Approximation of classes of periodic multivariable functions by linear positive operators
In an N-dimensional space, we consider the approximation of classes of translation-invariant periodic functions by a linear operator whose kernel is the product of two kernels one of which is positive. We establish that the least upper bound of this approximation does not exceed the sum of properly...
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| Дата: | 2006 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2006
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3430 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509519713402880 |
|---|---|
| author | Bushev, D. M. Kharkevych, Yu. I. Бушев, Д. М. Харкевич, Ю. І. |
| author_facet | Bushev, D. M. Kharkevych, Yu. I. Бушев, Д. М. Харкевич, Ю. І. |
| author_sort | Bushev, D. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:54:30Z |
| description | In an N-dimensional space, we consider the approximation of classes of translation-invariant periodic functions by a linear operator whose kernel is the product of two kernels one of which is positive. We establish that the least upper bound of this approximation does not exceed the sum of properly chosen least upper bounds in m-and ((N ? m))-dimensional spaces. We also consider the cases where the inequality obtained turns into the equality. |
| first_indexed | 2026-03-24T02:42:24Z |
| format | Article |
| fulltext |
UDK 517.5
D. M. Bußev, G. I. Xarkevyç (Volyn. un-t, Luc\k)
NABLYÛENNQ KLASIV PERIODYÇNYX
FUNKCIJ BAHAT|OX ZMINNYX LINIJNYMY
DODATNYMY OPERATORAMY
In an N-dimensional space, we consider the approximation of classes of periodic functions which are
invariant with respect to a displacement by a linear operator with kernel determined as the product of
two kernels, one of which is positive. We establish that the least upper bound of this approximation does
not exceed the sum of respectively chosen least upper bounds in m- and ( N – m )-dimensional spaces.
We also consider the cases in which the obtained inequality becomes the equality.
Vstanovleno, wo v N-vymirnomu prostori toçna verxnq meΩa nablyΩennq klasiv periodyçnyx
funkcij, invariantnyx vidnosno zsuvu, linijnym operatorom z qdrom, wo [ dobutkom dvox qder,
odne z qkyx [ dodatnym, ne perevywu[ sumy vidpovidno vybranyx toçnyx verxnix meΩ v m- i ( N –
m )-vymirnyx prostorax. Rozhlqnuto vypadky, v qkyx dlq otrymano] nerivnosti ma[ misce znak
rivnosti.
Vidomo, wo znaxodΩennq velyçyn toçnyx verxnix meΩ nablyΩennq klasiv peri-
odyçnyx funkcij linijnymy operatoramy u prostori rozmirnosti N ne zavΩdy
moΩna zvesty do obçyslennq vidpovidnyx velyçyn u prostori menßo] rozmirnos-
ti. V danij roboti qkraz i vkazano umovy, pry qkyx ce moΩlyvo zrobyty.
Dovedeno, wo v N-vymirnomu prostori toçna verxnq meΩa nablyΩennq kla-
siv periodyçnyx funkcij, invariantnyx vidnosno zsuvu, linijnym operatorom z
qdrom, wo [ dobutkom dvox qder, odne z qkyx m-vymirne i dodatne, ne perevywu[
sumy vidpovidno vybranyx toçnyx verxnix meΩ v m- i ( N – m )-vymirnyx prosto-
rax. Vstanovleno, wo v nerivnosti ma[ misce znak rivnosti dlq central\no-sy-
metryçnyx klasiv neperervnyx abo istotno obmeΩenyx funkcij, qki zadovol\nq-
gt\ we odnu dodatkovu umovu. Z oderΩanyx rezul\tativ u vyhlqdi naslidku
moΩna oderΩaty teoremu 1 z roboty [1].
Pokazano, wo nablyΩennq linijnym dodatnym operatorom z dovil\nym qdrom
zaleΩyt\ lyße vid odnovymirnyx skladovyx c\oho qdra, tobto zbiha[t\sq z na-
blyΩennqm deqkym linijnym dodatnym operatorom z qdrom, wo [ dobutkom od-
novymirnyx qder.
Nexaj C
N, LN
∞ i Lp
N
— prostory 2 π-periodyçnyx po koΩnij iz N-zminnyx
funkcij f ( x ) = f ( x1 , … , xN ), vidpovidno neperervnyx, sutt[vo obmeΩenyx i su-
movnyx v p-mu stepeni ( 1 ≤ p < ∞ ), z normamy
f C N = sup ( )
x
f x , f LN
∞
= sup ( )
x
f xvrai ,
f Lp
N =
1
2
1
π
∫
/N
P
p
p
N
f x d x( ) ,
de
PN =
i
N
=
∏ [ ]
1
0 2; π
— N-vymirnyj kub.
Poznaçymo çerez x
m = ( x1 , … , xm ), x
N
–
m = ( xm + 1 , … , xN ) i x = ( x1 , … , xN )
vidpovidno m-, ( N – m )- i N-vymirni vektory, koordynatamy qkyx [ dijsni çys-
la; çerez E
m
, E
N
–
m
vidpovidno mnoΩyny vsix m - i ( N – m )-vymirnyx vektoriv
© D. M. BUÍEV, G. I. XARKEVYÇ, 2006
12 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
NABLYÛENNQ KLASIV PERIODYÇNYX FUNKCIJ BAHAT|OX ZMINNYX … 13
k
m = ( k1 , … , ki , … , km ) , k
N
–
m = ( km + 1 , … , kN ) , koordynatamy qkyx [ cili
nevid’[mni çysla. S ( k
m
) i S ( k
N
– m ) — çyslo koordynat vektoriv k
m
i k
N
– m, wo
dorivnggt\ nulg, B
m
i B
N
– m — vidpovidno mnoΩyny najmoΩlyvißyx m- i ( N –
– m )-vymirnyx vektoriv, koΩna koordynata qkyx dorivng[ 0 abo 1; B ( k
m
) i
B ( k
N
– m ) — pidmnoΩyny mnoΩyn B
m
i B
N
– m taki, wo qkwo k j = 0, to i i j = 0,
de i
m = ( i1 , … , ij , … , im ) ∈ B ( k
m
) ,
a
k
i
m
m
=
1
21π
π
m
P
m
j
m
j j j
m
f t k t i dt∫ ∏
=
−
( ) cos
— koefici[nty Fur’[ funkci] f ( x
m
) , de i
m = ( i1 , … , ij , … , im ) ∈ B ( k
m
) ,
U f x
n
m
m
+ ( ; ; )Λ =
1
π
λm
P
m m
n
m
m
mf x t t dt∫ + +( ) ( , )Λ ,
U f x
n
N m
N m−
−( ; ; )Λ =
1
π
λN m
P
N m N m
n
N m
N m
N mf x t t dt−
− − −
−
−∫ +( ) ( , )Λ ,
U f x
n nm N m;
( ; ; )−
+ Λ =
1
π
λ λN
P
n
m
n
N m
N
m N mf x t t t dt∫ + + −
−( ) ( , ) ( , )Λ Λ ,
U f x
n n nm
N m
1; ; ;
( ; ; )…
+
− Λ =
1
1π
λ µ λN
P i
m
n i n
N m
N
i N mf x t t t dt∫ ∏+
=
+ −
−( ) ( , , ) ( , )Λ Λ ,
U f xn ii
+ ( ; ; ; )λ µ =
1
0
2
π
λ µ
π
∫ + +f x t t dti i n i ii
( ) ( , , )Λ
— linijni operatory vidpovidno z qdramy
Λ
n
m
m t+ ( ; )λ =
k E
s k
l B k
k
l
i
m
l
i i i
m m
m
m m
m
m
i k t l
∈ ∈ =
∑ ∑ ∏ − −
1
2
1
21
( )
( )
( ) cosλ π
≥ 0,
Λ
n
N m
N m t−
−( ; )λ =
=
k E
s k
l B k
k
l
i m
N
l
i i i
N m N m
N m
N m N m
N m
N m
i k t l
− −
−
− −
−
−
∈ ∈ = +
∑ ∑ ∏ − −
1
2
1
21
( )
( )
( ) cosλ π
,
Λn ii
t+ ( ; ; )λ µ =
1
2
+
k
n
k
n
i k
n
i
i
i ikt kt
=
−
∑ −( )
1
1
λ µ( ) ( )cos sin ≥ 0.
Operator U f x
n
m
m
+ ( ; ; )Λ moΩna zapysaty u vyhlqdi
U f x
n
m
m
+ ( ; ; )Λ =
=
k E
s k
l B k
k
l
i B k
k
i
j
m
j j j j
m m
m
m m
m
m
m m
m
m
a k t i l
∈ ∈ ∈ =
∑ ∑ ∑ ∏ − +
1
2 21
( )
( ) ( )
cos ( )λ π
.
Qkwo f ( x ) = f ( xi ) — funkciq odni[] zminno] xi , to
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
14 D. M. BUÍEV, G. I. XARKEVYÇ
U f x
n im
+ ( ; ; )Λ =
1
2 0m a +
+
1
2 1
1
1
m
k
n
k k i i k i i k k i i k i i
i
i
i i i i i i
a k x b k x a k x b k x−
=
−
∑ +( ) + −( )( )λ µcos sin sin cos ,
de
aki
= a ki0 0 0 0
0 0 0 0 0
, , , , , ,
, , , , , ,
… …
… …
, bki
= a ki0 0 0 0
0 0 1 0 0
, , , , , ,
, , , , , ,
… …
… …
,
(1)
λki
= λ 0 0 0 0
0 0 0 0 0
, , , , , ,
, , , , , ,
… …
… …
ki
, µki
= λ 0 0 0 0
0 0 1 0 0
, , , , , ,
, , , , , ,
… …
… …
ki
.
Vvedemo poznaçennq
U f xn ii
+ ( ; ; )Λ = 2 1m
n iU f xm
− + ( ; ; )Λ =
1
π
λm
P
i i n
m
m
mf x t t dt∫ + +( ) ( ; )Λ =
=
1
0
2
π
λ
π
∫ + ( )+f x t t dti i n i ii
( ) ;Λ ,
de
Λn ii
t+ ( )λ; =
1
2
+
k
n
k i i k i i
i
i
i i
k t k t
=
−
∑ −( )
1
1
λ µcos sin ≥ 0,
a λki
i µki
vyznaçagt\sq formulamy (1).
Nexaj A — bud\-qkyj linijnyj operator, qkyj vidobraΩa[ mnoΩynu M ⊂ X
N
( X
N = C
N, LN
∞ , Lp
N ) v X
N
, mnoΩyna M [ invariantnog vidnosno zsuvu, tobto z
vklgçennq f ( x ) ∈ M vyplyva[, wo f ( x + t ) ∈ M i G M A
X N( , ) =
= sup ( )
f M
Xf A f N
∈
− — nablyΩennq mnoΩyny M operatorom A u prostori X
N
.
Poznaçymo çerez M
X m , M
X N m− i M xi
pidmnoΩyny funkcij mnoΩyny M,
qki otrymugt\sq z M, qkwo zafiksuvaty zminni vidpovidno ( xm + 1 , … , xN ) ,
( x1 , … , xm ) i ( x1 , … , xi – 1, xi + 1 , … , xN ).
U roboti dovedeno, wo qkwo mnoΩyna M [ invariantnog vidnosno zsuvu, to
G M U
n n X
m N m N
,
; −
+( ) ≤ G M U
X n X
m m m, +( ) + G M U
X n X
N m N m N m− − −( ), , (2)
i rozhlqnuto vypadky, pry qkyx v nerivnosti (2) ma[ misce znak rivnosti.
Qkwo zafiksuvaty zminni x xm1
0 0, ,…( ) = xm
0 abo x xm N+ …( )1
0 0, , = x N m
0
−
, to z
oznaçennq norm vyplyva[
f C N = sup ,
x
m N m
Cm
N mf x x
0
0
−( ) − = sup ,
x
m N m
CN m
mf x x
0
0
−
−( ) , (3)
f LN
∞
= sup ,
x
m N m
Lm
N mf x x
0
0vrai −( )
∞
− = sup ,
x
m N m
LN m
mf x x
0
0
− ∞
−( )vrai , (4)
f LP
N ≤ sup ,
x
m N m
Lm P
N mf x x
0
0
−( ) − , (5)
f LP
N ≤ sup ,
x
m N m
LN m P
mf x x
0
0
−
−( ) . (6)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
NABLYÛENNQ KLASIV PERIODYÇNYX FUNKCIJ BAHAT|OX ZMINNYX … 15
Lema 1. Qkwo mnoΩyna M ⊂ X
N [ invariantnog vidnosno zsuvu, to vykonu-
[t\sq nerivnist\ (2).
Dovedennq. Vvedemo dopomiΩni linijni operatory
U f x
nm ;
( ; ; )∞
+ Λ =
1
π
λm
P
m m N m
n
m
m
mf x t x t dt∫ + − +( , ) ( , )Λ ,
U f x
nN m− ∞; ( ; ; )Λ =
1
π
λN m
P
m N m N m
n
N m
N m
N mf x x t t dt−
− − −
−
−∫ +( , ) ( , )Λ .
Vykorystovugçy teoremu Fubini, nevid’[mnist\ qdra Λ
n
m
m t+ ( ; )λ i vyznaçen-
nq operatoriv U f x
nm ;
( ; ; )∞
+ Λ , U f x
nN m− ∞; ( ; ; )Λ , otrymu[mo
f x U f x
n n X
m N m N
( ) ( ; ; )
;
− −
+ Λ ≤
≤
1
π
λm
P
m m N m
n
m m
Xm
m
N
f x f x t x t dt∫ − +( )− +( ) ( ; ) ( , )Λ +
1
π
λm
P
n
m
m
m t∫ +Λ ( , ) ×
×
1
π
λN m
P
m m N m
n
N m N m m
XN m
N m
N
f x t x f x t t dt dt−
− − −
−
−∫ + − +( )
( , ) ( ) ( , )Λ =
= f x U f x
n X
m N
( ) ( ; ; )
;
− ∞
+ Λ +
+
1
π
λm
P
n
m m m N m
n
m m N m m
Xm
m N m
N
t f x t x U f x t x dt∫ + −
∞
−+ − +( )( )−Λ Λ( , ) ( , ) ; ; ( , )
;
.
(7)
Iz nerivnosti (7), vykorystovugçy uzahal\nenu nerivnist\ Minkovs\koho, (dyv.,
napryklad, [2, c. 22]), znaxodymo
G M U
n nm N m,
; −
+( ) ≤ sup ( ) ( ; ; )
;
f M
n X
f x U f xm N
∈
∞
+−
Λ +
+
1
π
λm
P
n
m m m N m
n
m m N m
X
m
m
m N m N
t f x t x U f x t x dt∫ + −
∞
−+ − +( )( )
−Λ Λ( , ) ( ; ) ; ; ,
;
.
(8)
Iz nerivnosti (8) i spivvidnoßen\ (3) – (6) vyplyva[
G M U
n nm N m,
; −
+( ) ≤ sup sup , ; ; ,
;
f M x
m N m
n
m N m
XN m
m m
f x x U f x x
∈
−
∞
+ −
−
( ) − ( )( )
0
0 0Λ +
+
1
π
λm
P
n
m
m
m t∫ +Λ ( , ) ×
× sup sup , ; ; ,
;
f M x t
m m N m
n
m m N m
X
m
m m
N m N m
f x t x U f x t x dt
∈ +
−
∞
−+( ) − +( )( )− −
0 0
0 0 0 0Λ .
Oskil\ky f ( x ) pry fiksovanyx x N m
0
−
i xm
0 + tm
0 naleΩyt\ vidpovidno mnoΩy-
nam M
X m i M
X N m− , to, vykorystovugçy oznaçennq operatoriv U f x
nm ;
( ; ; )∞
+ Λ
i U f x
nN m− ∞;
( ; ; )Λ , U f x
n
m
m
+ ( ; ; )Λ , U f x
n
N m
N m−
−( ; ; )Λ , otrymu[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
16 D. M. BUÍEV, G. I. XARKEVYÇ
G M U
n n X
m N m N
,
; −
+( ) ≤ sup ( ) ( ; ; )
f M
n
m
X
X m
m mf x U f x
∈
+− Λ +
+
1
π
λm
P
n
m
m
m t∫ +Λ ( , ) sup ( ) ( ; ; )
f M
n
N m
X
m
X
N m
N m N mf x U f x dt
∈
−
−
− −− Λ =
= G M U
X n X
m m m, +( ) + G M U
X n X
N m N m N m− − −( ), .
Lemu 1 dovedeno.
Naslidok 1. Qkwo mnoΩyna M ⊂ X
N [ invariantnog vidnosno zsuvu, to
G M U
n n n Xm
N m N
,
, , ;1 …
+
−( ) ≤
i
m
x
n X
G M Ui
i
=
+∑ ( )
1
, ( , )λ µ + G M U
X n X
N m N m N m− − −( ), .
Ma[ misce taka teorema.
Teorema 1. Nexaj mnoΩyna M ∈ X
N ( X
N = C
N, LN
∞) [ invariantnog vidnos-
no zsuvu i central\no-symetryçnog, tobto z vklgçennq f ( x ) ∈ M vyplyva[,
wo – f ( x ) ∈ M. Qkwo z vklgçennq f ( x
m
) ∈ M
X m ⊂ M i g ( x
N
–
m
) ∈ M
X N m− ⊂ M
vyplyva[, wo f x g xm N m( ) ( )+( )− ∈ M, to
G M U
n n X
m N m N
,
; −
+( ) = G M U
X n X
m m n, +( ) + G M U
X n X
N m N m N m− − −( ), . (9)
Dovedennq. Prypustymo, wo G M U
X n X
m m m, +( ) = ∞ abo
G M U
X n X
N m N m N m− − −( ), = ∞. Oskil\ky M
X m ⊂ M i M
X N m− ⊂ M, to
G M U
n n X
m N m N
,
; −
+( ) ≥ G M U
X n n X
m m N m N
,
; −
+( ) = G M U
X n X
m m m, +( ) = ∞
abo
G M U
n n X
m N m N
,
; −
+( ) ≥ G M U
X n X
N m N m N m− − −( ), .
Z lemy 1 vyplyva[ (9).
Nexaj G M U
X n X
m m m, +( ) < ∞ i G M U
X n X
N m N m N m− − −( ), < ∞. Todi za oznaçen-
nqm toçno] verxn\o] meΩi isnugt\ funkci] ϕk ( x
m
) ∈ M
X m i φ ( x
N
–
m
) ∈ M
X N m−
taki, wo magt\ misce spivvidnoßennq
G M U
X n X
m m m, +( ) = ϕ ϕk
m
n k
m
X
x U xm m( ) ( ; ; )− + Λ
i
G M U
X n X
N m N m N m− − −( ), = φ φk
N m
n k
N m
X
x U xN m N m( ) ( ; ; )− −− − −Λ ,
abo isnugt\ poslidovnosti funkcij ϕk
mx( ){ } ∈ M
X m i φk
N mx( )−{ } ∈ M
X N m− ,
k = 1, 2, 3, … , dlq qkyx
G M U
X n X
m m m, +( ) = lim ( ) ( ; ; )
k
k
m
n k
m
X
x U xm m
→∞
+−ϕ ϕΛ (10)
i
G M U
X n X
N m N m N m− − −( ), = lim ( ) ( ; ; )
k
k
m
n k
N m
X
x U xN m N m
→∞
−− − −φ φΛ . (11)
Pokladagçy fk ( x ) = ϕ φk
m
k
N mx x( ) ( )+( )− ∈ M ta vraxovugçy central\nu
symetrig mnoΩyny M i te, wo X
N = C
N
abo X
N = LN
∞ , otrymu[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
NABLYÛENNQ KLASIV PERIODYÇNYX FUNKCIJ BAHAT|OX ZMINNYX … 17
G M U
n n X
m N m N
,
; −
+( ) ≥ lim ( ) ( ; ; )
;k
k n n X
f x U f xm N m N→∞
+− − Λ =
= lim ( ) ( ; ; )
k
k
m
n k
m
X
x U xm m
→∞
+−ϕ ϕΛ +
+ lim ( ) ( ; ; )
k
k
N m
n k
N m
X
x U xN m N m
→∞
− −− − −φ φΛ ,
a ce razom iz spivvidnoßennqmy (10), (11), (2) i dovodyt\ (9).
Vidmitymo, wo isnu[ mnoΩyna M ⊂ C
N
, invariantna vidnosno zsuvu i cent-
ral\no-symetryçna, dlq qko] iz vklgçen\ f ( x
m
) ∈ M
X m i g ( x
N
–
m
) ∈ M
X N m−
ne vyplyva[, wo f x g xm N m( ) ( )+( )− ∈ M.
Nexaj, napryklad, M = Hω ( t, z ) ⊂ C
2
— mnoΩyna neperervnyx 2 π-periodyç-
nyx po koΩnij iz zminnyx funkcij f ( x, y ), dlq qkyx ω ( f; t; z ) ≤ ω ( t, z ), de
ω ( f; t; z ) ≤ sup ( , ) ( , )
,h t z
Cf x h y f x y
≤ ≤
+ + ∂ −
∂
2 ,
a ω ( t, z ) — funkciq typu modulq neperervnosti. Vidomo (dyv., napryklad, [3,
c. 124]), wo
max ( , ); ( , )ω ωt z0 0{ } ≤ ω ( t, z ) ≤ ω ( t, 0 ) + ω ( 0, z ).
Qkwo ω ( t, z ) ≠ ω ( t, 0 ) + ω ( 0, z ), to f ( t ) ∈ ω ( t, 0 ) ∈ Hω ( t, 0 ) = M
x
i g ( z ) =
= ω ( 0, z ) ∈ Hω ( 0, z ) = M
y
, ale f t g z( ) ( )+( ) ∉ M.
Naslidok 2. Qkwo M — central\no-symetryçna mnoΩyna, invariantna
vidnosno zsuvu, dlq qko] iz vklgçen\ f ( xi ) ∈ M xi ⊂ M , i = 1, m , i
g ( x
N
–
m
) ∈ M
X N m− ⊂ M vyplyva[, wo
i
m
i
N mf x g x=
−∑ +( )1
( ) ( ) ∈ M, to
G M U
n n n Xm
N m N
,
, , ;1 …
+
−( ) =
i
m
x
n x
G M Ui
i i=
∑ ( )
1
, ( , )λ µ + G M U
X n X
N m N m N m− − −( ), ,
de X
N = C
N
abo X
N = LN
∞ .
Z naslidku 1, mirkugçy, qk i pry dovedenni teoremy 1, oderΩu[mo naslidok 2.
Poznaçymo çerez H N
ω( )1 klas funkcij f ( x ) ∈ C
N
, qki zadovol\nqgt\ umovu
f x f x( ) ( )− ′ ≤
i
N
i i ix x
=
∑ − ′( )
1
1ω( )
,
a çerez H N
ω( )2 klas funkcij f ( x ) ∈ C
N
, dlq qkyx
f x h f x f x h( ) ( ) ( )+ − + −2 ≤
i
N
i ih
=
∑ ( )
1
2ω( )
.
Tut ωi it
( )( )1
i ωi it
( )( )2
— dovil\ni fiksovani funkci] typu moduliv neperervnos-
ti perßoho i druhoho porqdkiv vidpovidno. Oskil\ky klasy H i
N
ω( ) , i = 1, 2, zado-
vol\nqgt\ umovy naslidku 2, to, vykorystavßy joho, moΩna otrymaty, napryk-
lad, teoremu 1 z [1].
Teorema 2. Qkwo mnoΩyna N zadovol\nq[ umovy naslidku 2 i, krim toho, iz
vklgçennq f ( x ) ∈ M vyplyva[, wo ( f ( x ) + C) ∈ M, de C — dovil\na stala,
to
G M U
n n X
m N m N
,
; −
+( ) =
i
m
x
n X
G M Ui
i
=
+∑ ( )
1
1, ( )Λ + G M U
X n X
N m N m N m− − −( ), . (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
18 D. M. BUÍEV, G. I. XARKEVYÇ
Dovedennq. Oskil\ky mnoΩyna M
X m ∈ X
N
[ invariantnog vidnosno zsuvu,
X
N = C
N
*abo X
N = LN
∞ i dlq koΩno] funkci] f ( x
m
) ∈ M
X m vyplyva[, wo
( f ( x
m
) + C) ∈ M
X m , to
G M U
X n X
m m m, +( ) = sup ( ) ,
f M
m
P
m
n
m
X m m
mf t t dt
∈
+∫ ( )
0
1
π
λΛ , (13)
de M
X m
0
— pidmnoΩyna funkcij mnoΩyny M
X m takyx, wo f ( 0 ) = f ( 0, … , 0 ) =
= 0. Vnaslidok toho, wo mnoΩyny M xi
zadovol\nqgt\ ti Ω umovy, wo i mno-
Ωyna M
X m , vraxovugçy oznaçennq operatoriv U f xn ii
+ ( ; ; )Λ , ma[mo
G M Ux
n X
i
i
, ( )+( )Λ 1 = sup ( ) ,
f M
i n i i
xi
i
f t t dt
∈
+∫ ( )
0
1
0
2
π
λ
π
Λ , (14)
de M xi
0 — pidmnoΩyna funkcij mnoΩyny M xi
takyx, wo f ( 0 ) = 0.
Vykorystovugçy teoremu Fubini, ma[mo
1
π
λm
P
m
n
m
m
mf t t dt∫ + ( )( ) ,Λ =
=
1 1
01
0
2
1 2 2 1 2
1
π π
λ
π
m
P
m m n
m
m
m
mf t t t f t t t dt dt dt−
+
−
∫ ∫ … − …( )
…( , , , ) ( , , , ) ( , )Λ +
+
P
m m n
m
m
m
mf t t t f t t t dt dt dt dt
−
∫ ∫ … − …( )
…+
1
1
0 0 0
0
2
2 3 3 2 1 3π
λ
π
( , , , , ) ( , , , , ) ( , )Λ + …
… +
P
m n
m
m m
m
mf t t dt dt dt
−
∫ ∫ …
…
+
−
1
1
0 0 0 0
0
2
1 1π
λ
π
( , , , , , ) ( , )Λ . (15)
Qkwo f ( t1 , … , tm ) ∈ M
X m
0
, to pry fiksovanyx ( t2 , … , tm ), ( t3 , … , tm ), … , tm
vidpovidno ( f ( t1 , t2 , … , tm ) – f ( 0, t2 , … , tm ) ) ∈ Mt
0
1
, ( f ( 0, t2 , t3 , … , tm ) – f ( 0, 0,
t3 , … , tm ) ) ∈ Mt
0
2 , … , f ( 0, 0, 0 , … , tm ) ∈ Mtm
0 . Todi, vykorystovugçy (15), ne-
vid’[mnist\ qdra Λ
n
m
m t+ ( , )λ i oznaçennq operatoriv U f xn ii
+ ( ; ; )Λ , otrymu[mo
sup ( ) ,
f M
m
P
m
n
m
X m m
mf t t dt
∈
+∫ ( )
0
1
π
λΛ ≤
i
m
f M
m
P
i n
m
ti
m
mf t t dt
= ∈
∑ ∫ ( )
1 0
1
sup ( ) ,
π
λΛ =
. =
i
m
f M
i n i i
ti
i
f t t dt
= ∈
+∑ ∫ ( )
1 0
2
0
1
sup ( ) ,
π
λ
π
Λ . (16)
Z (13), (16), (14) i teoremy 1 vyplyva[, wo
G M U
n n X
m N m N
,
; −
+( ) ≤
i
m
x
n X
G M Ui
i
=
+∑ ( )
1
1, ( )Λ + G M U
X n X
N m N m N m− − −( ), .
Dovedennq spravedlyvosti rivnosti (12) analohiçne dovedenng rivnosti (9).
Teoremu 2 dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
NABLYÛENNQ KLASIV PERIODYÇNYX FUNKCIJ BAHAT|OX ZMINNYX … 19
ZauvaΩennq. Dlq klasiv funkcij, wo zadovol\nqgt\ umovy teoremy 2, na-
blyΩennq linijnym dodatnym operatorom iz dovil\nym qdrom zaleΩyt\ vid od-
novymirnyx dodankiv c\oho qdra. Tomu, zhidno z naslidkom 2, dlq takyx klasiv
funkcij nablyΩennq dodatnym operatorom iz dovil\nym qdrom zbiha[t\sq z na-
blyΩennqm dodatnym operatorom z qdrom, wo [ dobutkom odnovymirnyx qder.
Vykorystovugçy dovedeni tverdΩennq, moΩna, napryklad, znaxodyty ocinky
zverxu abo asymptotyçni rivnosti dlq velyçyn G M U
n n X
m N m N
,
; −
+( ) , qkwo vidomi
ocinky zverxu abo asymptotyçni rivnosti vidpovidno dlq velyçyn
G M U
X n X
m m m, +( ) , G M U
X n X
N m N m N m− − −( ), , G M Ux
n X
i
i
, ( )+( )Λ , i = 1, m .
Poznaçymo çerez
Fl ( f , x1 ) =
1
0
2
1 1 1 1π
π
∫ + +f x t t dtl( ) ( )Λ
operator Fej[ra z qdrom
Λl t+ ( )1 =
sin
sin
2
1
1
2
2 2
lt
l t
/
/
( )
( )
,
çerez
Sn, m ( f , x2 , x3 ) =
1
2 2 2 3 3 2 3 2 3
2
π
P
n mf x t x t D t D t dt dt∫∫ + +( , ) ( ) ( )
operator Fur’[, de
Dk ( t ) =
sin
sin
k t
t
+( )
( )
/
/
1 2
2 2
— qdro Dirixle, çerez
Fl Sn, m ( f , x
3
) =
1
3
3 3 3 3
3
π
P
l n mf x t t dt∫ + +( ) ( ), ,Λ
operator z qdrom Λl n m t, , ( )+ 3 = Λl n mt D t D t+ ( ) ( ) ( )1 2 3 , qke [ dobutkom qdra Fej[ra i
qder Dirixle. Todi z naslidku 2 vyplyva[ G H F Sl n m Cω( ) , ,1 3
3( ) = G H Fl Cω( ) ,1 1
1( ) +
+ G H Sn m Cω( ) , ,1 2
2( ) .
ZauvaΩymo, wo z asymptotyçnymy rivnostqmy dlq velyçyn G H Sn m Cω( ) , ,1 2
2( )
i G H Fl Cω( ) ,1 1
1( ) , a takoΩ vidpovidnog bibliohrafi[g moΩna detal\niße oznajo-
mytys\ v monohrafi] O.VI.VStepancq [4].
1. Zaderej P. V. O pryblyΩenyy peryodyçeskyx funkcyj mnohyx peremenn¥x poloΩytel\-
n¥my polynomyal\n¥my operatoramy // Yssledovanyq po teoryy pryblyΩenyq funkcyj y
yx pryloΩenyj. – Kyev, 1978. – S. 85 – 88.
2. Besov O. V., Yl\yn V. T., Nykol\skyj S. M. Yntehral\n¥e predstavlenyq funkcyj y teore-
m¥ vloΩenyq. – M.: Nauka, 1975. – 480 s.
3. Tyman A. F. Teoryq pryblyΩenyq funkcyj dejstvytel\noho peremennoho. – M.: Fyzmat-
hyz, 1960. – 624 s.
4. Stepanec A. Y. Ravnomern¥e pryblyΩenyq tryhonometryçeskymy polynomamy. – Kyev:
Nauk. dumka, 1981. – 339 s.
OderΩano 14.06.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
|
| id | umjimathkievua-article-3430 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:42:24Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3c/91cb1f397937cfc671d19ec8443e8f3c.pdf |
| spelling | umjimathkievua-article-34302020-03-18T19:54:30Z Approximation of classes of periodic multivariable functions by linear positive operators Наближення класів періодичних функцій багатьох змінних лінійними додатними операторами Bushev, D. M. Kharkevych, Yu. I. Бушев, Д. М. Харкевич, Ю. І. In an N-dimensional space, we consider the approximation of classes of translation-invariant periodic functions by a linear operator whose kernel is the product of two kernels one of which is positive. We establish that the least upper bound of this approximation does not exceed the sum of properly chosen least upper bounds in m-and ((N ? m))-dimensional spaces. We also consider the cases where the inequality obtained turns into the equality. Встановлено, що в $N$-внмірному просторі точна верхня межа наближення класів періодичних функцій, інваріантних відносно зсуву, лінійним оператором з ядром, що є добутком двох ядер, одне з яких є додатним, не перевищує суми відповідно вибраних точних верхніх меж в $m$- і $(N - m)$-вимірних просторах. Розглянуто випадки, в яких для отриманої нерівності має місце знак рівності. Institute of Mathematics, NAS of Ukraine 2006-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3430 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 1 (2006); 12–19 Український математичний журнал; Том 58 № 1 (2006); 12–19 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3430/3600 https://umj.imath.kiev.ua/index.php/umj/article/view/3430/3601 Copyright (c) 2006 Bushev D. M.; Kharkevych Yu. I. |
| spellingShingle | Bushev, D. M. Kharkevych, Yu. I. Бушев, Д. М. Харкевич, Ю. І. Approximation of classes of periodic multivariable functions by linear positive operators |
| title | Approximation of classes of periodic multivariable functions by linear positive operators |
| title_alt | Наближення класів періодичних функцій багатьох змінних лінійними додатними операторами |
| title_full | Approximation of classes of periodic multivariable functions by linear positive operators |
| title_fullStr | Approximation of classes of periodic multivariable functions by linear positive operators |
| title_full_unstemmed | Approximation of classes of periodic multivariable functions by linear positive operators |
| title_short | Approximation of classes of periodic multivariable functions by linear positive operators |
| title_sort | approximation of classes of periodic multivariable functions by linear positive operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3430 |
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