Approximation of Continuous Functions of Low Smoothness by de la Vallee-Poussin Operators
We study some problems of the approximation of continuous functions defined on the real axis. As approximating aggregates, the de la Vallee-Poussin operators are used. We establish asymptotic equalities for upper bounds of the deviations of the de la Vallee-Poussin operators from functions of low sm...
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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 | 2020-03-18T19:59:42Z |
| description | We study some problems of the approximation of continuous functions defined on the real axis. As approximating aggregates, the de la Vallee-Poussin operators are used. We establish asymptotic equalities for upper bounds of the deviations of the de la Vallee-Poussin operators from functions of low smoothness belonging to the classes \(\hat C^{\bar \psi } \mathfrak{N}\). |
| first_indexed | 2026-03-24T02:45:40Z |
| format | Article |
| fulltext |
UDK 517.5
V. Y. Rukasov, E. S. Sylyn (Slavqn. ped. un-t)
PRYBLYÛENYE NEPRERÁVNÁX FUNKCYJ NEBOL|ÍOJ
HLADKOSTY OPERATORAMY VALLE PUSSENA
We study some problems of the approximation of continuous functions defined on the real line. As
approximating aggregates, we use the de la Vallée Poussin operators. We establish asymptotic equalities
for upper bounds of deviations of the de la Vallée Poussin operators from functions of small smoothness
belonging to the classes ̂C
ψ
� .
Vyvçagtsq deqki pytannq nablyΩennq neperervnyx funkcij, vyznaçenyx na dijsnij osi. V qkos-
ti nablyΩugçyx ahrehativ vykorystovugt\sq operatory Valle Pussena. Vstanovlggt\sq
asymptotyçni rivnosti dlq verxnix meΩ vidxylen\ operatoriv Valle Pussena vid funkcij malo]
hladkosti klasiv Ĉψ
� .
V rabote [1] vveden¥ klass¥ Ĉψ� sledugwym obrazom. Pust\ � — mnoΩest-
vo v¥pukl¥x vnyz pry vsex v ≥ 1 funkcyj ψ ( v ) takyx, çto lim ( )v v→∞ ψ = 0.
KaΩdug funkcyg ψ ∈ � doopredelym na [ 0; 1 ) tak, çtob¥ poluçennaq
funkcyq (kotorug, po-preΩnemu, oboznaçaem ψ ( ⋅ ) ) b¥la neprer¥vna dlq lg-
boho v ≥ 0, ψ ( 0 ) = 0 y ee proyzvodnaq ψ ′ ( v ) = ψ ′ ( v + 0 ) ymela ohranyçennug
varyacyg na promeΩutke [ 0; ∞ ). Tohda � oboznaçaet mnoΩestvo takyx funk-
cyj. PodmnoΩestvo funkcyj ψ, dlq kotor¥x
1
∞
∫ ψ( )t
t
dt < ∞,
oboznaçym � ′.
Pust\ ψk ∈ �, k = 1, 2, tohda ψk + y ψk – — sootvetstvenno çetnoe y neçet-
noe prodolΩenye funkcyy ψk , k = 1, 2. Dlq par¥ ( ψ1 , ψ2 ) opredelym funk-
cyg ψ :
ψ =df
ψ1 + + i ψ2 – . (1)
Pry πtom sootvetstvugwee preobrazovanye Fur\e funkcyy ψ ymeet vyd ψ̂ =
= ψ̂1+ + i ψ̂2−, hde preobrazovanye Fur\e ponymaetsq v ob¥çnom sm¥sle:
ˆ( )h t =
1
2π
R
ixth x e dx∫ −( ) .
Dalee, pust\ 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
.
Çerez ̂C
ψ� oboznaçym podmnoΩestvo neprer¥vn¥x funkcyj f ∈ L̂ , predsta-
vym¥x ravenstvom
f ( x ) = A0 +
R
x t t dt∫ +ϕ ψ( ) ˆ ( ) =df
A0 + ϕ ∗ ψ̂ , (2)
v kotorom A0 — nekotoraq postoqnnaq, yntehral ponymaetsq kak predel po
rasßyrqgwymsq symmetryçn¥m promeΩutkam, ϕ ∈ �, � ⊂ L̂ . Sleduq
© V. Y. RUKASOV, E. S. SYLYN, 2005
394 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
PRYBLYÛENYE NEPRERÁVNÁX FUNKCYJ NEBOL|ÍOJ HLADKOSTY … 395
A.HY.HStepancu [2], funkcyg ϕ ( ⋅ ) v (2) naz¥vagt ψ -proyzvodnoj funkcyy
f ( ⋅ ) y oboznaçagt f ψ ( )⋅ .
Dlq pryblyΩenyq funkcyj f ∈ ̂C
ψ� budem yspol\zovat\ operator¥ vyda
Vσ, c = Vσ, c ( f, x, Λσ, c ) = A0 + f ψ ∗ λ ψσ,c
̂
,
hde
λσ, c ( t ) =
1 0
0
, ,
, ,
, .
≤ ≤
−
−
≤ ≤
≤
t c
t
c
c t
t
σ
σ
σ
σ
Takye operator¥ rassmatryvalys\ A. Y. Stepancom v rabotax [1 – 4].
Oboznaçym
ρσ, c ( f; x ) =df
f ( x ) – Vσ, c ( f; x ), σ > 0, c > 0.
Cel\g rabot¥ qvlqetsq yzuçenye asymptotyçeskoho povedenyq pry σ → ∞
velyçyn
� �ˆ ; ,C V c
ψ
σ( ) = sup ( ; )
ˆ
, ˆ
f C
c C
f
∈
⋅
ψ
ρσ
�
, (3)
hde v kaçestve mnoΩestva � yspol\zugt edynyçn¥j ßar SM prostranstva su-
westvenno ohranyçenn¥x funkcyj M (v πtom sluçae Ĉ Mψ = Ĉ∞
ψ), a takΩe
klass¥ Hω : Hω = ϕ ω ϕ ω∈ ≤{ }ˆ : ( ; ) ( )C t t , hde Ĉ — podmnoΩestvo neprer¥vn¥x
funkcyj yz L̂ , ω ( ϕ; t ) — modul\ neprer¥vnosty funkcyy ϕ ( ⋅ ), ω ( t ) — fyk-
syrovann¥j modul\ neprer¥vnosty (v πtom sluçae Ĉψ� = Ĉ Hψ
ω).
Approksymacyonn¥e svojstva operatorov Vσ, c pry c = σ – 1 yssledovan¥
A.HY. Stepancom v rabotax [1 – 5], pry 0 < c ≤ σ – 1 — odnym yz avtorov dannoj
stat\y [6, 7]. V peryodyçeskom sluçae analohyçnaq zadaça dlq summ Fur\e re-
ßena v [8, 9], dlq summ Valle Pussena — v [10, 11].
Narqdu s operatoramy Vσ, c ( f; x ) v rabote [1] vveden¥ operator¥ vyda
V fcσ,
* ( ) = V f xc cσ σ,
*
,
*( , , )Λ = A0 + f ψ ∗ λ ψσ,
*
c
̂
,
hde
λσ,
* ( )c t =
λ σ
σ
ψ σ
ψ
σ
σ, , , , ,
( )
( )
, .
c t c
t c
c
t
t
c t
∈[ ] ∞[ ]
− −
−
( ) ≤ ≤
0
1
∪
sign (4)
Sleduq [2], yz mnoΩestva � v¥delym podmnoΩestva �0 y � C . KaΩdoj
funkcyy ψ ∈ � sopostavym funkcyy η ( t ) = ψ ψ− /( )1 2( )t y µ ( t ) = t t t/ −( )η( ) ,
t ≥ 1. Tohda � 0 = ψ µ ψ∈ < <{ }� : ( , )0 1t K , � C = ψ ∈{ � : 0 2< ≤K tµ ψ( , ) ≤
≤ K3 } , Ki = const, i = 1, 2, 3 . Polahagt �0 = � 0 ∩ � , ′� 0 = � 0 ∩ � ′, � C =
= � C ∩ �.
Sformulyruem osnovnoe utverΩdenye πtoj rabot¥.
Teorema. Pust\ ψ1 ∈ �0 y ψ2 ∈ ′� 0 . Tohda dlq lgb¥x σ y h = h ( σ ),
σ > h ≥ 1, pry σ → ∞
� ˆ ; ,C V h∞ −( )ψ
σ σ =
2 2
π
ψ
σ
∞
∫ ( )t
t
dt +
4
2π
ψ σ σ
( ) ln
h
+ O ( 1 ) A ( ψk , σ, h ), (5)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
396 V. Y. RUKASOV, E. S. SYLYN
� ˆ ; ,C H V h
ψ
ω σ σ−( ) =
= Θω
π
π
ω
σ
ψ σ ψ σ
π
σ ω
σ
1 2 2 2
0
1
1
2 2
0
2
∫ ∫ ∫
+
∞ /
t
s st ds dt
h
t
t dt( )sin
( )
ln sin +
+ O ( 1 ) A ( ψk , σ, h )ω
σ
1
−
h
, (6)
hde
A ( ψk , σ, h ) =
k
k kh
=
∑ − −( )
1
2
ψ σ ψ σ( ) ( ) + ψ σ( ) , Θω ∈ [ 2 / 3; 1 ],
pryçem Θω = 1, esly ω ( t ) — v¥pukl¥j modul\ neprer¥vnosty, y O ( 1 ) — ve-
lyçyna, ravnomerno ohranyçennaq po σ y h.
Dokazatel\stvo. Pust\ f ∈ ̂C
ψ�, tohda dlq velyçyn¥ ρσ, c ( f, x ) ymeet
mesto ravenstvo
ρσ, c ( f, x ) =
R
cf x t t dt∫ +ψ
στ( ) ( ),
*̂ +
R
cf x t d t dt∫ +ψ
σ( ) ˆ ( ), ,
hde τσ,
* ( )c t = 1 −( )λ ψσ,
* ( ) ( )c t t ,
ˆ ( ),d tcσ =
1
2( )
( ) ( ) ( ) ( ) ( )
σ π
ψ ψ σ ψ ψ σ
σ
−
− −( ) + − − −( )[ ]∫ −
c
s c s e s e dt
c
ist ist
.
Uprostym eho, v¥delyv hlavn¥e çasty y ocenyv ostatky. RassuΩdaq po sxeme,
yzloΩennoj v rabote [12, c. 218 – 235], yspol\zuq rezul\tat¥ rabot¥ [7] y pola-
haq c = σ – h, a ∈ ( 0; π σ / h ), poluçaem
ρσ, σ – h ( f, x ) = –
ψ σ
π
δ σ
σ π
1( )
( )
sin
a t h
x t
t
t
dt
/ /≤ ≤
∫ + +
+
ψ σ
π
δ σ
σ π
2( )
( )
cos
a t h
x t
t
t
dt
/ /≤ ≤
∫ + +
+
1
2π
δ ψ
σ σt a
x t s st ds dt
≤
∞
/
∫ ∫+( ) ( )sin + O ( 1 ) A ( ψk , σ, h ) ζ ( � ),
hde
δ ( x + t ) =
f x f x t f C H
f x t f C
ψ ψ ψ
ω
ψ ψ
( ) ( ), ˆ ,
( ), ˆ ,
− + ∈
+ ∈
∞
esly
esly
ζ ( � ) =
ω
σ
ψ
ω
ψ
1
1
−
∈
∈
∞
h
f C H
f C
, ˆ ,
, ˆ .
esly
esly
Yspol\zuq ravenstvo a sin α – b cos α = a b2 2+ −sin( )γ α , γ = arctg ( b / a ),
naxodym
ρσ, σ – h ( f, x ) = –
ψ σ
π
δ σ γ
σ π
σ( )
( )
sin( )
a t h
x t
t
t
dt
/ /≤ ≤
∫ + −
+
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
PRYBLYÛENYE NEPRERÁVNÁX FUNKCYJ NEBOL|ÍOJ HLADKOSTY … 397
+
1
2π
δ ψ
σ σt a
x t s st ds dt
≤
∞
/
∫ ∫+( ) ( )sin + O ( 1 ) A ( ψk , σ, h ) ζ ( � ),
hde γσ = arctg ( ψ2 ( σ ) / ψ1 ( σ ) ).
Najdem verxnye hrany
� �ˆ ; ,C V h
ψ
σ σ−( ) .
Pust\ [12, c. 232] xk = ( k π + γσ ) / σ, tk = xk – π / 2 σ, k = 0, ± 1, ± 2, … , σ ∈ R;
k0 — takoe znaçenye k, dlq kotoroho tk0
— blyΩajßaq sprava ot toçky ( a +
+ π ) / σ toçka, v kotoroj sin ( σ t – γσ ) = 1, a k1 — naybol\ßee yz znaçenyj k
takyx, çto tk < π / h; k2 — takoe çyslo, çto tk2
— blyΩajßaq sleva ot toçky
– ( a + π ) / σ toçka sredy tex, v kotor¥x sin ( σ t – γσ ) = – 1, a k3 — naymen\ßee yz
znaçenyj takyx, çto tk > – π / h. Opredelym funkcyg lσ ( t ) posredstvom ra-
venstv lσ ( t ) = xk , t ∈ [ tk , tk + 1 ], k = k0 , … , k1 – 1, k = k3 , k3 + 1, … , k2 – 1, i3, 1 =
= [ t3 , t2 ] ∪ [ t0 , t1 ].
Uçyt¥vaq ynvaryantnost\ klassov ̂C
ψ� otnosytel\no sdvyha arhumenta,
naxodym
� ˆ ; ,C V h∞ −( )ψ
σ σ ≤
t a
s st ds dt
≤
∞
/
∫ ∫
σ σ
π
ψ1
2( )sin +
+
ψ σ
π
σ γσ
σ
( ) sin( )
( )
,i
t
l t
dt
3 1
∫ −
+ O ( 1 ) A ( ψk , σ, h ). (7)
Sleduq rassuΩdenyqm yz rabot¥ [12, c. 236], ubeΩdaemsq, çto dlq funkcyy
ϕ∗
( t ), sovpadagwej na mnoΩestve −[ ]/ /a aσ σ; ∪ i3, 1 s funkcyej ϕσ ( t ), hde
ϕσ ( t ) =
sign
sign
σ
σ
σ
ψ
σ
σ γ
∞
∫ ≤
− ∈
2
3 1
( )sin , ,
sin( )
( )
, ,,
s st ds t
a
t
l t
t i
sootnoßenye (7) obratytsq v ravenstvo.
Dlq zaverßenyq dokazatel\stva formul¥ (5) ostaetsq ubedyt\sq, çto
1
2π
ψ
σ σt a
s st ds dt
≤
∞
/
∫ ∫ ( )sin =
2 2
π
ψ
σ
∞
∫ ( )t
t
dt + O ( 1 ) ψ σ( ) ,
i
t
l t
dt
3 1,
sin( )
( )∫ −σ γσ
σ
=
4
π
σ
ln
h
+ O ( 1 ).
Dlq πtoho vospol\zuemsq sootnoßenyqmy (5.5.4) y (5.5.5) yz [12, c. 236], pry do-
kazatel\stve kotor¥x peryodyçnost\ funkcyy f ( t ) y vklgçenye σ ∈ N, po su-
westvu, ne yspol\zovalys\.
DokaΩem teper\ formulu (6). Pust\ f ∈ Ĉ Hψ
ω , tohda, uçyt¥vaq opredele-
nye funkcyy lσ ( t ), naxodym
� ˆ , ,C H V h
ψ
ω σ σ−( ) ≤
ψ σ
π
ϕ σ γ
ϕ
σ
ω
( )
sup ( )sin( )
k k
k
k H t
t
x
t t dt
k
k
=
−
∈
∑ ∫
−
+
3
2 11 1
+
+
k k
k
k H t
t
x
t t dt
k
k
=
−
∈
∑ ∫
+
−
0
1 11 1
sup ( )sin( )
ϕ
σ
ω
ϕ σ γ +
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
398 V. Y. RUKASOV, E. S. SYLYN
+
1
0 2π
ϕ ϕ ψ
ϕ σ σω
sup ( ) ( ) ( )sin
∈ ≤
∞
/
∫ ∫−( )
H t a
t s st ds dt + A ( ψk , σ, h ).
V [12, c. 239, 240] b¥ly poluçen¥ neravenstva (5.5.16) y (5.5.17), pry πtom, po
suty, ne yspol\zovalos\ to, çto n ∈ N y ϕ — peryodyçeskaq funkcyq.
Poπtomu
� ˆ ; ,C H V h
ψ
ω σ σ−( ) ≤
ψ σ
π
ω σ
π σ
( )
( )sin
0
2 1 1
2
1 1
3
2
0
1/
∫ ∑ ∑
=
−
=
−
−
t t dt
x xk k
k
k k k
k
k
+
+
1
2
0
2π
ω ψ
σ
σ
a
t s st ds dt
/
∫ ∫
∞
( ) ( )sin + A ( ψk , σ, h ). (8)
Çtob¥ postroyt\ funkcyg f * ∈ Ĉ Hψ
ω , dlq kotoroj znaçenye ρσ, h ( f; x ) sov-
padaet s pravoj çast\g (8), budem rassuΩdat\ po analohyy s [12, c. 240; 241].
PoloΩym
ϕk ( t ) =
1
2
2
1
2
2 1
ω
ω
( ) , ; ,
( ) , ; ,
x t t t x
t x t x t
k k k
k k k
−( ) ∈[ ]
− −( ) ∈[ ]
+
k = k k3 2 1, − , k = k k0 1 1, − ,
ϕ+ ( t ) = ( ) ( )− −1 0k k
k tϕ –
1
2
2ω π
σ
ω
σ
−
a
, t ∈ t tk k; +[ ]1 , k = k k0 1 1, − ,
ϕ– ( t ) = ( ) ( )− − +1 2 1k k
k tϕ +
1
2
2ω π
σ
ω
σ
−
a
, t ∈ t tk k; +[ ]1 , k = k k3 2 1, − ,
ˆ ( )ϕ t =
1
2
2
1
2
2
1
2
2
0
0
0 1
2
3 2
ω σ
ω σ σ
ϕ
ω σ σ
ϕ
t t a
a t a t
t t t t
a t t a
t t t t
t
k
k k
k
k k
( ) ≤ /
/( ) ∈ /[ ]
∈[ ]
− /( ) ∈ − /[ ]
∈[ ]
+
−
, ,
, ; ,
( ), ; ,
, ; ,
( ), ; ,
.dlq ostal\n¥x
Funkcyq f *, ψ -proyzvodnaq kotoroj sovpadaet s funkcyej ˆ ( )ϕ t , qvlqetsq
yskomoj πkstremal\noj funkcyej, poskol\ku esly ω ( t ) — v¥pukl¥j modul\
neprer¥vnosty, to ˆ ( )ϕ t ∈ Hω y, kak pokaz¥vagt neposredstvenn¥e podsçet¥,
dlq funkcyy f * sootnoßenye (8) stanovytsq ravenstvom. Esly Ωe ω ( t ) —
proyzvol\n¥j modul\ neprer¥vnosty, to sootnoßenye (8) budet ravenstvom s
nekotor¥m mnoΩytelem Θω ∈ 2 3 1/[ ]; .
UbeΩdaqs\ v tom, çto
k k
k
kx=
−
∑
3
2 1 1
+
k k
k
kx=
−
∑
0
1 1 1
=
4
π
σ
ln
h
+ O ( 1 ),
zaverßaem dokazatel\stvo sootnoßenyq (6).
Sledstvye. Pust\ ψ1 ∈ �0 , ψ2 ∈ �C y limσ σ→∞ /h = 0. Tohda dlq
lgb¥x σ y h = h ( σ ), σ > h ≥ 1, pry σ → ∞ v¥polnqgtsq asymptotyçeskye
ravenstva
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
PRYBLYÛENYE NEPRERÁVNÁX FUNKCYJ NEBOL|ÍOJ HLADKOSTY … 399
� ˆ ; ,C V h∞ −( )ψ
σ σ =
4
2π
ψ σ σ
( ) ln
h
+ O ( 1 ) ψ σ( ) , (9)
� ˆ ; ,C H V h
ψ
ω σ σ−( ) =
2 2
2
0
2Θω
πψ σ
π
σ ω
σ
( )
ln sin
h
t
t dt
/
∫
+ O ( 1 ) ψ σ( ) ω
σ
1
,
dagwye reßenye zadaçy Kolmohorova – Nykol\skoho (sm. [8]) dlq operatorov
Valle Pussena na klassax Ĉ∞
ψ y Ĉ Hψ
ω sootvetstvenno.
Zametym, çto esly ψ 1 ∈ � C , to ravenstvo (9) sovpadaet s rezul\tatom
teorem¥ 2 [14].
1. Stepanets A. I., Wang Kunyang, Zhang Xirong. Approximation of locally integrable function on the
real line // Ukr. mat. Ωurn. – 1999. – 51, # 11. – S. 1549
–
1561.
2. 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.
3. 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.
4. Stepanec A. Y. PryblyΩenye operatoramy Fur\e funkcyj, zadann¥x na dejstvytel\noj
osy // Tam Ωe. – 1988. – 40, # 2. – S. 198
–
209.
5. Stepanec A. Y. PryblyΩenye v prostranstvax lokal\no yntehryruem¥x funkcyj // Tam
Ωe. – 1994. – 46, # 5. – S. 597
–
625.
6. Rukasov V. Y. PryblyΩenye operatoramy Valle Pussena funkcyj, zadann¥x na dejstvy-
tel\noj osy // Tam Ωe. – 1992. – 44, # 5. – S. 682
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| id | umjimathkievua-article-3607 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:40Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/68/3f66d588b50754f41cf5fae21ce26468.pdf |
| spelling | umjimathkievua-article-36072020-03-18T19:59:42Z Approximation of Continuous Functions of Low Smoothness by de la Vallee-Poussin Operators Приближение непрерывных функций небольшой гладкости операторами Валле Пуссена Rukasov, V. I. Silin, E. S. Рукасов, В. И. Силин, Е. С. Рукасов, В. И. Силин, Е. С. We study some problems of the approximation of continuous functions defined on the real axis. As approximating aggregates, the de la Vallee-Poussin operators are used. We establish asymptotic equalities for upper bounds of the deviations of the de la Vallee-Poussin operators from functions of low smoothness belonging to the classes \(\hat C^{\bar \psi } \mathfrak{N}\). Вивчаются деякі питання наближення неперервних функцій, визначених на дійсній осі. В якості наближуючих агрегатів використовуються оператори Валле Пуссена. Встановлюються асимптотичні рівності для верхніх меж відхилень операторів Валле Пуссена від функцій малої гладкості класів $C$. Institute of Mathematics, NAS of Ukraine 2005-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3607 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 3 (2005); 394–399 Український математичний журнал; Том 57 № 3 (2005); 394–399 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3607/3945 https://umj.imath.kiev.ua/index.php/umj/article/view/3607/3946 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 of Low Smoothness by de la Vallee-Poussin Operators |
| title | Approximation of Continuous Functions of Low Smoothness by de la Vallee-Poussin Operators |
| title_alt | Приближение непрерывных функций небольшой гладкости операторами Валле Пуссена |
| title_full | Approximation of Continuous Functions of Low Smoothness by de la Vallee-Poussin Operators |
| title_fullStr | Approximation of Continuous Functions of Low Smoothness by de la Vallee-Poussin Operators |
| title_full_unstemmed | Approximation of Continuous Functions of Low Smoothness by de la Vallee-Poussin Operators |
| title_short | Approximation of Continuous Functions of Low Smoothness by de la Vallee-Poussin Operators |
| title_sort | approximation of continuous functions of low smoothness by de la vallee-poussin operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3607 |
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