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

Bushev, Dmytro Mykolayovych

Kharkevych, Yurii I.

Бушев, Дмитро Миколайович

Харкевич, Юрій Іліодорович

Electronic Eastern European National University Institutional Repository

Ukrainian Mathematical Journal, Vol. 58, No. 1, 2006
APPROXIMATION OF CLASSES OF PERIODIC MULTIVARIABLE FUNCTIONS
BY LINEAR POSITIVE OPERATORS
D. M. Bushev  and  Yu. I. Kharkevych UDC 517.5
In an  N-dimensional space, we consider the approximation of classes of translation-invariant pe-
riodic 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 con-
sider the cases where the inequality obtained turns into the equality. 
It is known that the determination of least upper bounds for the approximation of classes of periodic func-
tions by linear operators in an  N-dimensional space is not always reducible to the calculation of the correspond-
ing quantities in a space of lower dimension.  In the present work, we indicate conditions under which this re-
duction is possible. 
In what follows, we prove that the least upper bound of 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  m-dimensional
and positive does not exceed the sum of properly chosen least upper bounds in  m-  and  ( N – m  )-dimensional
spaces.  We establish that the inequality obtained turns into the equality in the case of centrally symmetric
classes of continuous or essentially bounded functions that satisfy an additional condition.  The results obtained
yield, as a consequence, Theorem 1 from [1]. 
We show that the approximation by a linear positive operator with arbitrary kernel depends only on the
one-dimensional components of this kernel, i.e., it coincides with the approximation by a linear positive operator
whose kernel is the product of one-dimensional kernels. 
Let  C  N,  LN
∞
,  and  Lp
N
  be the spaces of functions  f ( x )  =  f ( x1 , … , xN )  that are  2π-periodic in each of  N
variables and are, respectively, continuous, essentially bounded, and summable to the  p th power  ( 1 ≤ p < ∞  )
with the norms 
f C N   =  sup ( )
x
f x ,      f LN
∞
  =  sup ( )
x
f xvrai ,
f LpN   =  
1
2
1
π
⎛⎝ ⎞⎠
⎛
⎝⎜
⎞
⎠⎟∫
/N
P
p
p
N
f x d x( ) ,
where 
PN  =  
i
N
=
∏ [ ]
1
0 2; π
is an  N-dimensional cube. 
Volyn University, Lutsk. 
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 12–19, January, 2006.  Original article submitted June 14,
2005. 
12 0041–5995/06/5801–0012      ©  2006    Springer Science+Business Media, Inc.
APPROXIMATION OF CLASSES OF PERIODIC MULTIVARIABLE FUNCTIONS BY LINEAR POSITIVE OPERATORS 13
Let  x m = ( x1 , … , xm ),  x N  – m = ( xm + 1 , … , xN ),  and  x = ( x1 , … , xN )  denote, respectively,  m-,  ( N – m  ) -,
and  N-dimensional vectors whose coordinates are real numbers.  Let  E m  and  E N – m  be the sets of all  m-  and
( N – m )-dimensional vectors  k m = ( k1 , … , ki , … , km )  and  k N – m = ( km + 1 , … , kN )  whose coordinates are non-
negative integer numbers, let  S ( k  m  )  and  S  ( k N – m )  denote the numbers of the coordinates of  k  m  and  k  N  – m
that are equal to zero, let  B m  and  B N – m  be the sets of all possible  m-  and  ( N – m  )-dimensional vectors each
coordinate of which is equal to either  0  or  1,  let  B ( k m )  and  B ( k N – m )  denote the subsets of the sets  B  m  and
B  N – m  such that if  k j = 0,  then  i j = 0,  where  i 
m
 = ( i1 , … , ij , … , im ) ∈ B ( k m ),  let 
ak
i
m
m
  =  
1
21π
π
m
P
m
j
m
j j j
m
f t k t i dt∫ ∏
=
−
⎛⎝ ⎞⎠( ) cos
where  i m = ( i1 , … , ij , … , im ) ∈ B ( k m ) ,  be the Fourier coefficients of the function  f ( x m ),  and let 
U f x
n
m
m
+ ( ; ; )Λ   =  1
π
λ
m
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 m
f 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
be the linear operators with the kernels 
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.
14 D. M. BUSHEV  AND  YU. I. KHARKEVYCH
The operator  U f x
n
m
m
+ ( ; ; )Λ   can be represented in the form 
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 .
If  f ( x ) = f ( xi )  is a function of the single variable  xi ,  then 
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 ,
where 
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 .
Denote 
U f xn ii
+ ( ; ; )Λ   =  2 1m
n iU f xm
− + ( ; ; )Λ   =  1
π
λ
m
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 ,
where 
Λn ii t
+ ( )λ;   =  12   +  k
n
k i i k i i
i
i
i i
k t k t
=
−
∑ −( )
1
1
λ μcos sin   ≥  0
and  λki   and  μki   are defined by (1). 
Let  A  be a linear operator that maps a set  M ⊂ X N  ( X N = C N, LN
∞
, Lp
N )  onto  X N,  let the set  M  be trans-
lation-invariant, i.e., the inclusion  f ( x ) ∈ M  implies that  f ( x + t ) ∈ M,  and let 
G M A X N( , )   =  sup ( )f M X
f A f N
∈
−
be the approximation of the set  M  by the operator  A  in the space  X N. 
By  MX m ,  MX N m− ,  and  M
xi
  we denote the subsets of functions from the set  M  obtained from  M  by
fixing the variables  ( xm + 1 , … , xN ) ,  ( x1 , … , xm ),  and  ( x1 , … , xi – 1, xi + 1 , … , xN ),  respectively. 
In what follows, we prove that if the set  M  is translation-invariant, then 
G M U
n n X
m N m N
,
; −
+( )   ≤  G M UX n Xm m m, +( )   +  G M UX n XN m N m N m− − −( ), . (2)
We also consider the cases where inequality (2) turns into the equality. 
APPROXIMATION OF CLASSES OF PERIODIC MULTIVARIABLE FUNCTIONS BY LINEAR POSITIVE OPERATORS 15
If we fix the variables  x xm1
0 0
, ,…( ) = xm0   or  x xm N+ …( )10 0, ,  = x N m0 − ,  then the definition of norms yields 
f C N   =  sup ,
x
m N m
Cm N m
f x x
0
0
−( )
−
  =  sup ,
x
m N m
CN m m
f 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 LPN   ≤  sup ,
x
m N m
Lm P
N mf x x
0
0
−( )
−
, (5)
f LPN   ≤  sup ,
x
m N m
LN m P
mf x x
0
0
−
−( ) . (6)
Lemma 1.  If a set  M ⊂ X N  is translation-invariant, then inequality (2) is true. 
Proof.  We introduce the auxiliary linear operators 
U f x
nm ;
( ; ; )
∞
+ Λ   =  1
π
λ
m
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 .
Using the Fubini theorem, the nonnegativity of the kernel  Λ
n
m
m t
+ ( ; )λ ,  and the definition of the operators
U f x
nm ;
( ; ; )
∞
+ Λ   and  U f x
nN m− ∞;
( ; ; )Λ ,  we get 
f x U f x
n n X
m N m N
( ) ( ; ; )
;
−
−
+ Λ   
≤  1
π
λ
m
P
m m N m
n
m m
Xm
m
N
f x f x t x t dt∫ − +( )− +( ) ( ; ) ( , )Λ   +  1π 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
λ
m
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)
16 D. M. BUSHEV  AND  YU. I. KHARKEVYCH
Using inequality (7) and the Minkowski generalized inequality (see, e.g., [2, p. 22], we obtain 
G M U
n nm N m
,
; −
+( )  ≤  sup ( ) ( ; ; );f M n Xf x U f xm N∈ ∞+−⎛⎝⎜ Λ   
+  
1
π
λ
m
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)
Inequality (8) and relations (3) – (6) yield 
G M U
n nm N m
,
; −
+( )  ≤  sup sup , ; ; ,;f M x m N m n m N m XN m m mf x x U f x x∈ − ∞+ −− ( ) − ( )( )0 0 0Λ   
+  
1
π
λ
m
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 .
For fixed  x N m0
−
  and  xm0  + t
m
0 ,  f ( x )  belongs to the sets  MX m   and  MX N m− ,  respectively.  Therefore, using
the definition of the operators  U f x
nm ;
( ; ; )
∞
+ Λ ,  U f x
nN m− ∞;
( ; ; )Λ ,  U f x
n
m
m
+ ( ; ; )Λ ,  and  U f x
n
N m
N m−
−( ; ; )Λ ,
we get 
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
λ
m
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 UX n Xm m m,
+( )   +  G M UX n XN m N m N m− − −( ), .
Lemma 1 is proved. 
Corollary 1.  If a set  M ⊂ X N  is translation-invariant, then 
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 UX n XN m N m N m− − −( ), .
The following theorem is true: 
APPROXIMATION OF CLASSES OF PERIODIC MULTIVARIABLE FUNCTIONS BY LINEAR POSITIVE OPERATORS 17
Theorem 1.  Suppose that a set  M  ∈  X  N   ( X  N  = C  N , LN
∞
)  is translation-invariant and centrally sym-
metric, i.e., the inclusion  f  ( x  ) ∈ M  implies that  – f ( x  ) ∈ M.  If the inclusions  f  ( x  m  )  ∈ MX m  ⊂  M   and
g ( x N – m )  ∈ MX N m−  ⊂ M  imply that  f x g x
m N m( ) ( )+( )−  ∈ M,  then 
G M U
n n X
m N m N
,
; −
+( )   =  G M UX n Xm m n, +( )   +  G M UX n XN m N m N m− − −( ), . (9)
Proof.  Assume that  G M UX n Xm m m,
+( )  = ∞  or  G M UX n XN m N m N m− − −( ),  = ∞.  Since  MX m  ⊂  M   and
MX N m−  ⊂ M,  we get 
G M U
n n X
m N m N
,
; −
+( )   ≥  G M UX n n Xm m N m N, ; −+( )   =  G M UX n Xm m m, +( )   =  ∞
or 
G M U
n n X
m N m N
,
; −
+( )   ≥  G M UX n XN m N m N m− − −( ), .
Lemma 1 yields relation (9). 
Let  G M UX n Xm m m,
+( )  < ∞  and  G M UX n XN m N m N m− − −( ),  < ∞.  Then, by the definition of least upper bound,
there exist functions  ϕk ( x m )  ∈ MX m   and  φ ( x 
N
 
–
 
m
 )  ∈ MX N m−   such that 
G M UX n Xm m m,
+( )   =  ϕ ϕk m n k m Xx U xm m( ) ( ; ; )− + Λ
and 
G M UX n XN m N m N m− − −( ),   =  φ φk N m n k N m Xx U xN m N m( ) ( ; ; )− −− − −Λ ,
or there exist sequences of functions  ϕk mx( ){ } ∈  MX m   and  φk N mx( )−{ } ∈  MX N m− ,  k = 1, 2, 3, …  ,  for
which 
G M UX n Xm m m,
+( )   =  lim ( ) ( ; ; )
k k
m
n k
m
X
x U xm m
→∞
+
−ϕ ϕΛ (10)
and 
G M UX n XN m N m N m− − −( ),   =  lim ( ) ( ; ; )k k m n k N m Xx U xN m N m→∞ −− − −φ φΛ . (11)
Setting  fk ( x ) = ϕ φk m k N mx x( ) ( )+( )−  ∈ M  and taking into account that the set  M  is centrally symmetric
and either  X N = C N  or  X N = LN
∞
,  we obtain 
18 D. M. BUSHEV  AND  YU. I. KHARKEVYCH
G M U
n n X
m N m N
,
; −
+( )   ≥  lim ( ) ( ; ; );k k n n Xf 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 ,
which, together with (10), (11), and (2), proves relation (9). 
Note that there exists a set  M ⊂ C N  that is translation-invariant, centrally symmetric, and such that the in-
clusions  f ( x m )  ∈ MX m   and  g ( x 
N
 
–
 
m
 )  ∈ MX N m−   do not imply that  f x
m( )(  + g x N m( )− ) ∈ M. 
Let, e.g.,  M = Hω ( t, z ) ⊂ C 2  be the set of functions  f ( x, y )  that are continuous,  2π-periodic in each vari-
able, and such that  ω ( f; t; z ) ≤ ω ( t, z ),  where 
ω ( f; t; z )  ≤  sup ( , ) ( , )
,h t z
Cf x h y f x y
≤ ≤
+ + ∂ −
∂
2
and  ω ( t, z )  is a function of the type of a modulus of continuity.  It is known that (see, e.g., [3, p. 124]) that 
max ( , ); ( , )ω ωt z0 0{ }  ≤  ω ( t, z )  ≤  ω ( t, 0 )  +  ω ( 0, z ).
If  ω ( t, z ) ≠ ω ( t, 0 ) + ω ( 0, z ),  then  f ( t ) ∈ ω ( t, 0 ) ∈ Hω ( t, 0 ) = M x  and  g  ( z ) = ω ( 0, z ) ∈ Hω ( 0, z ) = M y,
but  f t g z( ) ( )+( ) ∉ M. 
Corollary 2.  If a set  M   is centrally symmetric, translation-invariant, and such that the inclusions
f xi( )  ∈ M xi  ⊂ M,  i = 1, m ,  and  g ( x N – m )  ∈ MX N m−  ⊂ M  imply that  i
m
if x
=
∑( 1 ( )  + g x N m( )− ) ∈ M,  then 
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 UX n XN m N m N m− − −( ), ,
where  X N = C N  or  X N = LN
∞
. 
Corollary 2 is obtained from Corollary 1 by analogy with the proof of Theorem 1. 
Let  H N
ω( )1   denote the class of functions  f ( x ) ∈ C 
N
  that satisfy the condition 
f x f x( ) ( )− ′   ≤  
i
N
i i ix x
=
∑ − ′( )
1
1ω( )
APPROXIMATION OF CLASSES OF PERIODIC MULTIVARIABLE FUNCTIONS BY LINEAR POSITIVE OPERATORS 19
and let  H N
ω( )2   be the class of functions  f ( x ) ∈ C 
N
  for which 
f x h f x f x h( ) ( ) ( )+ − + −2   ≤  
i
N
i ih
=
∑ ( )
1
2ω( ) .
Here,  ωi it
( )( )1   and  ωi it( )( )2   are arbitrary fixed functions of the type of moduli of continuity of the first order
and the second order, respectively.  Since the classes  H iNω( ) ,  i = 1, 2,  satisfy the conditions of Corollary 2, we
can obtain, e.g., Theorem 1 in [1] as a consequence of Corollary 2. 
Theorem 2.  If a set  N  satisfies the conditions of Corollary 2 and, in addition, the inclusion  f ( x ) ∈  M
implies that  ( f ( x ) + C) ∈ M,  where  C  is an arbitrary constant, then 
G M U
n n X
m N m N
,
; −
+( )   =  
i
m
x
n X
G M Ui
i
=
+∑ ( )
1
1, ( )Λ   +  G M UX n XN m N m N m− − −( ), . (12)
Since the set  MX m  ∈ X 
N
  is translation-invariant,  X N = C N  or  X  N  = LN
∞
,  and  ( f ( x  m  ) + C) ∈  MX m   for
every function  f ( x m )  ∈ MX m ,  we get 
G M UX n Xm m m,
+( )   =  sup ( ) ,
f M
m
P
m
n
m
X m m
mf t t dt
∈
+∫ ( )0 1π λΛ , (13)
where  MX m
0
  is the subset of functions of the set  MX m   for which  f ( 0 ) = f ( 0, … , 0 ) = 0.  By virtue of the fact
that the sets  M xi   satisfy the same conditions as the set  MX m ,  taking into account the definition of the opera-
tors  U f xn ii
+ ( ; ; )Λ   we get 
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)
where  M xi0   is the subset of functions of the set  M
xi
  for which  f ( 0 ) = 0. 
Using the Fubini theorem, we obtain 
1
π
λ
m
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 3
π
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 1
π
λ
π
( , , , , , ) ( , )Λ  . (15)
20 D. M. BUSHEV  AND  YU. I. KHARKEVYCH
If  f ( t1 , … , tm  ) ∈ MX m
0
,  then, for fixed  ( t2 , … , tm  ), ( t3 , … , tm ), … , tm  ,  we obtain  ( f ( t1 , t2 , … , tm ) –
f ( 0, t2 , … , tm ) ) ∈ Mt01 ,  ( f ( 0, t2 , t3 , … , tm ) – f ( 0, 0, t3 , … , tm ) ) ∈  Mt02 , … , f ( 0, 0, 0 , … , tm  ) ∈  Mtm0 ,  respec-
tively.  Then, taking into account relation (15), the nonnegativity of the kernel  Λ
n
m
m t
+ ( , )λ ,  and the definition of
the operators  U f xn ii
+ ( ; ; )Λ ,  we obtain 
sup ( ) ,
f M
m
P
m
n
m
X m m
mf t t dt
∈
+∫ ( )0 1π 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)
Relations (13), (16), and (14) and Theorem 1 yield 
G M U
n n X
m N m N
,
; −
+( )   ≤  
i
m
x
n X
G M Ui
i
=
+∑ ( )
1
1, ( )Λ   +  G M UX n XN m N m N m− − −( ), .
The proof of equality (12) is analogous to that of equality (9). 
Theorem 2 is proved. 
Remark 1.  For classes of functions that satisfy the conditions of Theorem 2, the approximation by a linear
positive operator with arbitrary kernel depends on the one-dimensional terms of this kernel.  Therefore, accord-
ing to Corollary 2, for such classes of functions the approximation by a positive operator with arbitrary kernel
coincides with the approximation by a positive operator whose kernel is the product of one-dimensional kernels. 
By using the statements proved, we can find, e.g., upper bounds or asymptotic equalities for the quantities
G M U
n n X
m N m N
,
; −
+( ) ,  provided that upper bounds or asymptotic equalities, respectively, are known for the quan-
tities  G M UX n Xm m m,
+( ) ,  G M UX n XN m N m N m− − −( ), ,  and  G M Ux n Xi i, ( )+( )Λ ,  i = 1, m . 
Consider the Fejér operator 
Fl ( f ,  x1 )  =  1
0
2
1 1 1 1
π
π
∫ + +f x t t dtl( ) ( )Λ
with the kernel 
Λl t
+ ( )1   =  
sin
sin
2
1
1
2
2 2
lt
l t
/
/
( )
( )  ,
the Fourier operator 
Sn, m ( f ,  x2 , x3 )  =  12 2 2 3 3 2 3 2 3
2
π P
n mf x t x t D t D t dt dt∫∫ + +( , ) ( ) ( )
APPROXIMATION OF CLASSES OF PERIODIC MULTIVARIABLE FUNCTIONS BY LINEAR POSITIVE OPERATORS 21
with the Dirichlet kernel 
Dk ( t )  =  sin
sin
k t
t
+( )
( )
/
/
1 2
2 2
,
and the operator 
Fl Sn, m ( f ,  x 3 )  =  13 3 3 3 3
3
π P
l n mf x t t dt∫ + +( ) ( ), ,Λ
with the kernel 
Λl n m t, , ( )+ 3   =  Λl n mt D t D t+ ( ) ( ) ( )1 2 3 ,
which is the product of the Fejér kernel and the Dirichlet kernels.  Then Corollary 2 yields 
G H F Sl n m Cω( ) , ,1 3
3( )   =  G H Fl Cω( ) ,1 11( )   +  G H Sn m Cω( ) , ,1 22( ) .
Note that additional information about asymptotic equalities for the quantities  G H Sn m Cω( ) , ,1 2
2( )   and
G H Fl Cω( ) ,1 1
1( )   and the corresponding bibliography can be found in [4]. 
REFERENCES
1. P. V. Zaderei,  “On the approximation of periodic multivariable functions by positive polynomial operators,” in: Studies on the
Theory of Approximation of Functions and Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences,
Kiev (1978). 
2. O. V. Besov, V. T. Il’in, and S. M. Nikol’skii,  Integral Representations of Functions and Imbedding Theorems [in Russian],
Nauka, Moscow (1975). 
3. A. F. Timan,  Theory of Approximation of Functions of Real Variables [in Russian], Fizmatgiz, Moscow (1960). 
4. A. I. Stepanets,  Uniform Approximation by Trigonometric Polynomials [in Russian], Naukova Dumka, Kiev (1981). 


APPROXIMATION OF CLASSES OF PERIODIC MULTIVARIABLE FUNCTIONS BY LINEAR POSITIVE OPERATORS

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APPROXIMATION OF CLASSES OF PERIODIC MULTIVARIABLE FUNCTIONS BY LINEAR POSITIVE OPERATORS

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