On the problem of approximation of functions by algebraic polynomials with regard for the location of a point on a segment
We obtain a correction of an estimate of the approximation of functions from the class W r H ω (here, ω(t) is a convex modulus of continuity such that tω '(t) does not decrease) by algebraic polynomials with regard for the location of a point on an interval.
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| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3226 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509278029217792 |
|---|---|
| author | Motornyi, V. P. Моторный, В. П. Моторный, В. П. |
| author_facet | Motornyi, V. P. Моторный, В. П. Моторный, В. П. |
| author_sort | Motornyi, V. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:39Z |
| description | We obtain a correction of an estimate of the approximation of functions from the class W r H ω (here, ω(t)
is a convex modulus of continuity such that tω '(t) does not decrease) by algebraic polynomials with regard for the location of a point on an interval. |
| first_indexed | 2026-03-24T02:38:33Z |
| format | Article |
| fulltext |
UDK 517.5
V. P. Motorn¥j (Dnepropetr. nac. un-t)
K VOPROSU O PRYBLYÛENYY FUNKCYJ
ALHEBRAYÇESKYMY MNOHOÇLENAMY
S UÇETOM POLOÛENYQ TOÇKY NA OTREZKE
We obtain a correction of an estimate of the approximation of functions from the class W Hr ω (here,
ω( )t is a convex modulus of continuity such that t t′ω ( ) does not decrease) by algebraic polynomials
with regard for the location of a point on an interval.
OderΩano utoçnennq ocinky nablyΩennq funkcij klasu W Hr ω
( ω( )t — opuklyj modul\ ne-
perervnosti, takyj, wo t t′ω ( ) ne spada[) alhebra]çnymy mnohoçlenamy z uraxuvannqm poloΩen-
nq toçky na vidrizku.
1. Vvedenye. Zadaça o pryblyΩenyy funkcyj, zadann¥x na otrezke – ,1 1[ ], al-
hebrayçeskymy mnohoçlenamy s uçetom poloΩenyq toçky vperv¥e b¥la ras-
smotrena y reßena S. M. Nykol\skym [1] dlq klassa W∞
1
-funkcyj, udovletvo-
rqgwyx uslovyg Lypßyca s konstantoj, ravnoj edynyce. S. M. Nykol\skyj
ukazal lynejn¥j metod L f xn( ; ) pryblyΩenyq alhebrayçeskymy mnohoçlena-
my funkcyj yz klassa W∞
1
takoj, çto
f x L f xn( ) – ( ; ) ≤ π
2
1 2– x
n
+ O
x n
n
ln
2
, x ∈[ ]– ,1 1 , (1)
y pokazal, çto konstantu
π
2
v neravenstve (1) umen\ßyt\ nel\zq.
∏tot rezul\tat S. M. Nykol\skoho otkr¥l vozmoΩnost\ pryblyΩenyq fun-
kcyj, zadann¥x na otrezke, alhebrayçeskymy mnohoçlenamy s uluçßenyem pry-
blyΩenyq u koncov otrezka y v to Ωe vremq asymptotyçesky nayluçßee na vsem
klasse dlq razlyçn¥x klassov neperyodyçeskyx funkcyj.
Vvedem sledugwye klass¥ funkcyj. Pust\ W r
∞ , r > 0, — klass funkcyj
fr , predstavym¥x na otrezke – ,1 1[ ], v vyde
f xr( ) = 1 1
1
1
Γ( )
( – ) ( ) ( )–
–
r
x t f t dt P xr
+ +∫ ,
hde Γ( )r — hamma-funkcyq ∏jlera, xr
+
–1
— useçennaq stepen\, funkcyq f t( )
yzmeryma y f t( ) ≤ 1 poçty vsgdu, a P x( ) — alhebrayçeskyj mnohoçlen
stepeny ne v¥ße r – 1[ ] ( a[ ] — celaq çast\ a). V sluçae cel¥x r πto klass
funkcyj f, (r – 1)-q proyzvodnaq kotor¥x absolgtno neprer¥vna, a f tr( ) ≤ 1
poçty vsgdu. Çerez W Hr ω
, r = 0, 1, … (W H0 ω = Hω), budem oboznaçat\
klass funkcyj f, r-q proyzvodnaq ( f ( )0 = f ) kotor¥x udovletvorqet uslovyg
f x f xr r( ) ( )( ) – ( )1 2 ≤ ω x x1 2–( ) ,
hde ω( )t — zadann¥j modul\ neprer¥vnosty.
Dlq klassa W r
∞ (r — celoe) A. F. Tyman (sm. [2, s. 310 – 314]) dokazal sle-
dugwee utverΩdenye.
Dlq lgboho natural\noho çysla r > 1 suwestvuet lynejn¥j metod
U f xn r, ( ; ) pryblyΩenyq funkcyj yz klassa W r
∞ takoj, çto dlq lgboj funk-
cyy f Wr∈ ∞ ymeet mesto neravenstvo
© V. P. MOTORNÁJ, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8 1087
1088 V. P. MOTORNÁJ
f x U f xn r( ) – ( ; ), ≤
K
n
x or
r
r
1 12– ( )( ) +
(2)
y konstantu Kr ( Kr — konstanta Favara) na klasse Wr
∞ umen\ßyt\
nel\zq.
Zametym, çto konstanta, opredelqgwaq ostatoçn¥j çlen v (2), zavysyt ot
funkcyy f.
Takym obrazom, dlq kaΩdoho natural\noho çysla r b¥l ukazan lynejn¥j
metod pryblyΩenyq, osuwestvlqgwyj asymptotyçesky nayluçßee pryblyΩe-
nye klassa Wr
∞ alhebrayçeskymy mnohoçlenamy v ravnomernoj metryke y v to
Ωe vremq kaΩdug funkcyg yz klassa Wr
∞ u koncov otrezka – ,1 1[ ] prybly-
Ωagwyj luçße. V rabotax N. P. Kornejçuka y A. Y. Polovyn¥ [3 – 5] ustanov-
leno, çto analohyçnaq ocenka s hlavn¥m çlenom, zavysqwym ot x, ymeet mesto
y v bolee obwem sluçae — kohda uçyt¥vaetsq povedenye modulq neprer¥vnosty
funkcyy yly modulq neprer¥vnosty proyzvodnoj. Odnako pryblyΩenye v πtom
sluçae osuwestvlqetsq nelynejn¥m metodom. Pryvedem osnovnoj rezul\tat
rabot¥ [5].
Pust\ ω( )t — v¥pukl¥j modul\ neprer¥vnosty. Tohda dlq lgboj funkcyy
f H∈ ω
suwestvuet posledovatel\nost\ alhebrayçeskyx mnohoçlenov
P f xn( ; ){ } stepeny n = 1, 2, … takaq, çto ravnomerno otnosytel\no vsex
x ∈ – ,1 1[ ] pry n → ∞ v¥polnqetsq neravenstvo
f x P f xn( ) – ( ; ) ≤ 1
2
1 2ω π
n
x–
+ o
n
ω 1
. (3)
Dlq neçetn¥x r A. A. Lyhun [6] poluçyl sledugwee obobwenye.
Dlq lgboho neçetnoho çysla r suwestvuet lynejn¥j metod pryblyΩenyq
Q f xn r, ( ; ) takoj, çto dlq lgboj funkcyy f, ymegwej neprer¥vnug proyzvod-
nug r-ho porqdka, v¥polnqetsq neravenstvo
f x Q f xn r( ) – ( ; ), ≤
K x
n
f x nr
r
r
2
1
1
2
2–
; – /( )
( )ω π + o n f nr r– ( ); /ω 1( )( ) ,
(4)
hde ω f tr( );( ) — modul\ neprer¥vnosty r-j proyzvodnoj funkcyy f x( ).
V ukazann¥x rabotax obobwenye teorem¥ S. M. Nykol\skoho soprovoΩda-
los\ ohrublenyem ostatoçnoho çlena. Poπtomu sledugwyj ßah, svqzann¥j s
razvytyem ukazann¥x yssledovanyj S. M. Nykol\skoho, sostoql v utoçnenyy os-
tatoçnoho çlena v neravenstvax (1) – (4). Perv¥m osuwestvyl eho V. N. Temlq-
kov [7]; on usylyl neravenstvo (1), ubrav ln n v ostatoçnom çlene. Pry πtom
pryblyΩenye funkcyj yz klassa W∞
1
uΩe osuwestvlqlos\ nelynejn¥m me-
todom.
Dlq lgboho natural\noho çysla r ≥ 2 R. M. Tryhubom [8] dokazan sledug-
wyj rezul\tat.
Dlq lgboj funkcyy f Wr∈ ∞ (r — natural\noe çyslo, bol\ßee yly ravnoe
2) suwestvuet posledovatel\nost\ alhebrayçeskyx mnohoçlenov p xn( ), n = r –
– 1, r,H… , udovletvorqgwyx neravenstvu
f x p xn( ) – ( ) ≤ K
x
nr
r
1 2–
+ c
x
n
r
r
r
1 2 1
1
–
–( )
+ ,
hde konstanta cr zavysyt ot r.
Sluçaj neceloho r rassmotren v rabotax [9, 10]. PryblyΩenye funkcyj yz
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
K VOPROSU O PRYBLYÛENYY FUNKCYJ ALHEBRAYÇESKYMY … 1089
klassa W Hr ω
alhebrayçeskymy mnohoçlenamy s uçetom poloΩenyq toçky na
otrezke yssledovalos\ v rabotax [11, 12], hde ustanovlen sledugwyj rezul\tat.
Pust\ ω( )t — v¥pukl¥j modul\ neprer¥vnosty takoj, çto funkcyq t t′ω ( )
ne ub¥vaet. Tohda dlq lgboj funkcyy f W Hr∈ ω
suwestvuet posledova-
tel\nost\ alhebrayçeskyx mnohoçlenov Q f xn
r( ; ) stepeny n = r, r + 1, … (pry
r = 0 n ≥ 1) takyx, çto v¥polnqetsq neravenstvo
f x Q f xn
r( ) – ( ; ) ≤
K x
n
K
K n
xr
r
r
r2
1 2
1
2
1 2–
–
+ω +
+ C
x
n
x
n n
n
n
r
r
r
1 1 1 12
1 2
2
1
–
–
ln
–
+
+
ω
, (5)
hde Kr — konstanta Favara, a velyçyna Cr zavysyt tol\ko ot r.
Ocenku (5) uluçßyt\ odnovremenno dlq vsex modulej neprer¥vnosty nel\-
zq. Odnako v sluçae r = 0 dlq modulej neprer¥vnosty ω( )t , kotor¥e medlen-
no stremqtsq k nulg, kohda t stremytsq k nulg, ostatoçn¥j çlen v neraven-
stve (5) v toçkax ± 1 moΩet ne stremyt\sq k nulg. Naprymer, esly ω( )t =
= 1 / (2 – ln )t dlq t ∈( ]0 1; y ω( )0 = 0, to ω( / ) ln1 2n n → 0,5 pry n → ∞ . V
nastoqwej rabote dokaz¥vaetsq, çto dlq x, raspoloΩenn¥x vblyzy koncov ot-
rezka – ,1 1[ ], proyzvedenye ln –n xω 1 2( / n + 1 2/n ) v neravenstve (5) moΩno
zamenyt\ na ω(ln / )n n2
.
Suwestvuet hypoteza, çto ln n v neravenstve (5) voobwe moΩno ubrat\.
Osnovn¥m rezul\tatom nastoqwej rabot¥ qvlqetsq sledugwaq teorema.
Teorema. Pust\ ω( )t — v¥pukl¥j modul\ neprer¥vnosty takoj, çto funk-
cyq t t′ω ( ) ne ub¥vaet. Tohda dlq lgboj funkcyy f W Hr∈ ω
suwestvuet
posledovatel\nost\ alhebrayçeskyx mnohoçlenov Q f xn
r( ; ) stepeny n = r, r +
+ 1, … (pry r = 0 n ≥ 1) takyx, çto v¥polnqetsq neravenstvo
f x Q f xn
r( ) – ( ; ) ≤
K x
n
K
K n
xr
r
r
r2
1 2
1
2
1 2–
–
+ω +
+ C
x
n
n
n
n
r
r
r
1 12
2–
ln+
ω
,
hde Kr — konstanta Favara, a velyçyna Cr zavysyt tol\ko ot r.
2. Neobxodym¥e opredelenyq y rezul\tat¥. Pust\
D tr( ) = 1 2
1π
π
cos –kt
r
kr
k
=
∞
∑ , r = 1, 2, … ,
— qdro Bernully y P tn
r( ) — tryhonometryçeskyj polynom stepeny ne v¥ße n
nayluçßeho L1-pryblyΩenyq D tr( ). Tohda ymeet mesto neravenstvo [10]
D t P t t dtr n
r( ) – ( ) sin
–π
π
∫ 2
≤ C
n
n
r r
ln
+1 . (6)
Zameçanye 1. Vsgdu v dal\nejßem absolgtn¥e konstant¥ budem obozna-
çat\ symvolom C, a konstant¥, zavysqwye ot parametra r, — symvolamy Cr ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1090 V. P. MOTORNÁJ
xotq v razn¥x mestax ony mohut ymet\ razlyçn¥e znaçenyq.
Lemma 1 [12]. Pust\ ω( )t — v¥pukl¥j modul\ neprer¥vnosty takoj, çto
t t′ω ( ) ne ub¥vaet. Tohda dlq lgb¥x poloΩytel\n¥x çysel b, s y τ ymeet
mesto neravenstvo
s bs b′ ′ω τω τ( ) – ( ) ≤ ′ω τ τ( ) –b s .
Pust\ x0 = 0 y pry n ≥ 5
xk = xk –1 + a
n
xk1 1
2– – , k = 1, 2, … , (7)
— toçky otrezka 0 1,[ ], hde a ∈[ ]1, π — nekotoroe postoqnnoe çyslo, kotoroe
pry dokazatel\stve teorem¥ budem v¥byrat\ v zavysymosty ot parametra r.
Oboznaçym çerez xN –1 naybol\ßug yz tex toçek, dlq kotor¥x v¥polnqetsq
neravenstvo xN –1 ≤ x , hde çyslo x = xn < 1 takoe, çto x + a
n
x1 2– = 1.
Esly xN –1 = x , to yz (7) sleduet, çto xN = 1, a esly xN –1 < x , to po oprede-
lenyg sçytaem, çto xN = 1. PoloΩym Ek = – , –x xk k+[ ]1 ∪ x xk k, +[ ]1 , k = 0,
1, … , N – 1.
Pust\ MH — klass funkcyj f, zadann¥x na otrezke −[ ]1 1; y udovletvorq-
gwyx uslovyg Lypßyca s konstantoj M : f x f x( ) – ( )1 2 ≤ M x1 – x2 . Obozna-
çym çerez φk a f x, ( ; ) funkcyg yz klassa M Hk , suwestvovanye kotoroj dlq za-
dannoj funkcyy f x( ) yz klassa Hω
( ω( )t — v¥pukl¥j modul\ neprer¥vnos-
ty) ustanovleno N. P. Kornejçukom [13, 14], takug, çto
f x f xk a( ) – ( ; ),φ ≤ ∆k , t ∈ −[ ]1 1; , (8)
hde Mk = ′
ω a
n
xk1 1
2– – , a
∆k = 1
2
1 11
2
1
2ω a
n
x M a
n
xk k k– – –– –
, k = 1, 2, … , N, a ∈[ ]1, π .
V rabote [12] dokazan¥ sledugwye utverΩdenyq.
Lemma 2. Ymegt mesto neravenstva
∆k – ∆k +1 < ( – )( – )M M x xk k k k+ +1 1 , k = 1, 2, … , N – 1.
Lemma 3. Pust\ ω( )t — v¥pukl¥j modul\ neprer¥vnosty. Tohda dlq lg-
boj funkcyy f H∈ ω
y lgboho çysla a ∈[ ]1, π suwestvuet posledovatel\-
nost\ absolgtno neprer¥vn¥x funkcyj ψn a f x, ( ; ){ } takyx, çto:
1) poçty vsgdu v¥polnqetsq neravenstvo
′ψn a f x, ( ; ) ≤ Mk +1, x Ek∈ , k = 0, 1, … , N – 1;
2) f x( ) – ψn a f x, ( ; ) ≤ ∆k , x Ek∈ ,, k = 0, 1, … , N – 1, hde Mk =
= ′
ω a
n
xk1 1
2– – , a ∆k = 1
2
1 1
2ω a
n
xk– –
– M a
n
xk k1 1
2– –
, k = 1, 2, …
… , N.
Dlq dokazatel\stva teorem¥ neobxodymo neskol\ko vydoyzmenyt\ lemmuH3.
Vvedem dlq πtoho ewe çyslo m kak naybol\ßee, dlq kotoroho v¥polnqetsq
neravenstvo 1 2– xm ≥ ln /n an.
Lemma 4. Pust\ v¥polnqetsq uslovye lemm¥H3. Tohda dlq lgboj funkcyy
f H∈ ω
y lgboho çysla a ∈[ ]1, π suwestvuet posledovatel\nost\ absolgt-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
K VOPROSU O PRYBLYÛENYY FUNKCYJ ALHEBRAYÇESKYMY … 1091
no neprer¥vn¥x funkcyj ˜ ( ; ),ψn a f x{ } takyx, çto:
1) poçty vsgdu v¥polnqetsq neravenstvo
˜ ( ; ),′ψn a f x ≤ Mk +1, x Ek∈ , k = 0, 1, … , m,
˜ ( ; ),′ψn a f x ≤ Mm +1, x ≥ xm +1;
2) f x( ) – ˜ ( ; ),ψn a f x ≤ ∆k , x Ek∈ , k = 0, 1, … , m, y f x( ) – ˜ ( ; ),ψn a f x ≤
≤ ∆m , x ≥ xm +1, hde çysla Mk y ∆k ymegt tot Ωe sm¥sl, çto y v lem-
meH3.
Dokazatel\stvo. Na otrezke – ;x xm m+ +[ ]1 1 poloΩym ˜ ( ; ),ψn a f x =
= ψn a f x, ( ; ) , hde ψn a f x, ( ; ) — funkcyq, suwestvovanye kotoroj sleduet yz
lemm¥H3, y opredelym funkcyg ˜ ( ; ),ψn a f x na otrezke x ∈ xm +( ]1 1; . Na otrezke
x ∈ – ; –1 1xm +[ ) πta funkcyq doopredelqetsq analohyçno. Yz dokazatel\stva
lemm¥H3 sleduet [12], çto v toçke xm +1 (tak budet y v lgboj toçke xi ∈( ]0 1; )
lybo ψn a mf x, ( ; )+1 = φm a mx+ +1 1, ( ), lybo ψn a mf x, ( ; )+1 < φm a mx+ +1 1, ( ). V per-
vom sluçae poloΩym ˜ ( ; ),ψn a f x = φm a x+1, ( ) dlq x ∈ xm +( ]1 1; . V sylu svojstv
funkcyj φk a x, ( ) y monotonnosty velyçyn ∆k yz neravenstva (8) sleduet
utverΩdenye lemm¥H4 dlq x ∈[ ]0 1; . Vo vtorom sluçae, kak sleduet yz dokaza-
tel\stva lemm¥H3 (sm. [12]), suwestvuet toçka xk takaq, çto ψn a kf x, ( ; ) =
= φk a kf x, ( ; ) , a na yntervalax ( ; )x xk j k j+ + +1 , j = 0, 1, … , m – k, funkcyq
ψn a f x, ( ; ) < φk j a f x+ +1, ( ; ) , pryçem na kaΩdom yz πtyx yntervalov funkcyq
ψn a f x, ( ; ) lynejna y ymeet proyzvodnug, ravnug Mk j+ +1 , t.He. na otrezkax
x xk j k j+ + +[ ]; 1 , j = 0, 1, … , m – k,
ψn a f x, ( ; ) = lk j, ≡ φk kx( ) + M x xk i k i k i
i
j
+ + +
=
∑ ( – )–1
1
+ M t xk j k j+ + +1( – ).
PoloΩym dlq x xm∈( ]+1 1; ˜ ( ; ),ψn a f x = min ( ), –l xk m k{ , φm a f x+ }1, ( ; ) . Oçevydno,
çto ˜ ( ; ),′ψn a f x ≤ Mm +1 dlq x xm∈( ]+1 1; . Ocenym uklonenye f x( ) –
– ˜ ( ; ),ψn a f x na πtom otrezke. Poskol\ku φm a f x+1, ( ; ) ≥ ˜ ( ; ),ψn a f x , v sylu (8)
˜ ( ; ),ψn a f x – f x( ) ≤ φm a f x+1, ( ; ) – f x( ) ≤ ∆m +1.
Esly ˜ ( ; ),ψn a f x = l xk m k, – ( ) , to, yspol\zuq (8), opredelenye funkcyy
˜ ( ; ),ψn a f x y neravenstvo φk a f x, ( ; ) ≤ φk a kf x, ( ; ) + M x xk k( – ), poluçaem
f x( ) – ˜ ( ; ),ψn a f x = f x( ) – φk a f x, ( ; ) + φk a f x, ( ; ) – ˜ ( ; ),ψn a f x ≤
≤ ∆k + φk kf x( ; ) + M x xk k( – ) – l xk m k, – ( ) =
= ∆k – ( – )( – )–M M x xk i k
i
m k
k i k i+
=
−
+ +∑
1
1 – ( – )( – )M M x xk j k m+ +1 ≤
≤ ∆k – ( – )( – )– –M M x xk i k i
i
m k
k i k i+ +
=
−
+ +∑ 1
1
1 .
Prymenqq teper\ lemmuH2, ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1092 V. P. MOTORNÁJ
f t( ) – ˜ ( ; ),ψn a f x < ∆k – ( – )–∆ ∆k i k i
i
m k
+ +
=
−
∑ 1
1
= ∆m .
Esly ˜ ( ; ),ψn a f x = φm f x+1( ; ) v nekotoroj toçke x , to v sylu (8) f t( ) –
– ˜ ( ; ),ψn a f x ≤ ∆m +1.
Lemma dokazana.
3. Dokazatel\stvo teorem¥. PreΩde vseho zametym, çto metod dokaza-
tel\stva faktyçesky sovpadaet s predloΩenn¥m v rabote [12], v çastnosty,
pryblyΩagwye mnohoçlen¥ opredelqgtsq podobno tomu, kak πto b¥lo sdelano
v ukazannoj rabote. Snaçala dokaΩem teoremu dlq r = 0. Pust\ f x H( ) ∈ ω y
˜ ( )ψ x = ˜ ( ; ),ψ πn f x — funkcyq, suwestvovanye kotoroj ustanovleno v lemmeH4.
Tak kak dlq x x xk k∈[ ]+, 1 1 2– x ≤ 1 2– xk , to ′
ω π
n
xk1 2– ≤
≤ ′
ω π
n
x1 2– y v sylu lemm¥H4
˜ ( )′ψ x ≤ ′
ω π
n
x1 2– , x x xm m∈[ ]+ +– ,1 1 , (9)
a dlq x ≥ xm +1, v sylu utverΩdenyq 1 lemm¥H4, s uçetom toho, çto 1 2– x <
< ln /n an, ymeet mesto neravenstvo
˜ ( )′ψ x ≤ ′( )ω ln /n n2
. (10)
Yz ocenok (9), (10) sledugt neravenstva
˜ ( )′ψ x ≤ ′
ω π
n
x1 2– , (11)
˜ ( )′ψ x ≤ ′( )ω ln /n n2 , x ∈[ ]– ,1 1 . (12)
Ocenka promeΩutoçnoho pryblyΩenyq funkcyy f x( ) funkcyej ˜ ( )ψ x polu-
çena v lemmeH4. Çtob¥ poluçyt\ pryblyΩenye ˜ ( )ψ x , rassmotrym alhebrayçes-
kyj mnohoçlen Q f xn
0( ; ) takoj, çto
Q f tn
0( ; cos ) = – ( – ) ˜ (cos ) sin
–
P t u u udun
1 ′∫ ψ
π
π
,
hde P tn
1( ) — tryhonometryçeskyj polynom stepeny ne v¥ße n nayluçßeho L1-
pryblyΩenyq qdra D t1( ) . Tohda posle zamen¥ x = cos t , t ∈[ ]0, π , yspol\zuq
neravenstvo (11), kak y v [12], poluçaem
˜ ( ) – ( ; )ψ x Q f xn
0 = ˜ (cos ) ( – ) ˜ (cos ) sin
–
ψ ψ
π
π
t P t u u udun+ ′∫ 1 =
=
–
( – ) – ( – ) ˜ (cos ) sin
π
π
ψ∫ { } ′D t u P t u u u dun1
1 ≤
≤
–
( – ) – ( – ) sin sin
π
π
ω π∫ ′
D t u P t u
n
u u dun1
1 ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
K VOPROSU O PRYBLYÛENYY FUNKCYJ ALHEBRAYÇESKYMY … 1093
≤ ′
∫ω π
π
π
n
t t D t u P t u dunsin sin ( – ) – ( – )
–
1
1 +
+
–
( – ) – ( – ) sin sin – sin sin
π
π
ω π ω π∫ ′
′
D t u P t u
n
u u
n
t t dun1
1 =
=
π ω π
2n n
t t′
sin sin + ∆ ( )t . (13)
Dlq sin t ≥ 1 1
2– –xN v [12] (sm. neravenstvo (17)) poluçena ocenka velyçyn¥
∆ ( )t :
∆ ( )t ≤ ′
ω π
n
t
n
n
sin
ln
2 = ′
ω π
n
x
n
n
1 2
2–
ln
. (14)
Pust\ toçka x̃ ∈ x xm m; +[ ]1 takaq, çto 1 2– x̃ = ln /n nπ , tohda dlq x ≤ x̃
′
ω π
n
x
n
n
1 2
2–
ln
≤ ′
ω π
π
ln lnn
n
n
n2 2 ≤ ω ln n
n2
. (15)
Ocenym uklonenye mnohoçlena Q f xn
0( ; ) ot funkcyy f x( ) dlq x ≤ x̃ .
Uçyt¥vaq neravenstva (13) – (15), ocenku promeΩutoçnoho pryblyΩenyq funk-
cyy f x( ) funkcyej ˜ ( )ψ x , poluçennug v lemmeH4, y monotonnost\ funkcyy
t t′ω ( ) , dlq x Ek∈ , k = 1, 2, … , m, kak y v rabote [12] (sm. neravenstvo (20)),
ymeem
f x Q f xn( ) – ( ; )0 ≤ f x x( ) – ˜ ( )ψ + ˜ ( ) – ( ; )ψ x Q f xn
0 ≤
≤ 1
2
1 2ω π
n
x–
+ C
n
x
n
n
′
ω π 1 2
2–
ln
. (16)
Otsgda dlq x ≤ x̃ sleduet ocenka
f x Q f xn( ) – ( ; )0 ≤ 1
2
1 2ω π
n
x–
+ C
n
n
ω ln
2
. (17)
Pust\ teper\ x ≥ x̃ . V πtom sluçae, yspol\zuq neravenstva (12) y (6), po-
luçaem
f x Q f xn( ) – ( ; )0 ≤ f x x( ) – ˜ ( )ψ + ˜ ( ) – ( ; )ψ x Q f xn
0 ≤
≤ ∆m nD t u P t u
n
u u du+ ′
∫
–
( – ) – ( – ) sin sin
π
π
ω π
1
1 ≤
≤ ∆m n
n
n
t D t u P t u du+ ′
∫ω
π
π
ln
sin ( – ) – ( – )
–
2 1
1 +
+ ′
∫ω
π
π
ln
( – ) – ( – ) sin – sin
–
n
n
D t u P t u u t dun2 1
1 ≤
≤ ∆m
n
n
n
n n
+ ′
ω πln ln
2 2 2
+ C
n
n
n
n
′
ω ln ln
2 2 ≤ C
n
n
ω ln
2
. (18)
Yz (17), (18) sleduet teorema dlq r = 0.
Zameçanye 2. Yz neravenstv (16), (18) vsledstvye v¥puklosty modulq ne-
prer¥vnosty (proyzvodnaq ne vozrastaet) sleduet neravenstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1094 V. P. MOTORNÁJ
f x Q f xn( ) – ( ; )0 ≤ 1
2
1 2ω π
n
x–
+ C
n
n
n
′
ω 1
2 2
ln
.
Suwestvugt moduly neprer¥vnosty takye, çto
′
ω 1
2 2n
n
n
ln
≤ C
n
ω 1
2
.
Naprymer, ω( )t = ( – ln )2 t α
, t ∈( ]0 1; , ω( )0 = 0, α ∈( ]0 1; . Poπtomu dlq takyx
modulej neprer¥vnosty ostatoçn¥j çlen v (17), (18) moΩno zamenyt\ na
C
n
ω 1
2
.
V sluçae r > 0 moΩno ohranyçyt\sq funkcyqmy yz klassa W Hr ω
, predsta-
vym¥my v vyde
f xr( ) = 1
1
1
1
( – )!
( – ) ( )–
–
r
x t f t dtr
x
∫ ,
hde f H∈ ω
. Pust\
I f u tr (cos ) ( )( ) = f u D t u dur(cos ) ( – )
0
2π
∫ .
PoloΩym
f tk (cos ) = (–sin ) (cos ) ( )t I f u tk
k ( ) + R tk ( ), k = 1, 2, … . (19)
Netrudno proveryt\, çto dlq lgboho k = 2, 3, … ymeet mesto ravenstvo
d
dx
R tk ( ) = –sin ( )–t R tk 1 + k t t I f u tk
k(–sin ) cos (cos ) ( )–1 ( ) , (20)
a
d
dx
R t1( ) = cos (cos ) ( ) –
sin
(cos )t I f u t
t
f t dt1
0
2
2
( ) ∫π
π
. (21)
Rekurrentnaq zavysymost\ (20), (21) pozvolqet najty (sm. [12], lemmuH8) ocenku
pryblyΩenyq funkcyy R tk ( ) , znaq pryblyΩenye funkcyj I f uk (cos )( ) ( )t y
R tk – ( )1 .
Lemma 5. Pust\ ω( )t — v¥pukl¥j modul\ neprer¥vnosty takoj, çto
funkcyq t t′ω ( ) ne ub¥vaet. Tohda dlq lgboj funkcyy f H∈ ω
y lgboho na-
tural\noho çysla r suwestvuet posledovatel\nost\ alhebrayçeskyx mnoho-
çlenov T f tn
r( ; ) stepeny n ≥ 2 takyx, çto v¥polnqetsq neravenstvo
I f u t T f tr n
r(cos ) ( ) – ( ; )( ) ≤
K
n
K
K n
tr
r
r
r2
2 1ω +
sin + C
n
n
n
r r
ω ln
2
, t ∈[ ]0, π , (22)
hde Kr — konstanta Favara, a velyçyna Cr zavysyt tol\ko ot r.
Dokazatel\stvo. Pust\ f H∈ ω
, ar =
2 1K
K
r
r
+
, r N∈ , y ˜ ( )ψ x ≡ ˜ ( ; ),ψn ar
f x
— funkcyq, postroennaq dlq f y dannoho çysla ar v sootvetstvyy s lemmojH4.
Predstavym I f ur (cos )( ) ( )t v vyde
I f u tr (cos ) ( )( ) = I f u u tr (cos ) – ˜ cos ( )ψ( )( ) + I u tr ˜ (cos ) ( )ψ( ) = V t1( ) + V t2( )
y pryblyzym kaΩdoe slahaemoe pravoj çasty. Pervoe budem approksymyrovat\
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
K VOPROSU O PRYBLYÛENYY FUNKCYJ ALHEBRAYÇESKYMY … 1095
tryhonometryçeskym polynomom
A tn
r( ) = P t u f u u dun
r( – ) (cos ) – (cos )
–
ψ
π
π
( )∫ ,
hde P tn
r( ) — tryhonometryçeskyj polynom stepeny ne v¥ße n nayluçßeho
L1-pryblyΩenyq qdra D tr( ).
Pust\ tk , k = 0, 1, … , N, — toçky otrezka 0 2, /π[ ] takye, çto cos tk = xk y
Fk = ( ; )t tk k+1 ∪ ( –π tk ; π – )tk +1 ∪ (–tk ; – )tk +1 ∪ (–π + +tk 1; – )π + tk , k = 0,
1, … , m – 1, Fm = (– ; )t tm m ∪ ( –π tm ; π) ∪ (–π ; tm – )π . Tohda, yspol\zuq
lemmuH4 y uçyt¥vaq çetnost\ funkcyj f u(cos ) y ˜ (cos )ψ u , poluçaem
V t A tn
r
1( ) – ( ) ≤ 1
2 0
1D t u P t u
a
n
tr n
r
Fk
m
r
k
k
( – ) – ( – ) sin –∫∑
=
{ω –
– ′
}ω a
n
t
a
n
t dur
k
r
ksin sin– –1 1 ≤ 1
2
ω αa
n
tr sin +( )
{ –
– ′ +( )
+( )} ∫ω α α
π
a
n
t
a
n
t D u P u dur r
r n
rsin sin ( ) – ( )
0
2
+
+
1
2 0
1
k
m
F
r n
r r
k
r
k
D t u P t u
a
n
t
a
n
t du
=
∑ ∫
+( )
{ }( – ) – ( – ) sin – sin–ω ω α +
+
a
n
D t u P t u
a
n
t tr
k
m
F
r n
r r
k
2 0=
∑ ∫ ′ +
{ +( – ) – ( – ) (sin ) (sin )ω α α –
– ′
}ω a
n
t t dur
k k(sin ) sin– –1 1 ≡ I1
0 + I1
1 + I1
2 , (23)
hde α — poloΩytel\naq konstanta, kotoraq v zavysymosty ot t budet oprede-
lena pozΩe. Oçevydno, çto
I1
0 =
K
n
K
K n
tr
r
r
r2
2 1ω α+ +
(sin ) –
K
n
K
K n
t tr
r
r
r
+
+
+′ +
+1
1
12
ω α α(sin ) (sin ) . (24)
Çtob¥ ocenyt\ I1
1
, rassmotrym dva sluçaq: a) sin t + α ≤ sin –tk 1 y b) sin t +
+ αH> sin –tk 1. V pervom sluçae raznost\ ω a
n
tr
ksin –1
– ω αa
n
tr (sin )+
neot-
rycatel\na y, v sylu teorem¥ LahranΩa y monotonnosty proyzvodnoj ′ω ( )t , ne
prev¥ßaet
a
n
a
n
tr r′ +
ω α(sin ) (sin –tk 1 – sin t – α), a vo vtorom otrycatel\na y,
sledovatel\no, men\ße
a
n
a
n
tr r′ +
ω α(sin ) sin –tk 1 – sin t – α . Takym obrazom,
uçyt¥vaq, çto u Fk∈ , y neravenstvo 0 < sin –tk 1 – sin tk ≤
2a
n
r ≤
2π
n
, ymeem
ω a
n
tr
ksin –1
– ω αa
n
tr (sin )+
≤
≤
a
n
a
n
t t u u tr r
k′ +
+ +[ ]ω α α(sin ) sin – sin sin – sin –1 ≤
≤
a
n
a
n
t
t u
n
r r′ +
+ +
ω α π α(sin ) sin
–
2
2
2
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1096 V. P. MOTORNÁJ
Yspol\zuq poslednee neravenstvo y neravenstvo (6), poluçaem
I1
1 ≤ C
a
n
t
n
n n n
r
r
r r r′ +
+ +{ }+ + +ω α α(sin )
ln
2 2 1
1
.
Dlq sin t ≥
ln n
n
poloΩym α = 0. Tohda
I1
1 ≤
C
n
a n
n
n
n n
r
r
r′
+{ }ω ln ln
2 2 2
1 ≤
C
n
n
n n
r
r ω ln
2 2
1
+{ }. (25)
Esly 0 ≤ sin t ≤
ln n
n
, poloΩym α = ln /n n . Tohda
I1
1 ≤
C
n
a n
n
n
n
n
n
r
r
r′
+ +{ }ω ln ln ln
2 2 2
1
≤
C
n
n
n
n
n
r
r ω ln ln
2 2
+{ }. (26)
Takym obrazom, dlq vsex t ∈[ ]0; π dlq I1
1
ymeet mesto ocenka (26).
Çtob¥ ocenyt\ I1
2
, prymenym k raznosty
′ +
+ω α αa
n
t tr (sin ) (sin ) – ′
ω a
n
t tr
k ksin sin– –1 1
lemmuH1, poloΩyv b =
a
n
r
, s = sin –tk 1, τ = sin t + α, y vospol\zuemsq tem, çto
u Fk∈ , a takΩe neravenstvom 0 < sin –tk 1 – sin tk ≤
2a
n
r ≤
2π
n
:
′ +
+ ′
ω α α ωa
n
t t
a
n
t tr r
k k(sin ) (sin ) – sin sin– –1 1 ≤
≤ ′ +
+ω α αa
n
t t tr
k(sin ) sin – sin –1 ≤
≤
a
n
a
n
t t u u tr r
k′ +
+ +[ ]ω α α(sin ) sin – sin sin – sin –1 ≤
≤
a
n
a
n
t
t u
n
r r′ +
+ +
ω α π α(sin ) sin
–
2
2
2
.
Yz posledneho neravenstva toçno tak Ωe, kak poluçen¥ ocenky (25), (26), dlq
vsex t ∈[ ]0; π poluçaem ocenku velyçyn¥ I1
2
:
I1
2 ≤
C
n
a n
n
n
n
n
n
r
r
r′
+ +{ }ω ln ln ln
2 2 2
1
≤
C
n
n
n
n
n
r
r ω ln ln
2 2
+{ }. (27)
Rassmotrym teper\ pryblyΩenye funkcyy V t2( ). Poskol\ku posle yntehry-
rovanyq po çastqm
V t2( ) = I ur ˜ (cos )ψ( ) = – ( – ) ˜ (cos ) sin
–π
π
ψ∫ + ′D t u u u dur 1 ,
dlq pryblyΩenyq funkcyy V t2( ) voz\mem tryhonometryçeskyj polynom B tn
r( )
vyda
B tn
r( ) = – ( – ) ˜ (cos ) sin
–π
π
ψ∫ + ′P t u u udun
r 1
,
hde P tn
r +1( ) — tryhonometryçeskyj polynom stepeny ne v¥ße n nayluçßeho
L1-pryblyΩenyq qdra D tr +1 ( ). Toçno tak Ωe, kak b¥ly dokazan¥ neravenstva
(11), (12), netrudno v¥vesty, çto
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
K VOPROSU O PRYBLYÛENYY FUNKCYJ ALHEBRAYÇESKYMY … 1097
˜ (cos )′ψ u ≤ min sin ,
ln′
′
{ }ω ωa
n
u
n
n
r , u ∈[ ]– ,π π . (28)
Yspol\zuq neravenstvo (28), dlq x ≤ x̃ poluçaem
V t B f xn
r
2( ) – ( ; ) ≤
–
( – ) – ( – ) sin sin
π
π
ω∫ + ′
D t u B t u
a
n
u u dur n
r r
1 ≤
≤ ′
∫ +ω
π
π
a
n
t t D t u B t u dur
r n
rsin sin ( – ) – ( – )
–
1 +
+
–
( – ) – ( – ) sin sin – sin sin
π
π
ω ω∫ + ′
′
D t u B t u
a
n
u u
a
n
t t dur n
r r r
1 =
=
K
n
a
n
t t tr
r
r
r
+
+ ′
+1
1 ω sin sin ( )∆ .
Ocenka velyçyn¥ ∆r t( ) osuwestvlqetsq toçno tak Ωe, kak y velyçyn¥ ∆ ( )t
dlq r = 0:
∆r t( ) ≤ C
a
n
t
n
n
n
n
n
n
r
r
r rmin sin
ln
,
ln ln′
′
{ }+ +ω ω2 2 .
Sledovatel\no,
V t B tn
r
2( ) – ( ) ≤
K
n
K
K n
t tr
r
r
r
+
+
+′
1
1
12
ω sin sin +
+ C
a
n
t
n
n
n
n
n
n
r
r
r rmin sin
ln
,
ln ln′
′
{ }+ +ω ω2 2 ≤
C
n
n
n
r
r ω ln
2
. (29)
Pust\ T f tn
r( ; ) = A tn
r( ) + B tn
r( ). Yz neravenstva
I f u t T f tr n
r(cos ) ( ) – ( ; )( ) ≤ V t A tn
r
1( ) – ( ) + V t B tn
r
2 ( ) – ( )
y ocenok (23), (24), (26), (27), (29) sleduet spravedlyvost\ lemm¥H5.
Zameçanye 3. Yz ocenok (24), (26), (27), (29) dlq modulej neprer¥vnosty,
ukazann¥x v zameçanyy 2, sleduet, çto velyçynu ω ln n
n2
v neravenstve (22)
moΩno zamenyt\ na ω 1
2n
.
V rabote [12] pokazano, çto yz sootnoßenyj (20), (21) sleduet suwestvovanye
dlq lgboho natural\noho çysla r posledovatel\nosty M tn
r( ){ } çetn¥x tryho-
nometryçeskyx polynomov stepeny ne v¥ße n ≥ 1 takyx, çto
R t M tr n
r( ) – ( ) ≤ C
t n
n
t n
n
r
r
r
ω sin /
(sin / ) –+
+
+
1
1 1
1 , t ∈[ ]0; π . (30)
S uçetom v¥puklosty modulq neprer¥vnosty pravug çast\ ocenky (30) moΩ-
no zamenyt\ na
C n
t n
n
r
r
r
ω 1 12
+(sin / )
.
Sledovatel\no,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1098 V. P. MOTORNÁJ
R t M tr n
r( ) – ( ) ≤ C n
t n
n
r
r
r
ω 1 12
+(sin / )
, t ∈[ ]0; π .
Yz lemm¥H5, ravenstva (19) y posledneho neravenstva dlq lgboj funkcyy
f H∈ ω
sleduet suwestvovanye alhebrayçeskoho mnohoçlena Q f xn
r( ; ) takoho,
çto dlq lgboho t ∈[ ]0, π
f t Q f tr n
r(cos ) – ( ; cos ) ≤
K t
n
K
K n
tr
r
r
r2
2 1sin
sin
+ω +
+ C
t n n n
n
r
r
r
(sin / ) ln /+ ( )1 2ω
,
a πto πkvyvalentno utverΩdenyg teorem¥.
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uslovyg Lypßyca // Yzv. AN SSSR. Ser. mat. – 1946. – 10. – S. 295 – 322.
2. Tyman A. F. Teoryq pryblyΩenyq funkcyj dejstvytel\noho peremennoho. – M.: Fyz-
mathyz, 1960. – 624 s.
3. Kornejçuk N. P., Polovyna A. Y. O pryblyΩenyy neprer¥vn¥x y dyfferencyruem¥x
funkcyj alhebrayçeskymy mnohoçlenamy na otrezke // Dokl. AN SSSR. – 1966. – 166, # 2.
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Lypßyca, alhebrayçeskymy mnohoçlenamy // Mat. zametky. – 1971. – 9, # 4. – S. 441 – 447.
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mnohoçlenamy // Ukr. mat. Ωurn. – 1972. – 24, # 3. – S. 328 – 340.
6. Lyhun A. A. O nayluçßem pryblyΩenyy dyfferencyruem¥x funkcyj alhebrayçeskymy
mnohoçlenamy // Yzv. vuzov. Matematyka. – 1980. – # 4. – S. 53 – 60.
7. Temlqkov V. N. PryblyΩenye funkcyj yz klassa W∞
1
alhebrayçeskymy mnohoçlenamy //
Mat. zametky. – 1981. – 29, # 4. – S. 597 – 602.
8. Tryhub R. M. Prqm¥e teorem¥ o pryblyΩenyy alhebrayçeskymy polynomamy hladkyx fun-
kcyj na otrezke // Tam Ωe. – 1993. – 54, # 6. – S. 113 – 121.
9. Motorn¥j V. P. PryblyΩenye yntehralov drobnoho porqdka alhebrayçeskymy mnohoçle-
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Poluçeno 15.12.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
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| id | umjimathkievua-article-3226 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:33Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/95/b2b71a474c1407baeab3b59110245595.pdf |
| spelling | umjimathkievua-article-32262020-03-18T19:48:39Z On the problem of approximation of functions by algebraic polynomials with regard for the location of a point on a segment К вопросу о приближении функций алгебраическими многочленами с учетом положения точки на отрезке Motornyi, V. P. Моторный, В. П. Моторный, В. П. We obtain a correction of an estimate of the approximation of functions from the class W r H ω (here, ω(t) is a convex modulus of continuity such that tω '(t) does not decrease) by algebraic polynomials with regard for the location of a point on an interval. Одержано уточнення оцінки наближення функцій класу W r H ω (ω(t) — опуклий модуль неперервності, такий, що tω '(t) не спадає) алгебраїчними многочленами з урахуванням положення точки на відрізку. Institute of Mathematics, NAS of Ukraine 2008-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3226 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 8 (2008); 1087–1098 Український математичний журнал; Том 60 № 8 (2008); 1087–1098 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3226/3198 https://umj.imath.kiev.ua/index.php/umj/article/view/3226/3199 Copyright (c) 2008 Motornyi V. P. |
| spellingShingle | Motornyi, V. P. Моторный, В. П. Моторный, В. П. On the problem of approximation of functions by algebraic polynomials with regard for the location of a point on a segment |
| title | On the problem of approximation of functions by algebraic polynomials with regard for the location of a point on a segment |
| title_alt | К вопросу о приближении функций алгебраическими многочленами с учетом положения точки на отрезке |
| title_full | On the problem of approximation of functions by algebraic polynomials with regard for the location of a point on a segment |
| title_fullStr | On the problem of approximation of functions by algebraic polynomials with regard for the location of a point on a segment |
| title_full_unstemmed | On the problem of approximation of functions by algebraic polynomials with regard for the location of a point on a segment |
| title_short | On the problem of approximation of functions by algebraic polynomials with regard for the location of a point on a segment |
| title_sort | on the problem of approximation of functions by algebraic polynomials with regard for the location of a point on a segment |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3226 |
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