Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities
We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of differentiable periodic functions by splines of defect 2. We obtain new sharp Jackson-type inequalities for the best approximations and the best one-sided approximations by splines of...
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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. |
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| datestamp_date | 2020-03-18T19:45:43Z |
| description | We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of differentiable periodic functions by splines of defect 2. We obtain new sharp Jackson-type inequalities for the best approximations and the best one-sided approximations by splines of defect 2. |
| first_indexed | 2026-03-24T02:36:29Z |
| 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)
NESYMMETRYÇNÁE PRYBLYÛENYQ
KLASSOV PERYODYÇESKYX FUNKCYJ
SPLAJNAMY DEFEKTA 2
Y NERAVENSTVA TYPA DÛEKSONA
We establish the exact values of the best (α, β)-approximations and the best one-sided approximations of
classes of differentiable periodic functions by splines of defect 2. We obtain new exact Jackson-type
inequalities for the best and best one-sided approximations by splines of defect 2.
Znajdeno toçni znaçennq najkrawyx (α, β)-nablyΩen\ i najkrawyx odnostoronnix nablyΩen\
klasiv dyferencijovnyx periodyçnyx funkcij splajnamy defektu 2. Otrymano novi toçni ne-
rivnosti typu DΩeksona dlq najkrawyx i najkrawyx odnostoronnix nablyΩen\ splajnamy
defektu 2.
1. Vvedenye. Pust\ Lp , 1 ≤ p ≤ ∞ , — prostranstva 2π-peryodyçeskyx funk-
cyj f : R R→ s sootvetstvugwymy normamy ⋅ Lp
= ⋅ p ; ′p = p p/( )− 1 ,
Cr , r = 0, 1, … , — prostranstvo r raz neprer¥vno dyfferencyruem¥x (nepre-
r¥vn¥x pry r = 0) 2π-peryodyçeskyx funkcyj.
Esly f Lp∈ y α, β — poloΩytel\n¥e çysla, to poloΩym
f f fp p; ,α β α β= ++ − ,
hde
f t f t± = ±{ }( ) max ( ), 0 .
Nayluçßym (α, β)-pryblyΩenyem funkcyy f Lp∈ mnoΩestvom H Lp⊂
v metryke Lp naz¥vaetsq velyçyna
E f H f hp
h H p( , ) : inf; , ; ,α β α β= −
∈
. (1)
Nayluçßee (α, β )-pryblyΩenye f Lp∈ podprostranstvom konstant budem
oboznaçat\ çerez E f p( ) ; ,α β .
Esly M Lp⊂ — nekotor¥j klass funkcyj, to velyçyna
E M H f hp
f M h H p( , ) inf; , ; ,α β α β= −
∈ ∈
sup (2)
naz¥vaetsq nayluçßym (α, β)-pryblyΩenyem klassa M mnoΩestvom H v met-
ryke Lp . Pry α = β = 1 velyçyn¥ (1) y (2) sovpadagt s ob¥çn¥m nayluçßym
Lp -pryblyΩenyem funkcyy f (oboznaçenye E f H p( , ) ) y klassa M (oboznaçe-
nye E M H p( , ) ) sootvetstvenno.
© V. F. BABENKO, N. V. PARFYNOVYÇ, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 11 1443
1444 V. F. BABENKO, N. V. PARFYNOVYÇ
Pust\ mnoΩestvo H Lp⊂ fyksyrovano. Sopostavym funkcyy f Lp∈
podmnoΩestva
H u t u H u t f t tf
+ = ∈ ≤ ≤ ≤{ }( ) : , ( ) ( ), 0 2π ,
H u t u H u t f t tf
− = ∈ ≥ ≤ ≤{ }( ) : , ( ) ( ), 0 2π .
Çerez E f H p
± ( , ) y E M H p
± ( , ) oboznaçym nayluçßye odnostoronnye
Lp -pryblyΩenyq funkcyy f y klassa M Lp∈ mnoΩestvom H sootvetstven-
no, t.>e.
E f H
f u u H H
H
p
p f f
f
±
± ±
±
=
− ∈{ } ≠ ∅
∞ = ∅
( , ) :
inf : , ,
, ,
y
E M H E f Hp
f M
p
±
∈
±=( , ) : sup ( , ) .
Velyçyn¥ E f H p
± ( , ) y E M H p
± ( , ) naz¥vagtsq nayluçßymy pryblyΩe-
nyqmy snyzu (+) y sverxu (–) funkcyy f Lp∈ y klassa M Lp⊂ soot-
vetstvenno.
V. F. Babenko [1, 2] (sm. takΩe [3], teorem¥ 1.4.10, 1.5.9) ustanovyl, çto esly
mnoΩestvo H Lp⊂ , 1 ≤ p < ∞, lokal\no kompaktno, to dlq lgboj funkcyy
f Lp∈ monotonno po α y β
lim ( , ) ( , ), ,β β→ ∞
+=E f H E f Hp p1 ,
lim ( , ) ( , ), ,α α→ ∞
−=E f H E f Hp p1 ,
a dlq lgboho mnoΩestva M Lp⊂ monotonno po α y β
lim ( , ) ( , ), ,β β→ ∞
+=E M H E M Hp p1 ,
(3)
lim ( , ) ( , ), ,α α→ ∞
−=E M H E M Hp p1 .
Çerez Lp
r
, r = 1, 2, … , oboznaçym mnoΩestvo 2π-peryodyçeskyx funkcyj, u
kotor¥x f f fr( ) ( ) :− =( )1 0
lokal\no absolgtno neprer¥vna, a f Lr
p
( ) ∈ .
Dlq vsex r = 1, 2, … , 1 ≤ p ≤ ∞ oboznaçym çerez Wp
r
klass funkcyj
f Lp
r∈ , u kotor¥x f p ≤ 1 .
Dlq natural\n¥x n y m çerez T n2 1− budem oboznaçat\ prostranstvo try-
honometryçeskyx polynomov porqdka ne v¥ße n – 1, a çerez S n m
k
2 , , k = 1, 2, —
prostranstvo 2π-peryodyçeskyx polynomyal\n¥x splajnov porqdka m defek-
ta k s uzlamy v toçkax k jπ / n, j ∈ Z .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 11
NESYMMETRYÇNÁE PRYBLYÛENYQ KLASSOV PERYODYÇESKYX FUNKCYJ… 1445
Pust\ ewe ϕ α βn m t, ( , ; ) , α, β > 0, m ∈ N , — 2π/n -peryodyçeskyj yntehral
porqdka m s nulev¥m srednym znaçenyem na peryode ot çetnoj 2π/n -pe-
ryodyçeskoj funkcyy ϕ α βn t, ( , ; )0 , kotoraq dlq t
n
∈
0,
π
opredelqetsq
sledugwym obrazom:
ϕ α β
α
πβ
α β
β
πβ
α β
πn t
t
n
n
t
n
, ( , ; )
,
( )
,
,
( )
0
0
=
≤ ≤
+
−
+
< ≤ ..
Pry α = β = 1 vmesto ϕ α βn m t, ( , ; ) budem pysat\ ϕn m t, ( ) .
Çerez
B t
kt m
k
m m
k
n
( )
( / )
=
−
=
∑ cos π 2
1
, m = 1, 2, … , (4)
oboznaçym qdro Bernully porqdka m (sm. [3, s. 36]) y poloΩym
�B t B tm
m
m( ) ( ) ( )= −2 1 .
Funkcyq
�Bm — πto (m – 1)-j peryodyçeskyj yntehral s nulev¥m srednym zna-
çenyem na peryode ot neçetnoj 2π-peryodyçeskoj funkcyy, ravnoj π – t dlq
t ∈ ( , )0 2π .
Otmetym (sm., naprymer, [3, s. 109]), çto pry m ≥ 2 y α → ∞
ϕ α1 1 0, ( , ; )m mB⋅ − →
∞
�
y
ϕ α1 1 1 1
1 0, ( , ; )⋅ − →�B .
Krome toho, pry vsex m ∈ N y α → ∞
E E Bm mϕ α1 1, ( , ; ) ( )⋅( ) →
∞ ∞�
. (5)
Yzvestno, çto dlq n, r, m ∈ N , m ≥ r – 1, p = 1, ∞
E W T
n
p
r
n p
r
r
,
,
2 1
1
−
∞( ) =
ϕ
(6)
y
E W S
n
p
r
n m p
r
r
, ,
,
2
1 1( ) = ∞
ϕ
. (7)
Ravenstvo (6) pry p = ∞ — πto rezul\tat Favara – Axyezera – Krejna (sm. [4 –
6]), pry p = 1 eho ustanovyl S.>M.>Nykol\skyj [7]. Ravenstvo (7) pry p = ∞ y
m = r – 1 ustanovleno V.>M.>Tyxomyrov¥m [8], a v ostal\n¥x sluçaqx —
A.>A.>Lyhunom [9].
Analohyçn¥e rezul\tat¥ dlq odnostoronnyx pryblyΩenyj klassov Wp
r
polynomamy poluçen¥ T.>Hanelyusom [10], a dlq nayluçßyx odnostoronnyx
pryblyΩenyj klassov Wp
r
splajnamy — V.>H.>Doronyn¥m y A.>A.>Lyhunom
[11]. Ymy b¥lo dokazano, çto pry n, m, r ∈ N , m ≥ r,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 11
1446 V. F. BABENKO, N. V. PARFYNOVYÇ
E W T E W S
E B
n
r
n
r
n m
r
r
±
−
± ∞( ) = ( ) =1 2 1 1 1 2
1
1
, ,
( )
,
�
. (8)
V. F. Babenko [1] ustanovyl, çto dlq n, r, m ∈ N , m ≥ r, pry vsex α, β > 0
E W T E W S
E
r
n
r
n m
r
1 2 1 1 1 2
1
1
1
, ,
(
; , , ; ,
,
−( ) = ( ) =
α β α β
ϕ αα β, ; )⋅( )∞
nr
. (9)
V rabote avtorov [12] poluçen, v çastnosty, analoh sootnoßenyq (7) pry p =
= 1 dlq nayluçßyx pryblyΩenyj splajnamy defekta 2: dlq n, r , m ∈ N , m ≥
≥ r, ymeet mesto ravenstvo
E W S
n
r
n m
r
r1 2
2
1
1
, ,
,( ) = ∞
ϕ
. (10)
V dannoj rabote m¥ najdem toçn¥e znaçenyq nayluçßyx (α, β )-pryblyΩe-
nyj E W Sr
n m1 2
2
1
, , ; ,
( ) α β
pry vsex α, β > 0 y nayluçßyx odnostoronnyx prybly-
Ωenyj E W Sr
n m
±( )1 2
2
1
, , dlq vsex n, r, m ∈ N , m ≥ r, a takΩe s pomow\g πtyx
rezul\tatov ustanovym nekotor¥e nov¥e neravenstva typa DΩeksona dlq nay-
luçßyx y nayluçßyx odnostoronnyx pryblyΩenyj v prostranstve L1 splaj-
namy yz S n m2
2
, (podrobnee o neravenstvax DΩeksona sm., naprymer, v [13]).
2. Nayluçßye ( , )αα ββ -pryblyΩenyq klassov W r
1 .
Teorema 1. Pust\ n, r, m ∈ N , m ≥ r, α, β > 0. Tohda
E W S
E
n
r
n m
r
r1 2
2
1
1
,
( , ; )
, ; ,
,( ) =
⋅( )∞
α β
ϕ α β
.
Pry α = β = 1 yz teorem¥ 1 sleduet (10). Krome toho, yz teorem¥ 1 v sylu
(3) y (5) v¥tekaet takoe sledstvye.
Sledstvye 1. Pust\ n, r, m ∈ N , m ≥ r. Tohda
E W S
E B
n
r
n m
r
r
± ∞( ) =1 2
2
1
,
( )
,
�
. (11)
Vezde dalee budem polahat\ t
j
n
j =
2π
, j ∈ Z . M¥ takΩe budem yspol\zo-
vat\ oboznaçenye g S n m⊥ 2
2
, , kotoroe oznaçaet, çto dlq lgboho s S n m∈ 2
2
,
g t s t dt( ) ( )
0
2
0
π
∫ = .
Pry πtom vmesto g t dt( )
0
2π
∫ = 0 budem pysat\ g ⊥ 1.
Dlq dokazatel\stva teorem¥ nam ponadobytsq sledugwee utverΩdenye, do-
kazannoe v [12].
Lemma 1. Pust\ n , m ∈ N , g L∈ 1 , g ⊥ 1, gm — m-q 2π-peryodyçeskaq
pervoobraznaq ot funkcyy g. Tohda dlq toho çtob¥ v¥polnqlos\ sootnoße-
nye g S n m⊥ 2
2
, , neobxodymo y dostatoçno v¥polnenyq uslovyj
g t g t g tm m m n+ + += = … =1 1 1 2 1( ) ( ) ( )
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 11
NESYMMETRYÇNÁE PRYBLYÛENYQ KLASSOV PERYODYÇESKYX FUNKCYJ… 1447
y
′ = ′ = … = ′ =+ + +g t g t g tm m m n1 1 1 2 1 0( ) ( ) ( ) .
Dokazatel\stvo teorem¥ 1. Yspol\zuq teoremu dvojstvennosty dlq nay-
luçßeho nesymmetryçnoho L1-pryblyΩenyq podprostranstvom (sm. [3], teore-
ma 1.4.9), poluçaem
E W Sr
n m
f W gr
1 2
2
1 11 1 1
, sup sup, ; , ,; ,
( ) =
∈ ≤∞ − −α β
α β
gg S n m
f t g t dt
⊥
∫
2
2
0
2
,
( ) ( )
π
. (12)
Yspol\zuq lemmu 1, sootnoßenye (12) posle r-kratnoho yntehryrovanyq po
çastqm moΩem zapysat\ v vyde
E W Sr
n m
f
f
g W
1 2
2
1 1
1
1 1
, sup sup, ; , ,
; ,
( ) =
≤
⊥
∈
∞ −
α β
α β−−
⊥
∫
1
2
2
0
2
r
r
n mg S
f t g t dt
,
( )
,
( ) ( )
π
=
= sup
; ,
,
( ) ( ),
( )
g W
g t g t
g t
m
n
∈
=…=
′ =…= ′
∞ − −
+
α β1 1
1
1
1 gg t
f
f
m r
n
f t g t dt
( )
,
( )sup ( ) ( )
=
≤
⊥
− +∫
0
1
1
1
0
2
1
π
=
= sup
; ,
,
( ) ( ),
( )
g W
g t g t
g t
m
n
∈
=…=
′ =…= ′
∞ − −
+
α β1 1
1
1
1 gg t
m r
n
E g
( )
( )
=
− +
∞( )
0
1 . (13)
PokaΩem, çto dlq lgboj funkcyy g W m∈ ∞
+ 1
takoj, çto
g t g t g tn( ) ( ) ( )1 2= = … =
y
′ = ′ = … = ′ =g t g t g tn( ) ( ) ( )1 2 0 ,
pry kaΩdom
|
| id | umjimathkievua-article-3113 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:36:29Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/26/b6e175093c9c0fb2e919fd0a1d4e9826.pdf |
| spelling | umjimathkievua-article-31132020-03-18T19:45:43Z Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities Несимметричные приближения классов периодических функций сплайнами дефекта 2 и неравенства типа Джексона Babenko, V. F. Parfinovych, N. V. Бабенко, В. Ф. Парфинович, Н. В. Бабенко, В. Ф. Парфинович, Н. В. We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of differentiable periodic functions by splines of defect 2. We obtain new sharp Jackson-type inequalities for the best approximations and the best one-sided approximations by splines of defect 2. Знайдено точні значення найкращих (α, β)-наближень i найкращих односторонніх наближень класів диференційовних періодичних функцій сплайнами дефекту 2. Отримано нові точні нерівності типу Джексона для найкращих і найкращих односторонніх наближень сплайнами дефекту 2. Institute of Mathematics, NAS of Ukraine 2009-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3113 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 11 (2009); 1443-1454 Український математичний журнал; Том 61 № 11 (2009); 1443-1454 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3113/2974 https://umj.imath.kiev.ua/index.php/umj/article/view/3113/2975 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. Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities |
| title | Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities |
| title_alt | Несимметричные приближения классов периодических функций сплайнами дефекта 2 и неравенства типа Джексона |
| title_full | Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities |
| title_fullStr | Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities |
| title_full_unstemmed | Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities |
| title_short | Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities |
| title_sort | nonsymmetric approximations of classes of periodic functions by splines of defect 2 and jackson-type inequalities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3113 |
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