Approximation of Continuous Functions by de La Vallee-Poussin Operators
For $\sigma \rightarrow \infty$, we study the asymptotic behavior of upper bounds of deviations of functions blonding to the classes $\widehat{C}_{\infty}^{\overline{\Psi}}$ and $\widehat{C}^{\overline{\Psi}} H_{\omega}$ from the so-called Vallee Poussin operators. We find asymptotic equalities th...
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| Date: | 2005 |
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| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2005
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3590 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509705278849024 |
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| author | Rukasov, V. I. Silin, E. S. Рукасов, В. И. Силин, Е. С. Рукасов, В. И. Силин, Е. С. |
| author_facet | Rukasov, V. I. Silin, E. S. Рукасов, В. И. Силин, Е. С. Рукасов, В. И. Силин, Е. С. |
| author_sort | Rukasov, V. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:22Z |
| description | For $\sigma \rightarrow \infty$, we study the asymptotic behavior of upper bounds of deviations of functions blonding to the classes
$\widehat{C}_{\infty}^{\overline{\Psi}}$ and $\widehat{C}^{\overline{\Psi}} H_{\omega}$ from the so-called Vallee Poussin operators.
We find asymptotic equalities that, in some important cases, guarantee the solution of the Kolmogorov - Nikol's'kyi problem for the Vallee Poussin operators on the classes
$\widehat{C}_{\infty}^{\overline{\Psi}}$ and $\widehat{C}^{\overline{\Psi}} H_{\omega}$.
|
| first_indexed | 2026-03-24T02:45:21Z |
| format | Article |
| fulltext |
UDK 517.5
V. Y. Rukasov, E. S. Sylyn (Slavqn. ped. un-t)
PRYBLYÛENYE NEPRERÁVNÁX FUNKCYJ
OPERATORAMY VALLE PUSSENA
For σ → ∞, we study the asymptotic behavior of upper bounds of deviations of functions blonding to
the classes Ĉ∞
ψ and Ĉ Hψ
ω from the so-called Vallée Poussin operators. We find asymptotic
equalities that, in some important cases, guarantee the solution of the Kolmogorov – Nikol’s’kyi problem
for the Vallée Poussin operators on the classes Ĉ∞
ψ and Ĉ Hψ
ω .
Vyvça[t\sq asymptotyçna povedinka pry σ → ∞ verxnix meΩ vidxylen\ funkcij klasiv Ĉ∞
ψ
i
Ĉ Hψ
ω vid tak zvanyx operatoriv Valle Pussena. Znajdeno asymptotyçni rivnosti, qki v deqkyx
vaΩlyvyx vypadkax zabezpeçugt\ rozv’qzok zadaçi Kolmohorova – Nikol\s\koho dlq operatoriv
Valle Pussena na klasax Ĉ∞
ψ
ta Ĉ Hψ
ω .
V nastoqwej stat\e yzuçagtsq vopros¥, svqzann¥e s pryblyΩenyem neprer¥v-
n¥x funkcyj cel¥my funkcyqmy πksponencyal\noho typa. A. Y. Stepanec [1]
vvel klass¥ L̂
ψ� sledugwym obrazom. Oboznaçym çerez L̂ mnoΩestvo
funkcyj f, zadann¥x na dejstvytel\noj osy R y ymegwyx koneçnug normu
f = sup ( )
a R a
a
f t dt
∈
+
∫
2π
,
a çerez � mnoΩestvo funkcyj ψ ( t ), v¥pukl¥x vnyz pry vsex t ≥ 1 y ysçeza-
gwyx na beskoneçnosty. KaΩdug funkcyg ψ ∈ � prodolΩym na promeΩu-
tok [ 0, 1 ) takym obrazom, çtob¥ poluçennaq funkcyq ( kotorug, po-preΩnemu,
budem oboznaçat\ çerez ψ ( ⋅ ) ) b¥la neprer¥vna pry vsex t ≥ 0, ψ ( 0 ) = 0 y ee
proyzvodnaq ψ ′ ( t ) = ψ ′ ( t + 0 ) ymela ohranyçennug varyacyg na promeΩutke
[ 0, ∞ ) . MnoΩestvo takyx funkcyj oboznaçym çerez �. PodmnoΩestvo funk-
cyj ψ, dlq kotor¥x
ψ( )t
t
dt
1
∞
∫ < ∞ ,
oboznaçym çerez � ′.
Pust\, dalee, ψk ∈ �, k = 1, 2, y pod ψk+ y ψk− budem ponymat\ çetnoe y
neçetnoe prodolΩenye funkcyy ψk, k = 1, 2. Dlq par¥ ( ψ1, ψ2 ) opredelym
funkcyg ψ :
ψ
df= ψ ψ1 2+ −+ i . (1)
Pry πtom sootvetstvugwee preobrazovanye Fur\e funkcyy ψ ymeet vyd
ψ̂ = ˆ ˆψ ψ1 2+ −+ i , (2)
hde preobrazovanye Fur\e ponymaetsq v ob¥çnom sm¥sle:
ˆ ( )h t =
1
2π
h x e dxixt
R
( ) −∫ .
Tohda çerez L̂ψ
oboznaçym mnoΩestvo funkcyj f ∈ L̂ , predstavym¥x ra-
venstvom
f ( x ) = A x t t dt
R
0 + +∫ ϕ ψ( ) ˆ ( )
df= A0 + ∗ϕ ψ̂ , (3)
© V. Y. RUKASOV, E. S. SYLYN, 2005
230 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
PRYBLYÛENYE NEPRERÁVNÁX FUNKCYJ OPERATORAMY … 231
v kotorom A0 — nekotoraq postoqnnaq; yntehral ponymaetsq kak predel po
rasßyrqgwymsq symmetryçn¥m promeΩutkam; ϕ ∈ L̂ .
Sleduq A. Y. Stepancu [2], funkcyg ϕ ( ⋅ ) v predstavlenyy (3) naz¥vagt
ψ -proyzvodnoj funkcyy f ( ⋅ ) y polahagt ϕ ( ⋅ ) = f ψ( )⋅ .
Esly f ∈ L̂ψ
y pry πtom ϕ ∈ � , hde � — nekotoroe podmnoΩestvo yz L̂ ,
to polahagt f ∈ L̂ψ�. PodmnoΩestvo neprer¥vn¥x funkcyj yz L̂ψ� obozna-
çaetsq çerez Ĉ
ψ�.
V kaçestve mnoΩestva � budem rassmatryvat\ edynyçn¥j ßar S∞ prost-
ranstva suwestvenno ohranyçenn¥x funkcyj M :
S∞ = { ϕ : ess sup | ϕ ( t ) | ≤ 1 }
y klass¥ Hω :
Hω = { ϕ ∈ C : | ϕ ( t ) – ϕ ( t ′ ) | ≤ ω ( | t – t ′ | ) ∀ t, t ′ ∈ R } ,
hde C — podmnoΩestvo neprer¥vn¥x funkcyj yz L̂ , ω ( t ) — proyzvol\n¥j
fyksyrovann¥j modul\ neprer¥vnosty. Pry πtom Ĉ Sψ
∞
df= Ĉ∞
ψ .
Yz predloΩenyq 10 (sm. [3]) vydym, çto ψ̂1+ y ψ̂2 − summyruem¥ na R, pry-
çem
ˆ ( )ψ1+ t = O t( )−2 y ˆ ( )ψ2 − t = O t( )−2 pry t → ∞ ,
otkuda na osnovanyy (2)
ˆ ( )ψ t dt
R
∫ < ∞
y, sledovatel\no, klass¥ Ĉ
ψ� sostoqt yz funkcyj f ( x ) , neprer¥vn¥x dlq
vsex x ∈ ( – ∞ , ∞ ) .
Sleduq [2], dlq vsex 0 ≤ c ≤ σ opredelym semejstvo funkcyj Λσ,c =
= { }, ( )λ σ c t , hde
λ σ, ( )c t =
1 0
0
, ,
, ,
, ,
≤ ≤
−
−
≤ ≤
≤
t c
t
c
c t
t
σ
σ
σ
σ
(4)
y dlq funkcyj ψ1, ψ2 ∈ � opredelym semejstvo funkcyj Λσ,c
∗ = { }, ( )λ σ c t∗ ,
hde
λ σ, ( )c t∗ =
λ σ
σ
ψ σ
ψ
σ
σ, , [ , ] [ , ],
( ( ))
, .
c t c
t c
c
t
t
c t
∈ ∞
− −
−
≤ ≤
0
1
∪
sign
( )
(5)
KaΩdoj funkcyy f ∈ L̂ψ
sopostavym operator¥
V fcσ, ( ) = V f xc cσ σ, ,( , , )Λ =
A f c0 + ∗ψ
σλ ψ,
%
, (6)
V fcσ, ( )∗ = V f xc cσ σ, ,( ), ,∗ ∗Λ =
A f c0 + ∗ ∗ψ
σλ ψ,
%
. (7)
V rabote [5] pokazano, çto pry dostatoçno obwyx predpoloΩenyqx operato-
r¥ V fcσ, ( ) y V fcσ, ( )∗
prynadleΩat mnoΩestvu εσ , hde pod εσ ponymaetsq
mnoΩestvo cel¥x funkcyj πksponencyal\noho typa σ, σ ≥ 0. V peryodyçes-
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
232 V. Y. RUKASOV, E. S. SYLYN
kom sluçae, pry σ = n ∈ N y c = n – p, p ∈ N, p < n, operator¥ V cσ, sovpa-
dagt s yzvestn¥my summamy Valle Pussena:
V f xn p, ( , )
df=
1
1
p
S f xk
k n p
n
( , )
= −
−
∑ ,
hde S f xk ( , ), k = 0, 1, … , — çastn¥e summ¥ porqdka k rqda Fur\e funkcyy
f ( x ) . Poπtomu v dal\nejßem budem naz¥vat\ V cσ, operatoramy Valle Pussena.
V nastoqwej stat\e yzuçaetsq asymptotyçeskoe povedenye pry σ → ∞ verx-
nyx hranej
� �( )ˆ , ,C V c
ψ
σ =
sup ( ) ( )
ˆ
,
f C
c C
f x V x
∈
−
ψ
σ
�
,
hde pod � ponymaetsq mnoΩestvo S∞ lybo Hω .
Yssleduem yntehral\n¥e predstavlenyq uklonenyj
ρσ, ( , )c f x
df= f x V xc( ) ( ),− σ .
Ymeet mesto sledugwaq lemma.
Lemma$1. Pust\ ψi ∈ � ′, i = 1, 2, ψ = ψ ψ1 2+ −+ i y ai = ai ( σ ) , i = 1, 2,
— dve proyzvol\n¥e neprer¥vn¥e pry vsex σ ≥ 1 funkcyy, dlq kotor¥x
σ ai ( σ ) ≥ ai ( 0 ) > 0. Tohda esly f ∈ Ĉ∞
ψ , to dlq lgb¥x σ y h = h ( σ ) , σ >
> h ≥ 1, v kaΩdoj toçke x ∈ R
ρσ σ, ( ; )−h f x = – ν ψ σ
π
δ σ
1
1
1 1
( )
( ; )
sin
x t
t
t
dt
m t Ma a≤ ≤
∫ +
+ ν ψ σ
π
δ σ
σ
ψ
σ
ψ
2
2
1 2
2 2
1 2( )
( ; )
cos
( ; ; ) ( ; ; ), ,x t
t
t
dt b a f x b a f x
m t M
h h
a a≤ ≤
∫ + + , (8)
hde
b a f xh i
i
σ
ψ
, ( ; ; ) = O h
t
t
dti
i
ai
( ) ( )
( )
/ ( )
1
1
ψ σ ψ σ
σ
− + +
∞
∫ +
+
ψ σ ψ σ
σ
i i
a
t
t
dt
i
( ) ( )/
( )
− −
∞
∫ 1
, i = 1, 2. (9)
Esly Ωe f ∈ Ĉ Hψ ω , to dlq lgb¥x σ y h = h ( σ )
, σ > h ≥ 1, v kaΩdoj
toçke x ∈ R
ρσ σ, ( ; )−h f x = – ν ψ σ
π
δ σ
1
1
1 1
( )
( ; )
sin
x t
t
t
dt
m t Ma a≤ ≤
∫ +
+ ν ψ σ
π
δ σ
σ
ψ
σ
ψ
2
2
1 2
2 2
1 2( )
( ; )
cos
( ; ; ) ( ; ; ), ,x t
t
t
dt d a f x d a f x
m t M
h h
a a≤ ≤
∫ + + , (10)
hde
d a f xh i
i
σ
ψ
, ( ; ; ) = O h
t
t
dti
i
ai
( ) ( )
( )
/ ( )
1
1
ψ σ ψ σ
σ
− + +
∞
∫ +
+
ψ σ ψ σ ω
σσ
i i
a
t
t
dt
h
i
( ) ( )/
( )
− −
−
∞
∫ 1 1
, i = 1, 2, (11)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
PRYBLYÛENYE NEPRERÁVNÁX FUNKCYJ OPERATORAMY … 233
δ ( x; t ) =
f x t f C
f x f x t f C H
ψ ψ
ψ ψ ψ
ω
( ), ˆ ,
( ) ( ), ˆ ,
+ ∈
− + ∈
∞esly
esly
mai
= min ( );a
hi σ π{ } , Mai
= max ( );a
hi σ π{ } ,
νi = sign a
hi( )σ π−{ } , i = 1, 2.
Dokazatel\stvo. Vvedem oboznaçenyq:
ρσ, ( , )c f x∗
df= f x V f xc( ) ( , ),− ∗
σ , (12)
τσ, ( )c t = ( ), ( ) ( )1 − λ ψσ c t t , τσ, ( )c t∗ = ( ), ( ) ( )1 − ∗λ ψσ c t t . (13)
Yz opredelenyq ψ -proyzvodnoj y operatorov V f xcσ, ( , ), V f xcσ, ( , )∗
sleduet
ρσ, ( , )c f x = f f xc
ψ
στ∗ ˆ ( , ), , ρσ, ( , )c f x∗ = f f xc
ψ
στ∗ ∗ˆ ( , ), .
S druhoj storon¥,
ρσ, ( , )c f x = ρσ σ, ,( , ) ( , )c cf x f x∗ + ∆ , (14)
hde
∆σ, ( , )c f x = V f x V f xc cσ σ, ,( , ) ( , )− ∗ . (15)
Yssleduem vnaçale velyçynu ρσ, ( , )c f x∗ . Sohlasno teoremeM1 yz rabot¥ [2],
dlq lgboho u ∈ Wp
2
y 0 ≤ p ≤ c < σ, hde
Wσ
2
df= ϕ ε
ϕ
σ∈
+
< ∞
−∞
∞
∫:
( )2
21
t
t
dt ,
ymeem u c∗ ˆ ,τσ = 0, poπtomu
ρσ, ( , )c f x∗ = ( ) ˆ ,f u c
ψ
στ− ∗ ∗ . (16)
Yz opredelenyq preobrazovanyq Fur\e y (5) sleduet
ˆ ( ),τσ c t∗ =
1
2π
τσ, ( )c
ist
R
s e ds∗ −∫ =
1
2π σ
ψ σ ψ σ
σ
c
ist ists c
c
e e ds∫ −
−
+ −−( )( ) ( ) +
+
1
2π
ψ ψ
σ
∞
−∫ + −( )( ) ( )s e s e dsist ist .
Yz sootnoßenyq (1) ymeem
ψ ψ( ) ( )s e s eist ist− + − = 2 1 2( )( )cos ( )sinψ ψs st s st+ , (17)
y poπtomu dalee naxodym
ˆ ( ),τσ c t∗ =
ψ σ
π σ π
ψ
σ
σ
1
1
1( )
cos ( )cos
c
s c
c
st ds s st ds∫ ∫−
−
+
∞
+
+
ψ σ
π σ π
ψ
σ
σ
2
2
1( )
sin ( )sin
c
s c
c
st ds s st ds∫ ∫−
−
+
∞
df= R t R t1 2( ) ( )+ . (18)
Polahaq c = σ – h y yntehryruq po çastqm, poluçaem
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
234 V. Y. RUKASOV, E. S. SYLYN
R t1( ) =
ψ σ
π
σ σ1
2
2 2 2 2( ) sin sin( ) sin/ /t
t
t ht ht
ht
+ −
–
–
ψ σ σ
π π
ψ
σ
1
1
1( ) sin
( )sin
t
t t
s st ds−
∞
∫ , (19)
R t2( ) =
ψ σ
π
σ σ2
2
2 2 2 2( ) cos cos( ) sin/ /− + −
t
t
t ht ht
ht
+
+
ψ σ σ
π π
ψ
σ
2
2
1( ) cos
( )cos
t
t t
s st ds+
∞
∫ . (20)
V¥polnqq preobrazovanyq, ymeem
R t1( ) =
ψ σ
π
σ σ1
2 2
1( ) sin
sin
cos
cos
ht ht
ht
t
ht
ht
t
− + −
–
–
ψ σ
π
σ
π
ψ
σ
1
1
1( )
sin ( )sin
t
t
t
s st ds− ′
∞
∫ , (21)
R t2( ) =
ψ σ
π
σ σ2
2 2
1( ) sin
cos
cos
sin− − + −
ht ht
ht
t
ht
ht
t +
+
ψ σ
π
σ
π
ψ
σ
2
2
1( )
cos ( )cos
t
t
t
s st ds− ′
∞
∫ . (22)
Pust\, dalee, ai = ai ( σ ) — proyzvol\n¥e funkcyy, neprer¥vn¥e pry vsex
σ > 0 y takye, çto σ ai ( σ ) > ai ( 0 ) > 0, i = 1, 2. Pust\ mai
= min ( );{ai σ
π/ }h , Mai
= max ( );{ / }a hi σ π . Tohda, ysxodq yz sootnoßenyj (16) – (22), zapy-
s¥vaem
ρσ, ( , )c f x∗ = –
ψ σ
π
δ σ1
1 1
( )
( ; )
sin
m t Ma a
x t
t
t
dt
≤ ≤
∫ +
+
1 1
1 1
1 1π
δ ψ
π
δ ψ
σ σ σ σt a t a
x t s st ds dt x t s
st
t
ds dt
≤
∞
≥
∞
∫ ∫ ∫ ∫− ′
( ) ( )
( ; ) ( ) cos ( ; ) ( )
sin
+
+
ψ σ
π
δ σ σ
π
π
1
2 2
1( )
( ; )
sin
sin
cos
cos
/
/
−
∫ − + −
h
h
x t
ht ht
ht
t
ht
ht
t dt –
–
ψ σ
π
δ σ ψ σ
π
δ σ
π
1
2
22 2 2 2
2 2
( )
( ; )
sin( ) sin ( )
( ; )
cos
/
/ /
t h m t M
x t
t ht ht
ht
dt x t
t
t
dt
a a≥ ≤ ≤
∫ ∫− + +
+
1 1
2 2
2 2π
δ ψ
π
δ ψ
σ σ σ σt a t a
x t s st ds dt x t s
st
t
ds dt
≤
∞
≥
∞
∫ ∫ ∫ ∫+ ′
( ) ( )
( ; ) ( ) sin ( ; ) ( )
cos
+
+
ψ σ
π
δ σ σ
π
π
2
2 2
1( )
( ; )
sin
cos
cos
sin
/
/
−
∫ − − + −
h
h
x t
ht ht
ht
t
ht
ht
t dt +
+
ψ σ
π
δ σ
π
2
2
2 2 2 2( )
( ; )
cos( ) sin
/
/ /
t h
x t
t ht ht
ht
dt
≥
∫ −
df= B a f x P a f xhσ
ψ
σ
ψ
, ( ; ; ) ( ; ; )1 1
1 1+ –
– R a f x a f x a f x B a f x P a f xh h hσ
ψ
σ
ψ
σ
ψ
σ
ψ
σ
ψγ µ1 1 1 2 2
1 1 1 2 2( ; ; ) ( ; ; ) ( ; ; ) ( ; ; ) ( ; ; ), , ,+ − + + +
+ R a f x a f x a f xh hσ
ψ
σ
ψ
σ
ψγ µ2 2 2
2 2 2( ; ; ) ( ; ; ) ( ; ; ), ,+ + . (23)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
PRYBLYÛENYE NEPRERÁVNÁX FUNKCYJ OPERATORAMY … 235
V rabote [6] dlq lgboj funkcyy f ∈ Ĉ∞
ψ
poluçen¥ ocenky
P a f xi
iσ
ψ ( ; ; ) ≤ K
t
t
dti
a
i
i
ψ σ ψ σ
σ
( )
( )
/ ( )
+ +
∞
∫
1
, (24)
R a f xi
iσ
ψ ( ; ; ) ≤ K
t
t
dti
a
i i
i
ψ σ ψ σ ψ σ
σ
( )
( ) ( )
( )
/+ − −
∞
∫ 1
, (25)
i = 1, 2,
esly Ωe f ∈ Ĉ Hψ
ω , to
P a f xi
iσ
ψ ( ; ; ) ≤ K
t
t
dti
a
i
i
ψ σ ψ σ ω
σσ
( )
( )
/ ( )
+ +
∞
∫
1
1
, (26)
R a f xi
iσ
ψ ( ; ; ) ≤ K
t
t
dti
a
i i
i
ψ σ ψ σ ψ σ ω
σσ
( )
( ) ( )
( )
/+ − −
∞
∫ 1 1
, (27)
i = 1, 2.
Ocenky ostavßyxsq yntehralov najden¥ v rabote [7]. Tak, dlq lgboj funkcyy
f ∈ Ĉ∞
ψ
γ σ
ψ
, ( ; ; )h i
i a f x ≤ K iψ σ( ) , (28)
µσ
ψ
, ( ; ; )h i
i a f x ≤ K iψ σ( ) , (29)
i = 1, 2,
a dlq f ∈ Ĉ Hψ
ω
γ σ
ψ
, ( ; ; )h i
i a f x ≤ K
hiψ σ ω
σ
( )
1
−
, (30)
µσ
ψ
, ( ; ; )h i
i a f x ≤ K
hiψ σ ω
σ
( )
1
−
, (31)
i = 1, 2.
Rassmotrym teper\ velyçynu ∆σ, ( ; )c f x , opredelennug sootnoßenyem (15)
pry c = σ – h. V peryodyçeskom sluçae pry σ = n ∈ N y h = p ∈ N dlq velyçy-
n¥ ∆n p f x, ( ; ) v rabote [8] poluçena ocenka (26). Analyzyruq ee dokazatel\stvo,
vydym, çto esly funkcyq f ( ⋅ ) zadana na vsej dejstvytel\noj osy, budet ymet\
mesto analohyçnoe sootnoßenye
∆σ, ( ; )h C
f x ≤ ω
σ
ψ σ ψ σ1
1
2
−
− −
=
∑h
h
i
i i( ( ) ( )) ∀ f ∈ Ĉ Hψ
ω . (32)
Esly f ∈ Ĉ∞
ψ , to spravedlyva ocenka
∆σ, ( ; )h C
f x ≤
i
i ih
=
∑ − −
1
2
( ( ) ( ))ψ σ ψ σ . (33)
Obæedynqq sootnoßenyq (14), (23) – (33), pryxodym k utverΩdenyg lemm¥.
KaΩdoj funkcyy ψ ∈ �, sleduq A. Y. Stepancu [9], postavym v sootvetst-
vye dve funkcyy:
η ( t ) = η ( ψ , t ) df= ψ ψ−
1 1
2
( )t y µ ( t ) = µ ( ψ , t ) df=
t
t tη( ) −
, t ≥ 1,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
236 V. Y. RUKASOV, E. S. SYLYN
s pomow\g kotor¥x yz � v¥delym podmnoΩestvo
F
df= { ψ ∈ � : η ′ ( t ) ≤ K } , t ≥ 1, η ′ ( t ) = η ′ ( t + 0 ) ,
hde K — postoqnnaq ( vozmoΩno, zavysqwaq ot funkcyy ψ ) .
Sformulyruem osnovnoj rezul\tat rabot¥.
Teorema$1. Pust\ ψi ∈ F , i = 1, 2, ψ = ψ ψ1 2+ −+ i y suwestvugt kons-
tant¥ K y K ′ takye, çto
0 < K ≤
η ψ σ σ
η ψ σ σ
( ; )
( ; )
1
2
−
−
≤ K ′ < ∞ . (34)
Tohda dlq lgb¥x çysel σ y h = h ( σ ) , σ > h ≥ 1, pry σ → ∞ spravedlyv¥
ravenstva
�( )ˆ ; ,C V h∞ −
ψ
σ σ =
4
12
ψ σ
π
η σ σ ψ σ( )
ln
( )
( ) ( )
− + −
h
O h , (35)
�( )ˆ ; ,C H V h
ψ
ω σ σ− =
2 2
2
0
2θ ψ σ
π
η σ σ ω
σ
ω
π
( )
ln
( )
sin
/−
∫h
t t dt +
+ O h
h
( ) ( )1
1ψ σ ω
σ
−
−
, (36)
hde O ( 1 ) — velyçyna, ravnomerno ohranyçennaq po σ y h, η ( σ ) est\ lybo
η ( ψ1 ; σ ) , lybo η ( ψ2 ; σ ) , θω ∈ [ 2 / 3 ; 1 ] , pryçem θω = 1, esly ω ( t ) — v¥-
pukl¥j modul\ neprer¥vnosty.
Dokazatel\stvo. V¥berem v kaçestve ai ( σ ) velyçynu
ai
∗( )σ =
1
η ψ σ σ( ; )i −
, i = 1, 2,
dlq kotoroj na osnovanyy uslovyq (34) v¥polnqetsq sootnoßenye
a
a
dt
t
1
2
( )
( )
σ
σ
∫ = ln
( )
( )
a
a
2
1
σ
σ
= O ( 1 ) ,
kotoroe pozvolqet zamenyt\ yntehral¥ v ravenstvax (8) y (10) yntehralamy, ko-
tor¥e berutsq po odynakov¥m promeΩutkam ma1
≤ t ≤ Ma1
yly ma2
≤ t ≤
≤ Ma2
. Pohreßnosty pry takyx zamenax ne prev¥sqt velyçyn ostatoçn¥x çle-
nov. Yspol\zuq formulu
a sin α – b cos α = a b2 2+ −sin( )α γ ,
hde γ = arctg /b a, yz lemm¥M1 poluçaem
ρσ σ, ( ; )−h f x = ν ψ σ
π
δ σ γσ
a
m t Ma a
x t
t
t
dt
( )
( ; )
sin ( )
≤ ≤
∫ −
+
+ O b a f xh i
i
i( ) ( ; ; ),1
1
2
σ
ψ
=
∑
∀ f ∈ Ĉ∞
ψ , (37)
ρσ σ, ( ; )−h f x = ν ψ σ
π
δ σ γσ
a
m t Ma a
x t
t
t
dt
( )
( ; )
sin ( )
≤ ≤
∫ −
+
+ O d a f xh i
i
i( ) ( ; ; ),1
1
2
σ
ψ
=
∑
∀ f ∈ Ĉ Hψ
ω . (38)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
PRYBLYÛENYE NEPRERÁVNÁX FUNKCYJ OPERATORAMY … 237
Zdes\ velyçyn¥ b a f xh i
i
σ
ψ
, ( ; ; ) y d a f xh i
i
σ
ψ
, ( ; ; ) opredelen¥ ravenstvamy (9) y (11)
sootvetstvenno, γσ = arctg
ψ
ψ
2
1
, a v kaçestve funkcyy a ( σ ) moΩet b¥t\ lgbaq
yz funkcyj ai
∗( )σ , i = 1, 2, νa = sign { a ( σ ) – π / h }
.
Klass¥ Ĉ
ψ � , hde � est\ S∞ yly Hω , ynvaryantn¥ otnosytel\no sdvyha
po arhumentu, sledovatel\no,
� �( )ˆ , ,C V c
ψ
σ =
sup ( ; )
ˆ
,
f C
c f
∈ ψ
ρσ
�
0 .
Esly f ∈ Ĉ
ψ�, to f ψ ∈ �. S druhoj storon¥, dlq lgboho ϕ ∈ � v klasse
Ĉψ� suwestvuet funkcyq f ( ⋅ ) takaq, çto poçty vezde f ψ( )⋅ = ϕ ( ⋅ ) . Poπto-
mu yz ravenstv (37) y (38) poluçaem
�( )ˆ ; ,C V h∞ −
ψ
σ σ = sup
( )
( )
sin ( )
ϕ
σψ σ
π
ϕ σ γ
∈ ≤ ≤∞
∫ −
S m t Ma a
t
t
t
dt +
+ O b a f xh i
i
i( ) ( ; ; ),1
1
2
σ
ψ
=
∑
, (39)
�( )ˆ ; ,C H V h
ψ
ω σ σ− = sup
( )
( ( ) ( ))
sin ( )
ϕ
σ
ω
ψ σ
π
ϕ ϕ σ γ
∈ ≤ ≤
∫ − −
H m t Ma a
t
t
t
dt0 +
+ O d a f xh i
i
i( ) ( ; ; ),1
1
2
σ
ψ
=
∑
. (40)
V rabote [7] poluçen¥ ravenstva
sup ( )
sin ( )
ϕ
σϕ σ γ
∈ ≤ ≤∞
∫ −
S m t Ma a
t
t
t
dt =
4
1
π
π
σ
ln
( )
( )
a h
O+ , (41)
sup ( ( ) ( ))
sin ( )
ϕ
σ
ω
ϕ ϕ σ γ
∈ ≤ ≤
∫ − −
H m t Ma a
t
t
t
dt0 =
=
2 2
1
0
2θ
π
π
σ
ω
σ
ω
π
ln
( )
sin ( )
/
a h
t
t dt O
+∫ , θω ∈ [ / ];2 3 1 . (42)
V rabote [10] pokazano, çto pry σ ∈ N y ψi ∈ F , i = 1, 2,
ψ σ ψ σ ψ σ
σ σ
i
a
i i
a
t
t
dt
t
t
dt
( ) ( ) ( )
/ ( ) ( )
/+ + − −
∗ ∗
∞ ∞
∫ ∫
1
1
≤ K iψ σ( ) , i = 1, 2. (43)
Vklgçenye σ ∈ N ne pryncypyal\no, tak kak πta ocenka verna dlq lgboho σ ≥
≥ 1. Obæedynqq sootnoßenyq (39), (40), (41) – (43), poluçaem utverΩdenye teo-
rem¥.
Oboznaçym çerez F∞ mnoΩestvo funkcyj ψ ∈ F , dlq kotor¥x
lim ( ; )( )
σ
η ψ σ σ
→∞
− = ∞ .
Pust\ ψi ∈ F∞ , i = 1, 2, y çysla h = h ( σ ) v¥bran¥ tak, çto
lim
( ; )
σ
η ψ σ σ
→∞
−i
h
= ∞ , i = 1, 2.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
238 V. Y. RUKASOV, E. S. SYLYN
Tohda (sm. [8])
ω
σ
1
−
h
= O( )1
1ω
σ
, σ → ∞ ,
ψ σi h( )− = O i( ) ( )1 ψ σ , i = 1, 2, σ → ∞ ,
y yz teorem¥M1 poluçaem takoe sledstvye.
Sledstvye. Pust\ ψ i ∈ F∞ , i = 1, 2, ψ = ψ ψ1 2+ −+ i y v¥polneno uslo-
vye (34). Tohda dlq lgb¥x çysel σ y h = h ( σ ) , σ > h ≥ 1, pry σ → ∞
spravedlyv¥ ravenstva
�( )ˆ ; ,C V h∞ −
ψ
σ σ =
4
12
ψ σ
π
η σ σ ψ σ( )
ln
( )
( ) ( )
− +
h
O , (44)
�( )ˆ ; ,C H V h
ψ
ω σ σ− =
2 2
2
0
2θ ψ σ
π
η σ σ ω
σ
ω
π
( )
ln
( )
sin
/−
∫h
t
t dt +
+ O( ) ( )1
1ψ σ ω
σ
, (45)
hde O( )1 — velyçyna, ravnomerno ohranyçennaq po σ y h , η σ( ) est\ lybo
η ψ σ( ; )1 , lybo η ψ σ( ; )2 , θω ∈ [ / ];2 3 1 , pryçem θω = 1, esly ω ( t ) — v¥puk-
l¥j modul\ neprer¥vnosty.
Ravenstva (44) y (45) dagt reßenye zadaçy Kolmohorova – Nykol\skoho dlq
operatorov Valle Pussena na klassax Ĉ∞
ψ
y Ĉ Hψ
ω .
1. Stepanec A. Y. PryblyΩenye operatoramy Fur\e funkcyj, zadann¥x na dejstvytel\noj
osy // Ukr. mat. Ωurn. – 1988. – 40, # 2. – S. 198 – 209.
2. Stepanets A. I., Wang Kunyang, Zhang Xirong. Approximation of locally integrable function on the
real line // Tam Ωe. – 1999. – 51, # 11. – S. 1549 – 1561.
3. Stepanec A. Y. Klass¥ funkcyj, zadann¥x na dejstvytel\noj osy, y yx pryblyΩenye ce-
l¥my funkcyqmy. I // Tam Ωe. – 1990. – 42, # 1. – S. 102 – 112.
4. Stepanec A. Y. Klass¥ funkcyj, zadann¥x na dejstvytel\noj osy, y yx pryblyΩenye ce-
l¥my funkcyqmy. II // Tam Ωe. – # 2. – S. 210 – 222.
5. Stepanec A. Y. PryblyΩenye v prostranstvax lokal\no yntehryruem¥x funkcyj // Tam
Ωe. – 1994. – 46, # 5. – S. 597 – 625.
6. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj: V 2Mt. – Kyev: Yn-t matematyky NAN Ukray-
n¥, 2002. – T. 2. – 468 s.
7. Rukasov V. Y. PryblyΩenye neprer¥vn¥x funkcyj operatoramy Valle Pussena // Ukr. mat.
Ωurn. – 2003. – 55, # 3. – S. 414 – 424.
8. Rukasov V. Y., Çajçenko S. O. PryblyΩenye klassov C Hψ
ω summamy Valle Pussena // Tam
Ωe. – 2002. – 54, # 5. – S. 681 – 691.
9. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj: V 2 t. – Kyev: Yn-t matematyky NAN Ukray-
n¥, 2002. – T. 1. – 426 s.
10. Stepanec A. Y. Klassyfykacyq peryodyçeskyx funkcyj y skorost\ sxodymosty yx rqdov
Fur\e // Yzv. AN SSSR. Ser. mat. – 1986. – 50, # 2. – S. 101 – 136.
Poluçeno 11.11.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
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| id | umjimathkievua-article-3590 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:21Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5b/310a2b59855942ee8970b7cff7ef5a5b.pdf |
| spelling | umjimathkievua-article-35902020-03-18T19:59:22Z Approximation of Continuous Functions by de La Vallee-Poussin Operators Приближение непрерывных функций операторами Балле Пуссена Rukasov, V. I. Silin, E. S. Рукасов, В. И. Силин, Е. С. Рукасов, В. И. Силин, Е. С. For $\sigma \rightarrow \infty$, we study the asymptotic behavior of upper bounds of deviations of functions blonding to the classes $\widehat{C}_{\infty}^{\overline{\Psi}}$ and $\widehat{C}^{\overline{\Psi}} H_{\omega}$ from the so-called Vallee Poussin operators. We find asymptotic equalities that, in some important cases, guarantee the solution of the Kolmogorov - Nikol's'kyi problem for the Vallee Poussin operators on the classes $\widehat{C}_{\infty}^{\overline{\Psi}}$ and $\widehat{C}^{\overline{\Psi}} H_{\omega}$. Вивчається асимптотична поведінка при $\sigma \rightarrow \infty$ верхніх меж відхилень функцій класів $\widehat{C}_{\infty}^{\overline{\Psi}}$ і $\widehat{C}^{\overline{\Psi}} H_{\omega}$ від так званих операторів Валле Пуссена. Знайдено асимптотичні рівності, які в деяких важливих випадках забезпечують розв'язок задачі Колмогорова - Нікольського для операторів Валле Пуссена на класах $\widehat{C}_{\infty}^{\overline{\Psi}}$ та $\widehat{C}^{\overline{\Psi}} H_{\omega}$. Institute of Mathematics, NAS of Ukraine 2005-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3590 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 2 (2005); 230–238 Український математичний журнал; Том 57 № 2 (2005); 230–238 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3590/3912 https://umj.imath.kiev.ua/index.php/umj/article/view/3590/3913 Copyright (c) 2005 Rukasov V. I.; Silin E. S. |
| spellingShingle | Rukasov, V. I. Silin, E. S. Рукасов, В. И. Силин, Е. С. Рукасов, В. И. Силин, Е. С. Approximation of Continuous Functions by de La Vallee-Poussin Operators |
| title | Approximation of Continuous Functions by de La Vallee-Poussin Operators |
| title_alt | Приближение непрерывных функций операторами Балле Пуссена |
| title_full | Approximation of Continuous Functions by de La Vallee-Poussin Operators |
| title_fullStr | Approximation of Continuous Functions by de La Vallee-Poussin Operators |
| title_full_unstemmed | Approximation of Continuous Functions by de La Vallee-Poussin Operators |
| title_short | Approximation of Continuous Functions by de La Vallee-Poussin Operators |
| title_sort | approximation of continuous functions by de la vallee-poussin operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3590 |
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