Approximation of some classes of periodic functions of many variables
We obtain the exact order of deviations of Fejér sums on the class of continuous functions. This order is determined by a given majorant of the best approximations.
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| Дата: | 2010 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2010
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2945 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508946979094528 |
|---|---|
| author | Tovkach, R. V. Товкач, Р. В. |
| author_facet | Tovkach, R. V. Товкач, Р. В. |
| author_sort | Tovkach, R. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:40:46Z |
| description | We obtain the exact order of deviations of Fejér sums on the class of continuous functions. This order is determined by a given majorant of the best approximations. |
| first_indexed | 2026-03-24T02:33:18Z |
| format | Article |
| fulltext |
UDK 517.5
R. V. Tovkaç (Volyn. nac. un-t, Luc\k)
NABLYÛENNQ DEQKYX KLASIV PERIODYÇNYX
FUNKCIJ BAHAT|OX ZMINNYX
We obtain the exact order of deviations of the Fejer sums on a class of continuous functions. This order
is defined by the given majorant of the best approximations.
Poluçen toçn¥j porqdok otklonenyj summ Fejera na klasse neprer¥vn¥x funkcyj, kotor¥j
opredelqetsq zadannoj maΩorantoj nayluçßyx pryblyΩenyj.
U danij roboti vstanovleno porqdok spadannq toçno] verxn\o] hrani vidxylen\
sum Fej[ra na klasax 2π-periodyçnyx funkcij bahat\ox zminnyx, wo vyzna-
çagt\sq obmeΩennqmy na poslidovnist\ najkrawyx nablyΩen\. Zokrema, odno-
vymirnyj rezul\tat S. B. St[çkina poßyreno na vypadok funkcij bahat\ox
zminnyx.
Rozhlqnemo prostir C T d( ) neperervnyx 2π-periodyçnyx po koΩnij zminnij
funkcij f x( ) , x T d∈ , T d = −( ]π π; d
z normog f C = max ( )
x
f x < ∞.
Poznaçymo çerez W mnoΩynu, wo sklada[t\sq z poliedriv V z racional\ny-
my verßynamy, zirkovyx vidnosno poçatku koordynat, qkyj [ vnutrißn\og toç-
kog V. Poznaçennq nV slid rozumity qk mnoΩynu toçok x takyx, wo
x
n
V∈ ,
tobto nV = x
x
n
V: ∈{ } .
Nexaj f ( )⋅ ∈ C T d( ) ,
S f V x c en k
i k x
k nV
( ; ; ) ( , )=
∈
∑ (1)
i
σn f V x( ; ; ) =
1
1 0n
S f V xk
k
n
+ =
∑ ( ; ; ) (2)
— vidpovidno çastynni sumy ]] rqdu Fur’[ i sumy Fej[ra.
Nexaj dali Tn V, — mnoΩyna tryhonometryçnyx polinomiv z harmonikamy z
nV, tobto
Tn V, = t t x a en n k
i k x
k nV
: ( ) ( , )=
∈
∑ ,
de ak — dovil\ni kompleksni çysla.
Poznaçymo çerez E fn V C, ( ) najkrawe nablyΩennq funkcij f ( )⋅ tryhono-
metryçnymy polinomamy z Tn V, v metryci C, tobto
E fn V C, ( ) = inf ( )
,t T
n C
n n V
f t x
∈
− .
Dlq zadano] poslidovnosti ε = εk{ } , k = 0, 1, 2, … , z monotonno spadnymy
çlenamy (tobto εk ↓ 0 , k → ∞ ) poznaçymo çerez C( )ε klas funkcij
f ∈ C T d( ) takyx, wo
E fn V C n, ( ) ≤ +ε 1 .
© R. V. TOVKAÇ, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8 1149
1150 R. V. TOVKAÇ
Metog dano] roboty [ vyznaçennq toçnoho porqdku spadannq velyçyny
U Cn ( ),ε σ( ) = sup ( ) ( ; ; )
( )
( )
f C
n C Tf x f V x d
∈
−
ε
σ .
Teorema. Dlq n ∈N ma[ misce spivvidnoßennq
B
n
k
k
n
1
1
01+ +
=
∑ ε ≤ U Cn ( ),ε σ( ) ≤
B
n
k
k
n
2
1
01+ +
=
∑ ε , (3)
de konstanty B1 i B2 zaleΩat\ vid rozmirnosti d prostoru i homoteta
V, ale ne zaleΩat\ vid n i f.
V odnovymirnomu vypadku take tverdΩennq dovedeno S.@B.@St[çkinym [2], a
pry d > 1 dlq V = Π = x x j:{ ≤ γ j , j = 1, 2,…, d} — S.@P.@Bajborodovym [3].
TakoΩ slid vidmityty rezul\taty O.@I.@Kuzn[covo] [4] dlq poliedriv.
Dovedennq. Zhidno z teoremog@2 v [1] dlq bud\-qkyx f ∈ C( )ε ma[mo
U fn ( ; )σ = f x f V xn C T d( ) ( ; ; ) ( )− σ ≤
B
n
E fk V
k
n
C
2
01+ =
∑ , ( ) ≤
≤
B
n
k
k
n
2
1
01+ +
=
∑ ε .
Tomu
U C
B
n
n k
k
n
( ),ε σ ε( ) ≤
+ +
=
∑2
1
01
. (4)
Dlq dovedennq oberneno] nerivnosti dlq klasu funkcij C( )ε
U C
n
n k
k
n
( ),ε σ ε( ) ≥
+ +
=
∑1
1
1
0
(5)
pobudu[mo funkcig f1( )⋅ ∈ C( )ε , ne zaleΩnu vid n, dlq qko]
U f
n
n k
k
n
( , )1 1
0
1
1
σ ε=
+ +
=
∑ . (6)
Nexaj ak > 0 i akk d∈∑ N < ∞. Todi dlq funkci]
f x0( ) = a k xk
k
j j
j
d
d∈ =
∑ ∏
N
cos
1
= a x
lV l Vl
j j
j
d
ν
ν
ν
∈ −=
∞
=
∑∑ ∏
\( )
cos
11 1
(7)
budemo maty
E fn V C, ( )0 ≤ f x S f V xn C0 0( ) ( ; ; )− ≤
≤ ak
k nVd∈
∑
N \
= a
kV k Vk n
ν
ν∈ −= +
∞
∑∑
\( )11
. (8)
Dlq velyçyny U fn ( , )0 σ znajdemo ocinku zverxu, vykorystavßy nerivnist\
(8):
U fn ( , )0 σ =
1
1
0 0
0n
f x S f V xk
k
n
C
+
−( )
=
∑ ( ) ( ; ; ) ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
NABLYÛENNQ DEQKYX KLASIV PERIODYÇNYX FUNKCIJ BAHAT|OX ZMINNYX 1151
≤
1
1
0 0
0n
f x S f V xk C
k
n
+
−
=
∑ ( ) ( ; ; ) ≤
1
1 110n
al
l V Vkk
n
+ ∈ −= +
∞
=
∑∑∑
ν νν \( )
=
=
1
1 1 1n
k a
k
n
kV k V+ = ∈ −
∑ ∑ ν
ν \( )
+ a
kV k Vk n
ν
ν∈ −= +
∞
∑∑
\( )11
. (9)
Krim toho, vraxovugçy (7), ma[mo
U fn ( , )0 σ ≥ f f Vn0 00 0( ) ( ; ; )− σ =
= a
k
n
a
kV k Vk kV k
ν
ν
ν
ν∈ −=
∞
∈ −
∑∑ − −
+
\( ) \(11 1
1
1 ))Vk
n
∑∑
=1
=
=
1
1 1 1n
k a
k kV k V+ = ∈ −
∑ ∑ ν
ν \( )
+ a
kV k Vk n
ν
ν∈ −= +
∞
∑∑
\( )11
. (10)
Z spivvidnoßen\ (9) i (10) vyplyva[
U fn ( , )0 σ =
1
1 1 1n
k a
k
n
kV k V+ = ∈ −
∑ ∑ ν
ν \( )
+ a
kV k Vk n
ν
ν∈ −= +
∞
∑∑
\( )11
. (11)
Poznaçymo çerez s n( ) kil\kist\ toçok k = ( ,k k1 2 , … , kd ) , ki ∈N , i = 1, d ,
u mnoΩyni nV n V\( )− 1 , tobto
s n
k nV n V
( )
\( )
=
∈ −
∑ 1
1
.
Teper vyznaçymo funkcig f x1( ) takym çynom:
f x1( ) = ( )
( )
cos
\( )
ε ε ν
ν
k k
k
i i
i
d
kV n Vs k
x− +
=
∞
=∈ −
∑ ∏∑1
1 11
1
. (12)
Dlq tak vyznaçeno] funkci] f x1( ) zhidno z (8) ma[mo
E fn V C, ( )1 ≤ ( )
( )
( )ε εk k
k n s k
s k− +
= +
∞
∑ 1
1
1
= εn+1 .
OtΩe, f1( )⋅ naleΩyt\ C( )ε .
PokaΩemo, wo
U fn ( , )1 σ = f x f V xn C1 1( ) ( ; ; )− σ =
1
1
1
0n
k
k
n
+ +
=
∑ ε .
Na osnovi rivnosti (11) znajdemo
U fn ( , )1 σ =
1
1
1
1
1n
k
s k
s kk k
k
n
+
− +
=
∑ ( )
( )
( )ε ε +
+ ( )
( )
( )ε εk k
k n s k
s k− +
= +
∞
∑ 1
1
1
=
1
1
1
0n
k
k
n
+ +
=
∑ ε .
Tobto dlq funkci] f x1( ) , vyznaçeno] rivnistg (12), vykonugt\sq spivvidnoßen-
nq (6), a otΩe, ma[ misce i spivvidnoßennq (5).
Teoremu dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
1152 R. V. TOVKAÇ
Naslidok. Qkwo f ( )⋅ naleΩyt\ C T d( ) , to dlq bud\-qkoho r ∈N
U fn ( , )σ ≤
B
n
f
k
r
k
n
3
01
1
1+ +
=
∑ ω ; , (13)
de B3 zaleΩyt\ vid rozmirnosti prostoru d, homoteta V i çysla r, a
ωr f
k
;
1
1+
=
= sup ( ) ( , ,
t t t
k
r r
d
C f x t x
1
2
2
2 2 1
1
1 11
+ + … + ≤
+
−− + …ν
ν ν dd d
r
C
t+
=
∑ ν
ν
)
0
.
Dovedennq. Oçevydno, wo dlq dovil\noho k ∈( )N ∪ 0 isnugt\ çysla µi ,
µi ∈( )N ∪ 0 , taki, wo
E f E fk V C Cd, , , ,( ) ( )≤ …µ µ µ1 2
, (14)
de
E f
d Cµ µ µ1 2, , ,
( )
…
= inf ( ) ( , , , )
, , ,
, , ,
t
d C
d
d
f x t x x x
µ µ µ
µ µ µ
1 2
1 2 1 2
…
− ……
— povne najkrawe nablyΩennq funkci] f x x( ,1 2 , … , xd ) ∈ C T d( ) tryhono-
metryçnymy polinomamy t x x
dµ µ µ1 2 1 2, , , ( , ,… , … , xd ) porqdku ≤ µi vidpovidno
po zminnyx xi , i = 1, 2, … , d. Teper skorysta[mos\ uzahal\nenog teoremog
DΩeksona (dyv. [5, s. 113], teorema 2.4.1), na osnovi qko]
E f B f
d C rµ µ µ ω ρ
1 2 3, , , ( ) ( ; )… ≤ , ρ µ µ µ= + + … +− − −
1
2
2
2 2
d . (15)
Vraxovugçy (3), (14) i (15), otrymu[mo spivvidnoßennq (13).
Naslidok dovedeno.
1. Zaderej N. M., Tovkaç R. V. NablyΩennq periodyçnyx funkcij bahat\ox zminnyx sumamy
Fej[ra // Teoriq nablyΩen\ funkcij ta sumiΩni pytannq: Zb. prac\ In-tu matematyky NAN
Ukra]ny. – 2010. – 7, # 1. – S. 341 – 347.
2. Steçkyn S. B. O pryblyΩenyy peryodyçeskyx funkcyj summamy Fejera // Tr. Mat. yn-ta
AN SSSR. – 1961. – 62. – S. 48 – 60.
3. Bajborodov S. P. O pryblyΩenyy funkcyj mnohyx peremenn¥x prqmouhol\n¥my summamy
Valle Pussena // Mat. zametky. – 1981. – 29, # 5. – S. 711 – 730.
4. Kuznecova O. Y. Syl\naq summyruemost\, neravenstva Sydona, yntehryruemost\ // Rqdy
Fur’[: teoriq i zastosuvannq : Zb. prac\ In-tu matematyky NAN Ukra]ny. – 1998. – 20. –
S.@142 – 150.
5. Tyman M. F. Approksymacyq y svojstva peryodyçeskyx funkcyj. – Dnepropetrovsk: Po-
lyhrafyst, 2000. – 320 s.
OderΩano 04.12.09,
pislq doopracgvannq — 03.06.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
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| id | umjimathkievua-article-2945 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:33:18Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d7/9134234ea225a0502a73946249ad6fd7.pdf |
| spelling | umjimathkievua-article-29452020-03-18T19:40:46Z Approximation of some classes of periodic functions of many variables Наближення деяких класів періодичних функцій багатьох змінних Tovkach, R. V. Товкач, Р. В. We obtain the exact order of deviations of Fejér sums on the class of continuous functions. This order is determined by a given majorant of the best approximations. Получен точный порядок отклонений сумм Фейера на классе непрерывных функций, который определяется заданной мажорантой наилучших приближений. Institute of Mathematics, NAS of Ukraine 2010-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2945 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 8 (2010); 1149–1152 Український математичний журнал; Том 62 № 8 (2010); 1149–1152 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2945/2640 https://umj.imath.kiev.ua/index.php/umj/article/view/2945/2641 Copyright (c) 2010 Tovkach R. V. |
| spellingShingle | Tovkach, R. V. Товкач, Р. В. Approximation of some classes of periodic functions of many variables |
| title | Approximation of some classes of periodic functions of many variables |
| title_alt | Наближення деяких класів періодичних функцій багатьох змінних |
| title_full | Approximation of some classes of periodic functions of many variables |
| title_fullStr | Approximation of some classes of periodic functions of many variables |
| title_full_unstemmed | Approximation of some classes of periodic functions of many variables |
| title_short | Approximation of some classes of periodic functions of many variables |
| title_sort | approximation of some classes of periodic functions of many variables |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2945 |
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