋅ − ⋅ +σ ≤ I xn n
q p
L Ip,
/ /
( )
˜
1 1−
+
. (3.14)
Yz ocenok (3.13), (3.14) y yzvestnoho neravenstva ai
q
i
m q
=∑( )1
1/
≤ aii
m
=∑ 1
, 1 ≤
≤ q ≤ ∞ , sleduet
˜( ) ( ; ˜; ), ( )
x x Is n L Iq
⋅ − ⋅ +
+
σ ≤
∨
=
− − ′ −∑c I x I
i
n
n i
s q s
n i
1
1 1 1
,
/ ( )
,( )˜ ;ω + I xn n
q p
L Ip
,
/ /
( )
˜1 1−
+
,
(3.15)
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KOLMOHOROVSKYE Y LYNEJNÁE POPEREÇNYKY … 1639
hde
∨
c = ( )( )!s − −1 1. Oçevydno, çto esly ˜
( )x L Ip +
= 0, to yz (3.15) sleduet ra-
venstvo
˜( ) ( ; ˜; ), ( )
x x Is n L Iq
⋅ − ⋅ +
+
σ = 0. (3.16)
Dalee budem predpolahat\, çto ˜
( )x L Ip +
> 0. Zafyksyrovav proyzvol\noe
çyslo β β: ( , , )= s p q , udovletvorqgwee neravenstvu
β > max ( ) , ( )( )/ / / /1 1 1 1 1 11 1+ − ′ + − ′ −{ }− −s s q s p q p , (3.17)
netrudno proveryt\, çto suwestvugt c c1 1= ( )β y c c2 2= ( )β takye, çto
c n n i1
1− −−β β( ) ≤ In i, ≤ c n n i2
1− −−β β( ) , i = 1, … , n – 1, n > 1. (3.18)
Qsno takΩe, çto I nn n, = −β . No tohda yz (3.15) y (3.18) sleduet neravenstvo
˜( ) ( ; ˜; ), ( )
x x Is n L Iq
⋅ − ⋅ +
+
σ ≤ c
n i
n
x I
i
n s q
s q
s
n i
∗
=
− − − ′
− ′
−∑ −
1
1 1 1
1
1( ) ˜ ;
( )( / )
( / )
( )
,( )
β
β ω +
+ ˜
( )
( / / )x nL I
q p
p +
− −β 1 1 , (3.19)
hde c* = c*
( β, s, q ) . Krome toho, yz dokazann¥x v [5] neravenstv (3.25) – (3.29) y
(3.31) sleduet
i
n s p
L I
s p
j
i
s
s
j
p p
c n i
x n
i j x I
p
=
−
∗
− − ′
− ′
=
−∑ ∑− − +
+1
1 1 1
1
1
1
1
1
( )
˜
( ) ˜ ;
( )( / )
( )
( / )
( )
/
( )
β
β ω ≤ 1, (3.20)
hde c* = c* ( β, s, q ) , a çysla ( )i j s− +1 opredelen¥ sohlasno (2.1).
Takym obrazom, zadaçu o pryblyΩenyy funkcyj x̃ splajnamy σs n x I, ( ; ˜; )⋅ + ,
blahodarq neravenstvam (3.19) y (3.20), m¥ svely k πkstremal\noj zadaçe vyda
f an−1( ; )ω → sup, ω ∈ −Ωs p n b, , ( )1 , (3.21)
v prostranstve R
n−1, hde funkcyq f an− ⋅1( ; ) y mnoΩestvo Ωs p n b, , ( )−1 oprede-
len¥ v sootvetstvyy s (2.2) y (2.3), a koordynat¥ fyksyrovann¥x vektorov
a a an= … −( , , )1 1 y b b bn= … −( , , )1 1 opredelqgtsq sledugwym obrazom:
ai : = ( )( )( / ) ( / )n i ns q s q− − − ′ − − ′β β1 1 1
, i = 1, … , n – 1, (3.22)
bi : = c x n i nL I
s p s p
p∗
− − − ′ − − ′
+
−˜ ( )( )
( )( / ) ( / )1 1 1 1β β
, i = 1, … , n – 1. (3.23)
Polahaq ai := 0, i = n, … , n – 1 + s, y prymenqq lemmuI2.1 (s zamenoj n na n –
– 1 ), ymeem
max ( ; )
, , ( )ω
ω
∈ −
−Ωs p n b nf a
1
1 ≤
i
n
k
s
k
i k i
p p
s
k
a b
=
−
=
+
−
′ ′
∑ ∑ −
1
1
0
1
1
1( )
/
. (3.24)
Pust\ n > s + 1 y
c s q k
k
s� : ( )( / )= − − ′ − +=∏ β 1 1 1
1
. Tohda pry i = 1, … , n – 1 –
– s poluçaem
k
s
k s q
s
k
n i k
=
− − ′∑ −
− −
0
1 11( ) ( )( )( / )β =
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1640 V. N. KONOVALOV
=
c n i d ds
s q s
s
�
0
1
0
1
1
1 1
1∫ ∫… − − −…− …− − ′ −( )( )( / )τ τ τ τβ ≤
≤ c n i s q s�( )( )( / )− − − ′ −β 1 1 . (3.25)
Esly Ωe i = n – s, … , n – 1, to oçevydno, çto
k
s
k s q
s
k
n i k
=
− − ′∑ −
− −
0
1 11 0( ) max{ , }( )( / )β ≤ 2 1 1s s qs( )( / )β− − ′ . (3.26)
Yspol\zuq pry i = 1, … , n – 1 – s sootnoßenyq (3.22), (3.23) y (3.25), ymeem
k
s
k
i k i
s
k
a b
=
+
−∑ −
0
11( ) ≤
c
c
x
n i
nL I
q p s
q pp
�
∗
− − −
−+
−˜ ( )
( )
( )( / / )
( / / )
β
β
1 1 1
1 1 . (3.27)
Polahaq c ss s q
� : ( )( / )= − − ′2 1 1β
y yspol\zuq pry i = n – s, … , n – 1 sootnoßenyq
(3.22), (3.23) y (3.26), poluçaem
k
s
k
i k i
s
k
a b
=
+
−∑ −
0
11( ) ≤
c
c
x nL I
q p
p
�
∗
− −
+
˜
( )
( / / )β 1 1 . (3.28)
No tohda yz (3.27) sleduet
i
n s
k
s
k
i k i
p p
s
k
a b
=
− −
=
+
−
′ ′
∑ ∑ −
1
1
0
1
1
1( )
/
≤
≤
c x
c n
n i
L I
q p
i
n s
q p s p
p
p
� ˜
( )
( )
( / / )
( )( / / )
/
+
∗
−
=
− −
− − − ′
′
∑ −( )
β
β
1 1
1
1
1 1 1
1
≤
≤ ˙ ˜
( )
( / / ) ( )( / / ) /c x nL I
q p q p s p
p +
− − + − − − + ′β β1 1 1 1 1 1 =
= ˙ ˜
( )
/c x nL I
s q
p +
− + ′1
, (3.29)
hde ˙ ˙( , , , )c c s p q= β . Otmetym, çto zdes\ yspol\zovano uslovye (3.17) dlq β.
Krome toho, yz (3.28) sleduet
i n s
n
k
s
k
i k i
p p
s
k
a b
= −
−
=
+
−
′ ′
∑ ∑ −
1
0
1
1
1( )
/
≤
≤
c
c
x nL I
i n s
n p
q p
p
�
∗ = −
− ′
− −
+ ∑
˜
( )
/
( / / )
1 1
1 11 β ≤
≤ ˙̇ ˜
( )
( / / )c x nL I
q p
p +
− −β 1 1 , (3.30)
hde ˙̇ : ( ) /c c c s p= ∗
− ′� 1 1 . Poπtomu pry n > s + 1 v sylu neravenstv (3.24), (3.29) y
(3.30) poluçaem
max ( ; )
, , ( )ω
ω
∈ −
−Ωs p n b nf a
1
1 ≤ ˜ ˙ ˙̇
( )
/ ( / / )x cn cnL I
s q q p
p +
− + ′ − −+( )1 1 1β . (3.31)
Ostaetsq ocenyt\ sverxu funkcyg f an− ⋅1( ; ) v sluçae 1 < n ≤ s + 1. Lehko
ubedyt\sq, çto v πtom sluçae yz (3.22) – (3.24) sleduet ocenka
max ( ; )
, , ( )ω
ω
∈ −
−Ωs p n b nf a
1
1 ≤
�
c x L Ip
˜
( )+
, (3.32)
hde
� �
c c s p q= ( , , , )β . Oçevydno takΩe, çto pry uslovyy (3.17) ymeet mesto nera-
venstvo
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KOLMOHOROVSKYE Y LYNEJNÁE POPEREÇNYKY … 1641
n q p− −β( / / )1 1 ≤ n s q− + ′1/ . (3.33)
Tohda yz (3.19) – (3.21) y (3.31) – (3.33) sleduet
˜( ) ( ; ˜; ), ( )
x x Is n L Iq
⋅ − ⋅ +
+
σ ≤ c x nL I
s q
p
˜
( )
/
+
− + ′1 , (3.34)
hde c = c ( β, s, p, q ) , n > 1, y ˜
( )x L Ip +
> 0. No v sylu (3.16) neravenstvo (3.34)
budet v¥polnqt\sq y v sluçae, kohda n > 1 y ˜
( )x L Ip +
= 0. Esly Ωe n = 1 y
˜
( )x L Ip +
≥ 0, to ocenka (3.34) sleduet yz (3.10). Takym obrazom, neravenstvo
(3.34) dokazano dlq vsex n ≥ 1 y funkcyj x̃ takyx, çto ˜
( )x L Ip +
≥ 0.
Analohyçno poluçaem ocenku
˜( ) ( ; ˜; ), ( )
x x Is n L Iq
⋅ − ⋅ −
−
σ ≤ c x nL I
s q
p
˜
( )
/
−
− + ′1 , (3.35)
hde splajn σs n x I, ( ; ˜; )⋅ − opredelen sohlasno (3.9), a c = c ( β, s, p, q ) . Obæedy-
nqq ocenky (3.34) y (3.35), ymeem
˜( ) ( ; ˜; ), ( )
x x Is n L Iq
⋅ − ⋅σ ≤ c x nL I
s q
p
˜
( )
/− + ′1 , (3.36)
hde c = c ( β, s, p, q ) .
Ocenka sverxu (3.36) dokazana lyß\ dlq funkcyj x̃ , kotor¥e udovletvorq-
gt uslovyg (3.7). Rassmotrym teper\ obwyj sluçaj, ne predpolahaq, çto funk-
cyq x L Is
p∈ +∆ ( ) udovletvorqet uslovyg (3.7). Polahaem
˜( ) : ( ) ( ; ; )x t x t t xs= − π 0 , t ∈ I ,
hde
πs
k
k
k
s
t x
x
k
t( ; ; ) :
( )
!
( )
0
0
0
1
=
=
−
∑ , t ∈ I .
Oçevydno, çto ˜ ( )( )x k 0 0= , k = 0, … , s – 1. Yspol\zuq lemmuI2.2, ymeem
˜
( )x L Ip
≤ ˜
( )c x L Ip
, (3.37)
hde ˜ ˜( , )c c s p= . Pust\
˜ ( ; ; ) : ( ; ˜; ) ( ; ; ), ,σ σ πs n s n st x I x I t x= ⋅ + 0 , t ∈ I .
Qsno, çto v sylu (3.36) y (3.37) v¥polnqetsq neravenstvo
x x Is n L Iq
( ) ˜ ( ; ; ), ( )
⋅ − ⋅σ ≤ c x nL I
s q
p( )
/− + ′1 ,
hde c = c ( β, s, p, q ) . TakΩe oçevydno, çto splajn¥ ˜ ( ; ; ),σs n x I⋅ sovpadagt so
splajnamy σs n x I, ( ; ; )⋅ , opredelenn¥my sohlasno (3.6). Tohda v obwem sluçae
budet spravedlyva ocenka
x x Is n L Iq
( ) ( ; ; ), ( )
⋅ − ⋅σ ≤ cn s q− + ′1/ , n ≥ 1, x B Is
p∈ +∆ ( ), (3.38)
hde c = c ( β, s, p, q ) .
Oboznaçym çerez
˙ : ˙ ( ), , ,Σ Σs n s n I= β prostranstvo kusoçno-polynomyal\n¥x
funkcyj σs n, : I � R , ymegwyx v toçke t0 0:= proyzvodnug σs n
s
,
( )( )−1 0
porqdka s – 1 y sovpadagwyx na kaΩdom yz promeΩutkov In i, , i = ± 1, … , ± n,
s alhebrayçeskymy mnohoçlenamy πs n iI( ; ),⋅ porqdka ≤ s. Oçevydno, çto po-
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1642 V. N. KONOVALOV
stoenn¥e v¥ße splajn¥ σs n x I, ( ; ; )⋅ + , x B Is
p∈ +∆ ( ), prynadleΩat prostranstvu
˙
,Σs n. Qsno takΩe, çto aff span( ) ( )∆ ∆+ +=s
p
s
pB B . ∏to sleduet yz toho, çto mno-
Ωestvo ∆+
s
pB I( ) soderΩyt funkcyg x t0 0( ) ≡ , t ∈ I . Pry πtom otobraΩenyq
A Bs
p s n: ˙( ) ,span ∆ Σ+ � , opredelenn¥e sohlasno (3.1) – (3.6), qvlqgtsq lynej-
n¥my, a dim ˙( ),Σs n = s n( )2 1− . No tohda yz (3.38) sleduet, çto
E Bs
p s n Lq
∆ Σ+( ), ˙
,
lin
≤ cn s q− + ′1/ , n ≥ 1, 1 ≤ q < p ≤ ∞ ,
hde c = c ( s, p, q ) . Qsno, çto yz πtoj ocenky lehko poluçyt\ sootnoßenyq
d Bn
s
p Lq
( )∆+
kol ≤ d Bn
s
p Lq
( )∆+
lin ≤ cn s q− + ′1/ , n ≥ 1, 1 ≤ q < p ≤ ∞ , (3.39)
hde c = c ( s, p, q ) . Na πtom zakançyvaetsq dokazatel\stvo ocenok sverxu v
sootnoßenyy (1.1) dlq sluçaq 1 ≤ q ≤ 2 .
Esly Ωe 2 < q < p ≤ ∞ , to ocenky (3.39) neobxodymo usylyt\. ∏toho moΩ-
no dobyt\sq za sçet yspol\zovanyq texnyky dyskretyzacyy.
PoloΩyv
w tn( ) : = w tnβ, ( ) : = n t n− − −
− +( )1 1
1 β β β( )/
, t ∈ I , n ≥ 1,
pokaΩem, preΩde vseho, çto
( ( ) ( ; ; )),
/
( )
x x I ws n n
q
L I
⋅ − ⋅ − ′σ 1
1
≤ cn s q− + ′1/ , n ≥ 1, x B Is
p∈ +∆ ( ), (3.40)
hde σs n x I, ( ; ; )⋅ — splajn¥, postroenn¥e v¥ße, a c = c ( β, s, p, q ) .
Esly n = 1, to ocenka (3.40) qvlqetsq tryvyal\noj. V sluçae n > 1 m¥
vnov\ svedem zadaçu ob ocenke sverxu pryblyΩenyq splajnamy k πkstremal\noj
zadaçe v prostranstve R
n−1. V¥çyslenyq provedem na yntervale I+ y predpo-
loΩym, çto ˜
( )x L Ip +
> 0, hde x̃ udovletvorqet uslovyg (3.7).
Netrudno ubedyt\sq v suwestvovanyy çysel c c1 1= ( )β y c c2 2= ( )β takyx,
çto budut v¥polnqt\sq neravenstva
c In i1 , ≤ w tn( ) ≤ c In i2 , , t In i∈ , , i = 1, … , n, n ≥ 1, (3.41)
y
c w tn1 2 ( ) ≤ w tn( ) ≤ c w tn2 2 ( ), t ∈ I , n ≥ 1. (3.42)
Polahaq q = 1 v neravenstvax (3.13) y (3.14) y uçyt¥vaq (3.41), ymeem
( ˜( ) ( ; ˜; )),
/
( ),
x x I ws n n
q
L In i
⋅ − ⋅ +
− ′σ 1
1
≤ c I x In i
s q s
n i3
1 1
,
/ ( )
,˜ ;
− ′ −( )ω ,
esly i = 1, … , n – 1, y
( ˜( ) ( ; ˜; )),
/
( ),
x x I ws n n
q
L In n
⋅ − ⋅ +
− ′σ 1
1
≤ c In n
q p
3
1 1
,
/ /−
,
hde c3 = c3 ( β, s, p, q ) . RassuΩdaq dalee, kak pry dokazatel\stve ocenky (3.19),
poluçaem neravenstvo
( ˜( ) ( ; ˜; )),
/
( )
x x I ws n n
q
L I
⋅ − ⋅ +
− ′
+
σ 1
1
≤ c
n i
n
x I
i
n s q
s q
s
n i
∗
=
− − − ′
− ′
−∑ −
1
1 1 1
1
1( ) ˜ ;
( )( / )
( / )
( )
,( )
β
β ω +
+ c x nr
L I
q p
p
∗ − −
+
˜( )
( )
( / / )β 1 1 ,
hde c* = c*
( β, s, p, q ) . Uçyt¥vaq neravenstvo (3.20), vnov\ pryxodym k πkstre-
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KOLMOHOROVSKYE Y LYNEJNÁE POPEREÇNYKY … 1643
mal\noj zadaçe vyda (3.21) dlq funkcyy f an− ⋅1( ; ) na mnoΩestve Ωs p n b, , ( )−1 ,
hde koπffycyent¥ vektorov a y b opredelqgtsq ravenstvamy (3.22) y (3.23).
A poskol\ku uΩe dokazano (sm. (3.31) – (3.33)), çto
max ( ; )
, , ( )ω
ω
∈ −
−Ωs p n b nf a
1
1 ≤ c x nL I
s q
p
˜
( )
/
+
− + ′1 ,
poluçaem
( ˜( ) ( ; ˜; )),
/
( )
x x I ws n n
q
L I
⋅ − ⋅ +
− ′
+
σ 1
1
≤ cn s q− + ′1/ ,
hde c = c ( β, s, p, q ) . Analohyçnoe neravenstvo ymeet mesto y dlq yntervala I– .
OsvoboΩdaqs\, kak y v¥ße, ot ohranyçenyj (3.7), dokaz¥vaem ocenku (3.40) v
obwem sluçae.
Pry kaΩdom n ∈ N oboznaçym çerez Σ Σ+ +=, , , ,: ( )s n s n Iβ prostranstvo funk-
cyj σs n I I, ( ; ) :⋅ + + � R , kotor¥e qvlqgtsq alhebrayçeskymy mnohoçlenamy
πs n iI( ; ),⋅ porqdka ≤ s na promeΩutkax In i, , i = 1, … , n. Oçevydno, çto po-
stroenn¥e v¥ße splajn¥ σs n x I, ( ; ; )⋅ + , x B Is
p∈ +∆ ( ), prynadleΩat πtomu prost-
ranstvu. Lehko proveryt\, çto dim( ), ,Σ+ s n = sn y Σ+, ,s n ⊆ Σ+, ,s n2 , n ≥ 1.
Pust\ toçky t tsn i sn i, , ,:= β , i = 0, 1, … , sn, opredelen¥ sohlasno (3.1), s za-
menoj n na sn. Ustanovym vzaymno odnoznaçn¥e sootvetstvyq meΩdu prost-
ranstvamy Σ+, ,s n y R
sn
s pomow\g lynejn¥x operatorov dyskretyzacyy
T T Isn sn s n s n sn
sn
+ + + += ⋅ → … ∈, , , , , ,: : ( ; ) ( , , )β σ τ τΣ ' 1 R ,
hde
τi : = ( ) ( ) ( ; )/ ( )/
, ,sn sn i t Iq q
s n sn i
− −
− +− +β β σ1 1
1 , i = 1, … , sn.
V obratn¥x otobraΩenyqx
T T Isn sn
sn
sn s n s n+
−
+
−
+ += = … → ⋅ ∈, , , , , ,: : ( , , ) ( ; )1 1
1β τ τ τ σR ' Σ
splajn¥ σs n I, ( ; )⋅ + opredelqgtsq odnoznaçno, v sylu uslovyj ynterpolqcyy
σs n sn it I, ,( ; )− +1 : = ( ) ( )/ ( )/sn sn iq q
i
β β τ− + − −1 1 , i = 1, … , sn. (3.43)
Netrudno ubedyt\sq v suwestvovanyy çysel c c s q1 1= ( , , )β y c c s q2 2= ( , , )β
takyx, çto dlq vsex σs n s nI, , ,( ; )⋅ ∈+ +Σ budut v¥polnqt\sq neravenstva
c T Isn s n lq
sn1 + +⋅, , ( ; )σ ≤ σs n L I
I
q
, ( )
( ; )⋅ +
+
≤ c T Isn s n lq
sn2 + +⋅, , ( ; )σ . (3.44)
Dejstvytel\no, pust\
σs n t I, ( ; )+ : = πs n it I( ; ), , t ∈ In, i
, i = 1, … , n,
hde πs n iI( ; ),⋅ — proyzvol\n¥e mnohoçlen¥ na In, i porqdka ≤ s. Oçevydno, çto
σs n L I
I
q
, ( )
( ; )⋅ +
+
=
i
n
s n i L I
q
q
I
q n i=
∑ ⋅
1
1
π ( ; ), ( )
/
,
. (3.45)
Qsno takΩe, çto
πs n i L I
I
q n i
( ; ), ( ),
⋅ ≤ I In i
q
s n i L In i
,
/
, ( )
( ; )
,
1
π ⋅
∞
, i = 1, … , n. (3.46)
Otmetyv, çto toçky tsn s i j, ( )− + −1 1, j = 1, … , s, prynadleΩat promeΩutku In i, ,
predstavym mnohoçlen¥ πs n iI( ; ),⋅ v sledugwem vyde:
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1644 V. N. KONOVALOV
πs n it I( ; ), = πs sn s i j n i s j n i
j
s
t I l t I( ; ) ( ; ), ( ) , , ,− + −
=
∑ 1 1
1
, t ∈ In, i
, i = 1, … , n, (3.47)
hde l Is j n i, ,( ; )⋅ — fundamental\n¥e mnohoçlen¥ LahranΩa porqdka s takye,
çto l t Is j sn s i j n i, , ( ) ,( ; )− + −1 1 = 1 y l t Is j sn s i k n i, , ( ) ,( ; )− + −1 1 = 0, k ≠ j . Vospol\zovav-
ßys\ predstavlenyqmy (3.47), netrudno proveryt\, çto
πs n i L I
I
n i
( ; ), ( ),
⋅
∞
≤
∨
− + −
=
∑
c t Is sn s i j n i
q
j
s q
π ( ; ), ( ) ,
/
1 1
1
1
, i = 1, … , n, (3.48)
hde
∨
c =
∨
c s q( , , )β . TakΩe netrudno proveryt\, çto
ˆ ( ( ) )
( )
c
sn s i j
sn
1
11 1− − − + −β
β ≤ In i, ≤ ˆ ( ( ) )
( )
c
sn s i j
sn
2
11 1− − − + −β
β , (3.49)
hde ˆ ˆ ( , )c c s1 1= β y ˆ ˆ ( , )c c s2 2= β , a j = 1, … , s y i = 1, … , n. No tohda yz (3.46),
(3.48) y (3.49) lehko poluçaem ocenky
πs n i L I
I
q n i
( ; ), ( ),
⋅ ≤
≤ c
sn s i j
sn
t I
j
s q
q s sn s i j n i
q
q
2
1
1
1 1
1
1 1
=
−
− + −∑ − − − +
( ( ) )
( )
( ; )
( )/
/ , ( ) ,
/β
β π ,
hde c2 = c2 ( β, s, q ) y i = 1, … , n. Podstavlqq πty ocenky v ravenstvo (3.45),
dokaz¥vaem spravedlyvost\ pravoj çasty neravenstv (3.44).
Ostaetsq ubedyt\sq v spravedlyvosty levoj çasty neravenstv (3.44). Ys-
pol\zuq lemmuI2.3, ymeem
πs n i L I
I
q n i
( ; ), ( ),
⋅ ≥ c I In i
q
s n i L In i
,
/
, ( )
( ; )
,
1
π ⋅
∞
, i = 1, … , n, (3.50)
hde c = c ( s, q ) . Krome toho, qsno, çto pry vsex i = 1, … , n v¥polnqgtsq
neravenstva
πs n i L I
I
n i
( ; ), ( ),
⋅
∞
≥ s t Iq
s sn s i j n i
q
j
s q
− ′
− + −
=
∑
1
1 1
1
1
/
, ( ) ,
/
( ; )π . (3.51)
No tohda yz (3.49) – (3.51) sleduet
πs n i L I
I
q n i
( ; ), ( ),
⋅ ≥
≥ c
sn s i j
sn
t I
j
s q
q s sn s i j n i
q
q
1
1
1
1 1
1
1 1
=
−
− + −∑ − − − +
( ( ) )
( )
( ; )
( )/
/ , ( ) ,
/β
β π ,
hde c c s q1 1= ( , , )β y i = 1, … , n. Podstavlqq πty ocenky v ravenstvo (3.45),
ustanavlyvaem spravedlyvost\ levoj çasty neravenstv (3.44).
S pomow\g neravenstv (3.41) analohyçno dokaz¥vaem, çto
˙ ( ; ), ,c T Isn s n lsn+ +⋅σ
1
≤ σs n n
q
L I
I w,
/
( )
( ; )⋅ +
− ′
+
1
1
≤ ˙̇ ( ; ), ,c T Isn s n lsn+ +⋅σ
1
, (3.52)
hde ˙ ˙( , , )c c s q= β y ˙̇ ˙̇ ( , , )c c s q= β .
Oboznaçaq σ σs n s nt x I t x I, ,( ; ; ) : ( ; ; )+ = , t ∈ I+ , y nν
ν:= 2 , ν ≥ 0, polahaem
δ
νs n t x I, ( ; ; )+ : =
δ ν
δ δ ν
ν ν
s
s n s n
t x I
t x I t x I
,
, ,
( ; ; ), ,
( ; ; ) ( ; ; ), ,
1 0
1
1
+
+ +
=
− ≥
−
t ∈ I+ . (3.53)
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KOLMOHOROVSKYE Y LYNEJNÁE POPEREÇNYKY … 1645
Oçevydno, çto δ
ν νs n s nx I, , ,( ; ; )⋅ ∈+ +Σ , ν ≥ 0. Krome toho, yz (3.40), (3.42) y (3.53)
sleduet, çto dlq kaΩdoj funkcyy x B Is
p∈ +∆ ( ) pry kaΩdom ν ≥ 1 budut v¥-
polnqt\sq neravenstva
δ
ν νs n n
q
L I
x I w,
/
( )
( ; ; )⋅ +
− ′
+
1
1
≤
∨
+
⋅ − ⋅ +
− ′c x x I ws n n
q
L I
( ( ) ( ; ; )),
/
( )
σ
ν ν
1
1
+
+ ˆ ( ( ) ( ; ; )),
/
( )
c x x I ws n n
q
L I
⋅ − ⋅
− − +
+
− ′σ
ν ν1 1 1
1 ≤
≤ c s q2 1− − ′( / ) ν, (3.54)
hde
∨
c =
∨
c s p q( , , , )β , ĉ = ˆ ( , , , )c s p qβ y c = c s p q( , , , )β .
Yspol\zuq neravenstva (3.52) y (3.54), ymeem
T x Isn s n lsn+ +⋅, , ( ; ; )
ν ν
δ ν
1
≤ c s q
∗
− − ′2 1( / ) ν, ν ≥ 1, x B Is
p∈ +∆ ( ), (3.55)
hde c* = c* ( β, s, p, q ) . No tohda yz (3.55) sleduet, çto pry kaΩdom ν ≥ 1 obraz
T x Isn s n+ +⋅, , ( ; ; )
ν ν
δ v R
snν
kaΩdoho yz splajnov δ
νs n x I, ( ; ; )⋅ + , x B Is
p∈ +∆ ( ),
prynadleΩyt oktaπdru
c bs q sn
∗
− − ′2 1
1
( / ) ν ν : = τ τ τν
ν
ν∈ ≤{ }∗
− − ′l csn
l
s q
sn1
1
1
2, ( / ) .
Dalee budem yspol\zovat\ standartn¥e, dlq metodov dyskretyzacyy, ras-
suΩdenyq. Zafyksyruem proyzvol\noe çyslo c* > 1 y posledovatel\nost\ na-
tural\n¥x çysel mν , ν ≥ 0, takyx, çto mν ≤ snν , ν ≥ 0. Oçevydno, çto suwe-
stvuet posledovatel\nost\ podprostranstv Mm snν ν⊆ R , ν ≥ 1, ymegwyx
sledugwye svojstva:
dim ( )Mmν ≤ mν , ν ≥ 1, (3.56)
y
E b Msn m
lq
sn1
ν ν
ν
,( )lin
≤ c d bm
sn
lq
sn
∗ ( )ν
ν
ν1
lin
, ν ≥ 1. (3.57)
Zafyksyruem proyzvol\noe c� > c*. Qsno, çto suwestvuet posledovatel\nost\
lynejn¥x otobraΩenyj A Mm
sn m
ν
ν ν: R → , ν ≥ 1, takyx, çto
sup
τ ν
τ τ
ν ν
∈
−
b
m lsn q
snA
1
≤
c E b Msn m
lq
sn� 1
ν ν
ν
,( )lin
, ν ≥ 1. (3.58)
No tohda v sylu (3.56) – (3.58) dlq kaΩdoho τ ν ν∈ ∗
− − ′c bs q sn2 1
1
( / )
ymeem
τ τ
ν ν− Am lq
sn ≤ c d bs q
m
sn
lq
sn
�2 1
1
− − ′ ( )( / ) ν
ν
ν
ν
lin
, ν ≥ 1, (3.59)
hde c c c c�
�= ∗
∗ .
Pust\ m0 ≤ s — fyksyrovannoe natural\noe çyslo, a Mm0
— fyksyrovan-
noe podprostranstvo yz R
s
takoe, çto dim ( )Mm0 ≤ s. Polahaem
M M
m m m0
0
, ,
:
…
=
∗
=
∗
ν ν
ν
ν
span ∪ , ν* ≥ 0.
Tohda oçevydno, çto dim ( )
, ,
M
m m0 … ∗ν ≤ mνν
ν
=
∗
∑ 0
, ν* ≥ 0.
Pust\
Σ+ +
−=, , ,:s n
m
sn
mT M
ν ν
ν ν1 , ν* ≥ 0, (3.60)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1646 V. N. KONOVALOV
hde T sn+
−
, ν
1
— otobraΩenyq, opredelenn¥e v (3.43). Qsno, çto Σ+, ,s n
m
ν
ν ⊆ Σ+, ,s nν
y
dim ( ), ,Σ+ s n
m
ν
ν = dim ( )Mmν , ν ≥ 0. Opredelyv s pomow\g prostranstv Σ+, ,s n
m
ν
ν
prostranstva splajnov
Σ Σ+
…
+
=
∗
∗
=
∗
, ,
, ,
, ,:s n
m m
s n
m
ν
ν ν
ν
ν
ν
0
0
span ∪ , ν* ≥ 0, (3.61)
otmetym, çto Σ Σ+
…
+∗
∗
∗⊆, ,
, ,
, ,s n
m m
s nν
ν
ν
0
, ν* ≥ 0, y
dim ( ), ,
, ,
Σ+
…
=
∗
∗
≤
∗
∑s n
m m
m
ν
ν
ν
ν
ν
0
0
, ν* ≥ 0. (3.62)
Pust\ A Mm
s m
0
0: R → — fyksyrovannoe lynejnoe otobraΩenye. Esly
σ
ν νs n s nI, , ,( ; )⋅ ∈+ +Σ , ν ≥ 0, to polahaem
A I T A T Im
s n sn m sn s n+ + +
−
+ +⋅ = ⋅ν
ν ν ν ν ν
σ σ, , , ,( ; ) : ( ; )1 , ν ≥ 0, (3.63)
hde Amν , ν ≥ 1, — lynejn¥e otobraΩenyq, udovletvorqgwye neravenstvam
(3.58). Qsno, çto vse otobraΩenyq Am
s n s n
m
+ + +→ν
ν ν
ν: , , , ,Σ Σ , ν ≥ 0, qvlqgtsq ly-
nejn¥my.
Oboznaçym çerez ∆+ +
s
pB I suΩenye klassa ∆+
s
pB I( ) na promeΩutok I+ .
Lehko ubedyt\sq, çto aff ( )∆+ +
s
pB I = span ( )∆+ +
s
pB I . Oçevydno takΩe, çto
kaΩdaq funkcyq x B Is
p∈ + +span ( )∆ predstavyma na I+ v vyde
x ( t ) = ( )( ) ( ; ; ) ( ; ; ), ,x t t x I t x Is n s n− +∗ +
=
+
∗
∑σ δ
ν ν
ν
ν
0
, ν* ≥ 0. (3.64)
Na podprostranstve span ( )∆+ +
s
pB I opredelym lynejn¥e operator¥
A x t A t x I
m m m
s n+
…
=
+ +
∗
=
∗
∑0
0
, ,
,( ) : ( ; ; )ν ν
ν
ν
ν
δ , t ∈ I+ , ν* ≥ 0, (3.65)
hde otobraΩenyq Am
+
ν
opredelen¥ sohlasno (3.63). Qsno, çto obraz¥
A x
m m
+
… ∗0, , ν
funkcyj x B Is
p∈ + +∆ prynadleΩat prostranstvu Σ+
…
∗
∗
, ,
, ,
s n
m m
ν
ν0
,
opredelennomu sohlasno (3.60) y (3.61), a samy otobraΩenyq A
m m
+
… ∗0, , ν , oprede-
lenn¥e sohlasno (3.63) y (3.65), qvlqgtsq lynejn¥my. Oçevydno, çto v sylu
(3.63) – (3.65) dlq vsex x B Is
p∈ + +∆ , t ∈ I+ y ν* ≥ 0 ymegt mesto ravenstva
x t A x t
m m
( ) ( )
, ,
− +
… ∗0 ν = ( )( ) ( ; ; ),x t t x Is n− ∗ +σ
ν
+
+
ν
ν
δ δ
ν ν ν ν ν
=
+ +
−
+ +
∗
∑ −
0
1( ), , , ,( ; ; ) ( ; ; )s n sn m sn s nt x I T A T t x I .
No tohda dlq x B Is
p∈ + +∆ y ν* ≥ 0 budut v¥polnqt\sq neravenstva
x A x
m m
L Iq
( ) ( )
, ,
( )
⋅ − ⋅+
… ∗
+
0 ν ≤ x x Is n
L Iq
( ) ( ; ; ),
( )
⋅ − ⋅∗ +
+
σ
ν
+
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KOLMOHOROVSKYE Y LYNEJNÁE POPEREÇNYKY … 1647
+
ν
ν
δ δ
ν ν ν ν ν
=
+ +
−
+ +
∗
+
∑ ⋅ − ⋅
0
1
s n sn m sn s n L I
x I T A T x I
q
, , , , ( )
( ; ; ) ( ; ; ) . (3.66)
Yspol\zuq ocenku (3.38), ymeem
x x Is n
L Iq
( ) ( ; ; ),
( )
⋅ − ⋅∗ +
+
σ
ν
≤ c
s q n
2
1− − ′ ∗( / ) ν , ν* ≥ 0, x B Is
p∈ +∆ ( ), (3.67)
hde c = c ( β, s, p, q ) . Krome toho, pry vsex ν ≥ 0 spravedlyv¥ ravenstva
T x I T A T x Isn s n sn m sn s n+ + +
−
+ +⋅ − ⋅, , , , ,( )( ; ; ) ( ; ; )
ν ν ν ν ν ν
δ δ1 =
= T x I A T x Isn s n m sn s n+ + + +⋅ − ⋅, , , ,( ; ; ) ( ; ; )
ν ν ν ν ν
δ δ . (3.68)
Uçyt¥vaq sootnoßenyq (3.44), (3.55), (3.59) y (3.68), dlq kaΩdoj funkcyy
x B Is
p∈ +∆ ( ) pry kaΩdom ν ≥ 1 poluçaem
δ δ
ν ν ν ν νs n sn m sn s n L I
x I T A T x I
q
, , , , ( )
( ; ; ) ( ; ; )⋅ − ⋅+ +
−
+ +
+
1 ≤
≤ c T x I T A T x Isn s n sn m sn s n lq
sn2
1
+ + +
−
+ +⋅ − ⋅( ), , , , ,( ; ; ) ( ; ; )
ν ν ν ν ν ν ν
δ δ =
= c T x I A T x Isn s n m sn s n lq
sn2 + + + +⋅ − ⋅, , , ,( ; ; ) ( ; ; )
ν ν ν ν ν νδ δ ≤
≤ c d bs q
m
sn
lq
sn2 1
1
− − ′ ( )( / ) ν
ν
ν
ν
lin
, (3.69)
hde c2 — postoqnnaq yz pravoj çasty (3.44), a c = c ( β, s, p, q ) . Esly Ωe ν = 0,
to yz (3.44) y (3.68) sleduet
δ δs s m s s L I
x I T A T x I
q
, , , , ( )
( ; ; ) ( ; ; )1
1
10
⋅ − ⋅+ +
−
+ +
+
≤
≤ c T x I A T x Is s m s s lq
s2 1 10+ + + +⋅ − ⋅, , , ,( ; ; ) ( ; ; )δ δ . (3.70)
No tohda dlq x B Is
p∈ +∆ ( ) y ν* ≥ 1 yz (3.66), (3.67), (3.69) y (3.70) sledugt ne-
ravenstva
x A x
m m
L Iq
( ) ( )
, ,
( )
⋅ − ⋅+
… ∗
+
0 ν ≤
≤ c c T x I A T x Is q
s s m s s lq
s2 1
1 10
− − ′
+ + + +
∗
+ ⋅ − ⋅( / )
, , , ,( ; ; ) ( ; ; )ν δ δ +
+ c d bs q
m
sn
lq
sn
ν
ν
ν
ν
ν
ν
=
− − ′
∗
∗
∑ ( )
1
1
12 ( / ) lin
, (3.71)
hde c = c ( β, s, p, q ) .
Pry vsex ν* ≥ 0 polahaem m0 : = s y opredelqem operator Am
s s
0
: R R→
kak toΩdestvennoe otobraΩenye. Tohda pry ν* = 0 v sylu (3.53), (3.66) y (3.70)
ymeet mesto neravenstvo
x A xm
L Iq
( ) ( )
( )
⋅ − ⋅+
+
0 ≤ c, (3.72)
hde c = c ( β, s, p, q ) .
Rassmotrym teper\ sluçaj, kohda s > 1 y ν* = 2n, n ≥ 1. V πtom sluçae
çysla mν opredelqem, polahaq
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1648 V. N. KONOVALOV
mν : =
s n
s n nn
2 0
2 1 22
ν
ν
ν
ν
, , , ,
, , , .
= …
= + …
−
(3.73)
Lehko proveryt\, çto
ν
ν
=
∑
0
2n
m ≤ 3 2s n , n ≥ 1, s > 1. (3.74)
Krome toho, otmetym, çto tak kak mν = snν , ν = 0, … , n, to
d bm
sn
lq
snν
ν
ν1( )lin
= d bsn
sn
lq
snν
ν
ν1( )lin
= 0, ν = 0, … , n. (3.75)
Yspol\zuq sootnoßenyq (3.71) y (3.75), poluçaem
x A xm m
L I
n
q
( ) ( ), ,
( )
⋅ − ⋅+
…
+
0 2 ≤ c s q n2 1 2− − ′( / ) +
+ c d b
n
n
s q
m
sn
lq
sn
ν
ν
ν
ν
ν
= +
− − ′∑ ( )
1
2
1
12 ( / ) lin
, (3.76)
hde c = c ( β, s, p, q ) . Pry πtom qsno, çto
2 1 2− − ′( / )s q n ≤ 2 1 2− −( / )s n , n ≥ 1. (3.77)
Krome toho, yz lemm¥ 2.4 y sootnoßenyj (3.73) sleduet
d bm
sn
lq
snν
ν
ν1( )lin
≤ ˜ ( / / )c n q2 2 1 1 2− + ν , ν = n + 1, … , 2n,
hde c̃ = c̃ ( s, q ) . No tohda
ν
ν
ν
ν
ν
= +
− − ′∑ ( )
n
n
s q
m
sn
l
d b
q
sn
1
2
1
12 ( / ) lin
≤ ˜ ( / )c n
n
n
s2 2
1
2
3 2−
= +
− −∑
ν
ν ≤ c s n2 1 2− −( / ) , (3.78)
hde c = c ( s, q ) . Podstavlqq ocenky (3.77) y (3.78) v (3.76), poluçaem
x A xm m
L I
n
q
( ) ( ), ,
( )
⋅ − ⋅+
…
+
0 2 ≤ c s n2 1 2− −( / ) , n ≥ 1, s > 1, (3.79)
hde c = c ( β, s, p, q ) .
Yz neravenstv (3.72) y (3.79) srazu Ωe sleduet
E Bs
p s
m n
L I
n
q
∆ Σ+ +
…( )
+
,
, ,
, ,
( )2
2
2
0
lin
≤ c s n2 1 2− −( / ) , n ≥ 0, s > 1, (3.80)
hde c = c ( β, s, p, q ) . Pry πtom v sylu (3.62) y (3.74) ymeem
dim ( )
, ,
, ,Σ
+
…
s
m m
n
n
22
0 2 ≤ 3 2s n , n ≥ 0, s > 1. (3.81)
Rassmotrym, nakonec, sluçaj, kohda s = 1 y ν* = λ n, n ≥ 1, hde λ : =
: = [ q / 2 + 1 ] — celaq çast\ çysla q / 2 + 1. V πtom sluçae çysla mν oprede-
lqem, polahaq
mν : =
2 0
2 12
ν ν
ν λ
, , , ,
, , , .
= …
= + …
n
n nn
(3.82)
Tohda qsno, çto
ν
λ
ν
=
∑
0
n
m ≤ λ ( )n n+ 1 2 , n ≥ 1. (3.83)
Oçevydno takΩe, çto
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
KOLMOHOROVSKYE Y LYNEJNÁE POPEREÇNYKY … 1649
d bm
n
lq
nν
ν
ν1( )lin
= d bn
n
lq
nν
ν
ν1( )lin
= 0, ν = 0, … , n. (3.84)
Yspol\zuq sootnoßenyq (3.71) y (3.84), poluçaem
x A x
m m
L I
n
q
( ) ( )
, ,
( )
⋅ − ⋅+
…
+
0 λ ≤ c c d bn q
n
n
q
m
n
lq
n2 2
1
1
−
= +
−+ ( )∑λ
ν
λ
ν
ν
ν
ν
/ / lin
, (3.85)
hde c = c ( β, p, q ) . Pry πtom qsno, çto
2−λn q/ ≤ 2 2−n/ , n ≥ 1. (3.86)
Krome toho, yz lemm¥ 2.4 y sootnoßenyj (3.82) sleduet
d bm
n
lq
nν
ν
ν1( )lin
≤ ˜ / /c n q2 22− ν , ν = n + 1, … , λ n ,
hde c̃ = ˜( )c q . No tohda
ν
λ
ν
ν
ν
ν
= +
−∑ ( )
n
n
q
m
n
l
d b
q
n
1
12 / lin
≤ c n nλ 2 2− / , (3.87)
hde c = c q( ). Podstavlqq ocenky (3.86) y (3.87) v (3.85), poluçaem
x A x
m m
L I
n
q
( ) ( )
, ,
( )
⋅ − ⋅+
…
+
0 λ ≤ cn n2 2− / , n ≥ 1, s = 1, (3.88)
hde c = c ( β, p, q ) . A yz neravenstv (3.72) y (3.88) srazu Ωe sleduet, çto v sluçae
s = 1 budet spravedlyva ocenka
E Bp
m m
L I
n
n
q
∆ Σ+ +
…( )
+
1
1 2
0,
, ,
, ,
( )
λ
λ
lin
≤ c n n( ) /+ −1 2 2, n ≥ 0, (3.89)
hde c = c ( β, p, q ) . Pry πtom v sylu (3.62) y (3.83) ymeem
dim
, ,
, ,
Σ
+
…( )1 2
0
λ
λ
n
nm m
≤ λ ( )n n+ 1 2 , n ≥ 0. (3.90)
Otmetym, çto podprostranstva Σ
+
…
, ,
, ,
s
m m
n
n
22
0 2
y Σ
+
…
, ,
, ,
1 2
0
λ
λ
n
nm m
opredelen¥ lyß\ na
promeΩutke I+ . Analohyçno m¥ opredelym podprostranstva Σ
−
…
, ,
, ,
s
m m
n
n
22
0 2
y
Σ
−
…
, ,
, ,
1 2
0
λ
λ
n
nm m
dlq promeΩutka I– . Qsno, çto tohda budut v¥polnqt\sq neravenstva
E Bs
p s
m m
L I
n
n
q
∆ Σ+ −
…( )
−
,
, ,
, ,
( )22
0 2
lin
≤ c s n2 1 2− −( / ) , n ≥ 0, s > 1, (3.91)
hde c = c ( β, s, p, q ) , y
dim
, ,
, ,Σ
−
…
s
m m
n
n
22
0 2 ≤ 3 2s n, n ≥ 0, s > 1. (3.92)
Esly Ωe s = 1, to
E Bp
m m
L I
n
n
q
∆ Σ+ −
…( )
−
1
1 2
0,
, ,
, ,
( )
λ
λ
lin
≤ c n n( ) /+ −1 2 2, n ≥ 0, (3.93)
hde c = c ( β, p, q ) , y
dim
, ,
, ,
Σ
−
…( )1 2
0
λ
λ
n
nm m
≤ λ ( )n n+ 1 2 , n ≥ 0. (3.94)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1650 V. N. KONOVALOV
Teper\ moΩno opredelyt\ neobxodym¥e podprostranstva splajnov na vsem
yntervale I. Pust\ σ+ +
…
+∈Σ
, ,
, , ( )
s
m m
n
n I
22
0 2
y σ− −
…
−∈Σ
, ,
, , ( )
s
m m
n
n I
22
0 2 , yly
σ λ
λ
+ +
…
+∈Σ
, ,
, ,
( )
1 2
0
n
nm m
I y σ λ
λ
− −
…
−∈Σ
, ,
, ,
( )
1 2
0
n
nm m
I . Polahaq σ σ( ) : ( )t t= + dlq t ∈( , )0 1 ,
σ σ( ) : ( )t t= − dlq t ∈ −( , )1 0 y σ σ σ( ) : ( ( ) ( ))/0 0 0 2= ++ − , opredelqem podprost-
ranstva Σ
s
m m
n
n I
,
, , ( )
22
0 2…
y Σ
1 2
0
,
, ,
( )λ
λ
n
nm m
I
…
na vsem yntervale I. Tohda yz (3.80),
(3.81), (3.91) y (3.92) sledugt neravenstva
E Bs
p s
m m
L I
n
n
q
∆ Σ+
…( ),
,
, ,
( )22
0 2
lin
≤ c s n2 1 2− −( / ) , n ≥ 0, s > 1, (3.95)
hde c = c ( β, s, p, q ) , y
dim
,
, ,Σ
s
m m
n
n
22
0 2…( ) ≤ 6 2 1s n + , n ≥ 0, s > 1. (3.96)
Esly Ωe s = 1, to yz (3.89), (3.90), (3.93) y (3.94) budut sledovat\ neravenstva
E Bp
m m
L I
n
n
q
∆ Σ+
…( )1
1 2
0,
,
, ,
( )
λ
λ
lin
≤ c n n( ) /+ −1 2 2, n ≥ 0, (3.97)
hde c = c ( β, p, q ) , y
dim
,
, ,
Σ
1 2
0
λ
λ
n
nm m…( ) ≤ 2 1 2 1λ ( )n n+ + , n ≥ 0. (3.98)
No tohda pry 2 < q < p ≤ ∞ yz (3.95) y (3.96) lehko poluçaem ocenky
d Bn
s
p Lq
( )∆+
kol ≤ d Bn
s
p Lq
( )∆+
lin ≤ cn s− +1 2/ , n ≥ 1, s ≥ 1, (3.99)
hde c = c ( β, s, p, q ) , a yz (3.97) y (3.98) — ocenky
d Bn p Lq
( )∆+
1 kol ≤ d Bn p Lq
( )∆+
1 lin ≤ cn n− +1 2 3 21/ /(ln( )) , n ≥ 1, (3.100)
hde c = c ( β, p, q ) . Neobxodym¥e ocenky sverxu v sootnoßenyqx (1.1) y (1.2)
sledugt yz ocenok (3.39), (3.99) y (3.100).
4. Dokazatel\stvo ocenok snyzu v teoreme,1.1. Dlq toho çtob¥ poluçyt\
neobxodym¥e ocenky snyzu v sootnoßenyqx (1.1) y (1.2), vnov\ vospol\zuemsq
texnykoj dyskretyzacyy. Razbyvaq ynterval I = ( – 1, 1 ) toçkamy
tn i,
∗ : = – 1 2+ i n/( ), i = 0, 1, … , 4n,
na ynterval¥
In i,
∗ : = ( ), ,,t tn i n i−
∗ ∗
1 , i = 1, … , 4n,
polahaem
Is n i, ,
∗ : = ( ), ,, ( ( ) )t t s nn i n i−
∗ ∗ −+ +1
12 1 , i = 1, … , 4n.
Oçevydno, çto In i,
∗ = ( )2 1n − y Is n i, ,
∗ = ( ( ) )2 1 1s n+ − , i = 1, … , 4n.
Pry s, n ∈ N y 1 ≤ q ≤ ∞ opredelqem operator¥ dyskretyzacyy
Ts n,
∗ : = Ts q n, ,
∗ : Lq � x → τ = ( , , )τ τ1 4… n ∈ R4n ,
zadavaq koordynat¥ τi , i = 1, … , 4n, vektora τ s pomow\g 4n ravenstv
τi : = 2 1
1
0
-s
s n i
q s k
s n i
k
s
I
I
s
k
x t k I dt
s n i
, ,
/
, ,( )
, ,
∗ − ′ − ∗
=
−
+( )
∑∫
∗
. (4.1)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
KOLMOHOROVSKYE Y LYNEJNÁE POPEREÇNYKY … 1651
Esly vospol\zovat\sq neravenstvom Hel\dera, to netrudno ubedyt\sq, çto
x L Iq( ) ≥ T xs n lq
n,
∗
4 , x L Iq∈ ( ) , 1 ≤ q ≤ ∞ . (4.2)
Polahaq
ϕ0, , ( )n i t : =
1
0
, ,
, ,
,
,\
t I
t I I
n i
n i
∈
∈
∗
∗ i = 1, … , 4n, (4.3)
opredelqem pry s ∈ N sledugwye funkcyy:
ϕs n i t, , ( ) : = ϕ τ τs n i
t
d−
−
∫ 1
1
, , ( ) , t ∈ I, i = 1, … , 4n. (4.4)
Pry s ∈ N polahaem
ψs n i t, , ( ) : = ϕ ϕs n i L I s n i t, , ( ) , , ( )
∞
−1
, t ∈ I, i = 1, … , 4n, (4.5)
y oboznaçaem çerez Ψs
n
s n i i
n4
1
4: { }, ,= =ψ sootvetstvugwyj nabor yz 4n funkcyj
ψs n i, , . Çerez S s
n
1
4+( )Ψ oboznaçym 1-sektor po systeme Ψs
n4 . Napomnym, çto
opredelenye (neotrycatel\n¥x) p -sektorov po zadann¥m systemam funkcyj
pryvedeno pered formulyrovkoj lemm¥ 2.5.
Oçevydno, çto ∆ Ψ+ ∞
+⊃s
s
nB S1
4( ) y, sledovatel\no,
d Bn
s
Lq
( )∆+ ∞
kol ≥ d Sn s
n
Lq
( ( ))1
4+ Ψ kol . (4.6)
Polahaq T Ts n s
n
s n s n i i
n
, , , ,: { }∗ ∗
==Ψ4
1
4ψ , çerez S Ts n r s
n
1
4+ ∗( ), ,Ψ oboznaçaem 1-sektor, po-
roΩdaem¥j systemoj Ts n r s
n
, ,
∗ Ψ4 . Qsno, çto v sylu (4.2) v¥polnqetsq neravenstvo
d Sn s
n
Lq
( ( ))1
4+ Ψ kol ≥ d S Tn s n r s
n
lq
n( ( )), ,1
4
4
+ ∗ Ψ kol . (4.7)
Uçyt¥vaq sootnoßenyq (4.3) – (4.5), ymeem
( ) , , , ,
, ,
−
+( )
− ∗
=
∑∫
∗
1
0
s k
s n i s n j
k
s
I
s
k
t k I dt
s n j
ψ =
=
I
I I
s n i
s
s s
s n j
s n j s n j
t t t dt dt dt
, ,
, , , ,
, ,
( ) ( )
∗
∗ ∗
∫ ∫ ∫… + +…+ …
0 0
1 1ψ =
=
I
I I
n i s
s n i L I
s
s n j
s n j s n j t t t
dt dt dt
, ,
, , , ,
, ,
, , ( )
( )
∗
∗ ∗
∞
∫ ∫ ∫…
+ +…+
…
0 0
0 1
1
ϕ
ϕ
.
No tohda
( ) , , , ,
, ,
−
+( )
− ∗
=
∑∫
∗
1
0
s k
s n i s n i
k
s
I
s
k
t k I dt
s n i
ψ =
Is n i
s
s n i L I
, ,
, , ( )
∗ +
∞
1
ϕ
, i = j, (4.8)
y
( ) , , , ,
, ,
−
+( )
− ∗
=
∑∫
∗
1
0
s k
s n i s n j
k
s
I
s
k
t k I dt
s n j
ψ = 0, i ≠ j . (4.9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1652 V. N. KONOVALOV
Pust\ E en i
i
n4
1
4: { }= = — standartnaq systema vektorov e1 : = ( 1, 0, … , 0 ) , …
… , e4n : = ( 0, … , 0, 1 ) v prostranstve R
4n . Tohda yz (4.1), (4.8) y (4.9) sleduet
Ts n s n i, , ,
∗ ψ = 2 12 1 1 1 1− − − − − − −+
∞
s q s q
s n i L I
s q is n e/ /
, , ( )
/( ) ϕ , i = 1, … , 4n. (4.10)
Krome toho, yz (4.3) y (4.4) sledugt neravenstva
ϕs n i L I, , ( )∞
≤ 2 2 1s n− − , i = 1, … , 4n. (4.11)
Pust\ c ss q s q∗ − + − − −= +: ( )/ /2 13 2 1 1
y a : = ( a1, … , a4n ) ∈ R
4n . Tohda yz (4.10) y
(4.11) sleduet, çto mnoΩestvo
c n S Es q n∗ − + ′ +1
1
4/ ( ) : = τ τ : , /= ≤
=
∗ − + ′∑a e a c ni
i
i
n
l
s q
n
1
4
1
1
4
soderΩytsq v 1-sektore S1
+
( ),Ts n s
n∗ Ψ4 , poroΩdaemom systemoj Ts n s
n
,
∗ Ψ4 .
Poπtomu
d S Tn s n s
n
l n( ( )),1
4
1
4
+ ∗ Ψ kol ≥ c n d S Es q
n
n
lq
n
∗ − + ′ +1
1
4
4
/ ( ( ))kol . (4.12)
Vospol\zovavßys\ lemmojI2.5, v kotoroj zamenym n na 4n, y poloΩyv m =
= n, poluçym
d S En
n
lq
n( ( ))1
4
4
+ kol ≥ c n q
∗
− − +( / / )1 2 1 , 1 ≤ q ≤ ∞ , (4.13)
hde c∗ = c q∗( ). No tohda yz (4.6), (4.7), (4.12) y (4.13) sleduet neravenstvo
d Bn
s r
Lq
( )∆+ ∞
kol ≥ cn s q− + ′min{ / , / }1 1 2 , n ≥ 1, 1 ≤ q ≤ ∞ , (4.14)
hde c = c ( s, q ) . Esly Ωe 1 ≤ q < p ≤ ∞ , to ocenky
d Bn
s
p Lq
( )∆+
lin ≥ d Bn
s
p Lq
( )∆+
kol ≥ cn s q− + ′min{ / , / }1 1 2 , n ≥ 1, s ≥ 1, (4.15)
hde c = c ( s, p, q ) , neposredstvenno sledugt yz (4.14). Na πtom dokazatel\stvo
ocenok snyzu v sootnoßenyqx (1.1) y (1.2) zaverßeno. Ostaetsq lyß\ obæedy-
nyt\ ocenky sverxu, ustanovlenn¥e v tret\em punkte, s ocenkamy snyzu (4.15),
çtob¥ zaverßyt\ dokazatel\stvo teorem¥I1.1.
1. Bullen P. S. A criterion for n-convexity // Pacif. J. Math. – 1971. – 36. – P. 81 – 98.
2. Roberts A. W., Varberg D. E. Convex functions. – New York: Acad. Press, 1973. – 300 p.
3. Pečarić J. E., Proschan F., Tong Y. L. Convex functions, partial orderings, and statistical applica-
tions // Math. Sci. and Eng. – Boston: Acad. Press, 1992. – 187.
4. Konovalov V. N. Formosoxranqgwye popereçnyky typa Kolmohorova klassov s-monoton-
n¥x yntehryruem¥x funkcyj // Ukr. mat. Ωurn. – 2004. – 55, # 7. – S. 901 – 926.
5. Konovalov V. N. Shape preserving widths of Kolmogorov-type of the classes of positive,
monotone, and convex integrable functions // E. J. Approxim. – 2004. – 10, # 1-2. – P. 93 – 117.
6. Konovalov V. N., Leviatan D. Kolmogorov and linear widths of weighted Sobolev-type classes on
a finite interval, II // J. Approxim. Theory. – 2001. – 113. – P. 266 – 297.
7. Konovalov V. N., Leviatan D. Shape-preserving widths of weighted Sobolev-type classes of
positive, monotone and convex functions on a finite interval // Constr. Approxim. – 2003. – 19. –
P. 23 – 58.
8. Konovalov V. N., Leviatan D. Shape preserving widths of Sobolev-type classes of s-monotone
functions on a finite interval // Isr. J. Math. – 2003. – 133. – P. 239 –268.
9. De Vore R. A., Lorentz G. G. Constructive approximation. – Berlin: Springer-Verlag, 1993. –
449 p.
10. Hluskyn E. D. Norm¥ sluçajn¥x matryc y popereçnyky koneçnomern¥x mnoΩestv // Mat.
sb. – 1986. – 120, # 1. – S. 180 – 189.
Poluçeno 26.10.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
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| id | umjimathkievua-article-3715 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:47:36Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cd/8ff2f99a7d7d15947ad18abf5310dbcd.pdf |
| spelling | umjimathkievua-article-37152020-03-18T20:02:57Z Kolmogorov and linear widths of classes of s-monotone integrable functions Колмогоровские и линейные поперечники классов s-монотонных интегрируемых функций Konovalov, V. N. Коновалов, В. Н. Коновалов, В. Н. Let $s \in \mathbb{N}$ and let $\Delta^s_+$ be the set of functions $x \mapsto \mathbb{R}$ on a finite interval $I$ such that the divided differences $[x; t_0, ... , t_s ]$ of order $s$ of these functions are nonnegative for all collections of $s + 1$ distinct points $t_0,..., t_s \in I$. For the classes $\Delta^s_+ B_p := \Delta^s_+ \bigcap B_p$ , where $B_p$ is the unit ball in $L_p$, we obtain orders of the Kolmogorov and linear widths in the spaces $L_q$ for $1 \leq q < p \leq \infty$. Нехай $s \in \mathbb{N}$ i $\Delta^s_+$ — множина функцій $x \mapsto \mathbb{R}$ на скінченному інтервалі $I$ таких, що поділені різниці $[x; t_0, ... , t_s ]$ порядку $s$ цих функцій є невід'ємними для всіх наборів з $s + 1$ різних точок $t_0,..., t_s \in I$. Для класів $\Delta^s_+ B_p := \Delta^s_+ \bigcap B_p$, де $B_p$ — одинична куля в $L_p$, знайдено порядки у просторах $L_q$ при $1 \leq q < p \leq \infty$ колмогоровських і лінійних поперечників. Institute of Mathematics, NAS of Ukraine 2005-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3715 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 12 (2005); 1633–1652 Український математичний журнал; Том 57 № 12 (2005); 1633–1652 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3715/4155 https://umj.imath.kiev.ua/index.php/umj/article/view/3715/4156 Copyright (c) 2005 Konovalov V. N. |
| spellingShingle | Konovalov, V. N. Коновалов, В. Н. Коновалов, В. Н. Kolmogorov and linear widths of classes of s-monotone integrable functions |
| title | Kolmogorov and linear widths of classes of s-monotone integrable functions |
| title_alt | Колмогоровские и линейные поперечники классов s-монотонных интегрируемых функций |
| title_full | Kolmogorov and linear widths of classes of s-monotone integrable functions |
| title_fullStr | Kolmogorov and linear widths of classes of s-monotone integrable functions |
| title_full_unstemmed | Kolmogorov and linear widths of classes of s-monotone integrable functions |
| title_short | Kolmogorov and linear widths of classes of s-monotone integrable functions |
| title_sort | kolmogorov and linear widths of classes of s-monotone integrable functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3715 |
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