In this paper we study approximative properties of modified Szasz-Mirakyan operators for functions of two variables from exponential weight spaces. We present theorems giving a degree of approximation by these operators for exponential bounded functions

Herzog, Monika

Portal Czasopism Naukowych (E-Journals)

MONIKA HERZOG*1
APPROXIMATION OF FUNCTIONS OF TWO 
VARIABLES FROM EXPONENTIAL WEIGHT SPACES
APROKSYMACJA FUNKCJI DWÓCH ZMIENNYCH 
Z WYKŁADNICZYCH PRZESTRZENI WAGOWYCH
*  Dr Monika Herzog,  Instytut Matematyki, Wydział  Fizyki, Matematyki  i  Informatyki,  Politechnika 
Krakowska.
A b s t r a c t
In  this  paper  we  study  approximative  properties  of  modified  Szasz-Mirakyan  operators 
for  functions of  two variables  from exponential weight  spaces. We present  theorems giving 
a degree of approximation by these operators for exponential bounded functions.
Keywords: linear positive operators, Bessel function, modulus of continuity, degree of 
approximation
S t r e s z c z e n i e
W artykule przedstawiono aproksymacyjne własności zmodyfikowanych operatorów typu Sza-
sza-Mirakyana dla funkcji dwóch zmiennych z wykładniczych przestrzeni wagowych. Wypro-
wadzono  twierdzenia podające  rząd aproksymacji  funkcji ograniczonych wykładniczo przez 
operatory tego typu.
Słowa kluczowe: dodatni operator liniowy, funkcja Bessela, moduł ciągłości, rząd aproksymacji
41. Introduction
Let us denote by C R0( )  a set of all real-valued functions continuous on R0 0= +∞[ ); . In 
paper [1] we investigated operators of Szasz-Mirakyan type defined as follows
where Γ is the Euler-gamma function and Iv the modified Bessel function defined by the 
formula ([6], p. 77)
We studied approximative properties of these operators in exponential weight spaces
where wp was the exponential weight function defined as follows:
for x ∈ R0.
In the spaces we introduced the norm:
and we established ([1], Corollary 1)  that operators  An
ν  are  linear, positive, bounded and 
transform the space Ep into Er for some r > p.
Notice that a certain modification of  An
ν  give a linear, positive and bounded operator  Ln
ν
transforming the space Ep into Er (Theorem 2.1), namely:
In the present paper we introduce an analogy of the presented theorems for f ∈ Ep, 
but now we consider the bivariate version of the operator Ln
ν
I z
k k
z k
k
ν
ν
ν
( )= ( )
+( ) + +( )
+
=
∞
∑ 2
2
0 1 1
1
Γ Γ
( )
E f C R w f Rp p= ∈ ( ){ }0 0: ,is uniformly continuous and bounded on 
f w x f x x R
p p
= ( ) ( ) ∈{ }sup : ( )0 3
L f x I nx k k
f k
n p
x
n
nx k v
k
ν
ν ν;
,( )= ( )
( )
+( ) + +( ) +






+
=
∞
∑1
1 1
22
2
0 Γ Γ
>
( ) =





0
0 0
4
;
, .
( )
f x
A f x I nx k k
f k
n
x
n
nx k v
k
ν
ν ν;
,( )= ( )
( )
+( ) + +( )





 >
+
=
∞
∑1
1 1
2
0
2
2
0 Γ Γ
;
, ,f x0 0( ) =





w x e p Rp
px( )= ∈− +, ( )2
5for n, m ∈ N, v, μ ∈ R0 and f ∈ Ep,q, where:
and wp,q is the exponential weight function
for  x y R,( )∈ 02 .
It is easy to verify that Ep,q is a normed space with the norm:
Moreover, we use the weighted modulus of continuity defined as follows:
where:
for  x y h d R, , , .( ) ( )∈ 02
The note was inspired by the results of [3–5] which investigate approximation prob-
lems for bivariate operators.
We shall present theorems giving a degree of approximation of function f ∈ Ep,q by op-
erators Ln m,
,ν µ .
2. Auxiliary results
At the beginning we recall preliminary results which we immediately obtain from paper 
[1] and definition (4).
L f x y
I nx I my k k
n m
nx k v my
,
, ; , ( )ν µ
ν µ ν
( )=
( ) ( )
( )
+( ) + +( )
+
5
1 1
1 1
2
2
2
Γ Γ
( )
+( ) + +( ) + +





 > >
+
=
∞
=
∞
∑∑
2
00 1 1
2 2
0 0
j
jk j j
f k
n p
j
m q
x y
µ
µΓ Γ
, , , ;
1
1 1
2
0 0 0
2
2
0I nx k k
f k
n p
x y
nx k v
kν ν( )
( )
+( ) + +( ) +





 > =
+
=
∞
∑Γ Γ , , , ;
, , , ;
1
1 1
0
2
0 0
2
2
0I my j j
f j
m q
y x
my j
jµ
µ
µ( )
( )
+( ) + +( ) +





 > =
+
= Γ Γ
∞
∑
( ) = =












 f x y0 0 0, , ,
E f C R w fp q p q, ,:= ∈ ( )02  is uniformly continuous and bounded on R02{ },
w x y e p q Rp q
px qy
, , , , ( )( )= ∈− +( ) + 6
f w x y f x y x y R
p q p q, ,
sup , , ; , ( )= ( ) ( ) ( )∈{ }02 7
ω f E t s f h t d sp q h d p q, ; , sup : , , , ( ), , ,( )= ∈[ ] ∈[ ]{ }∆ 0 0 8
∆h d f x y f x h y d f x y, , , ,( )= + +( )− ( )
6Lemma 2.1 ([1], Lemma 8) For each v ∈ R0 there exists a positive constant M(v) such 
that for all n ∈ N and x ∈ R0 we have:
By elementary calculations we get
Lemma 2.2 For each n ∈ N, v ∈ R0, p ∈ R+ and x ∈ R0.
Lemma 2.3 For each n ∈ N, v ∈ R0, p ∈ R+ and x ∈ R0.
Similarly to Lemma 6 ([1]), using basic properties of modified Bessel function (1), we 
can prove
Lemma 2.4 For each v ∈ R0 and p ∈ R+ there exists a positive constant M (v, p) such that 
for all n ∈ N and z ∈ R0 we have:
I nx
I nx
Mν
ν
ν+
( )
( )
≤ ( )1 ,
nx
I nx
I nx
Mν
ν
ν+
( )
( )
− ≤ ( )1 1 .
L x L t x x n
n p
I nx
I nx
n
n p
A t x
L t
n n n
n
ν ν ν
ν
ν
ν
1 1
1
2
; , ; ; ,( )= ( )=
+
( )
( )
=
+
( )+
;x x n
n p
I nx
I nx
x n
n p
I nx
I nx( )= +






( )
( )
+
+( )
( )
( )
=+ +2
2
2
2
12ν
ν
ν
ν
n
n p
A t x
L t x x n
n p
A t x p
n
x
L
n
n n
n
+





 ( )
−( )=
+
( )−





2
ν
ν ν
ν
; ,
; ; ,
t x x n
n p
A t x x x p
n
A t x x p
n
xn n−( )( )= +





 −( )( )− −( )+


2
2
2
2; ; ;ν ν









2
L e x
I nx
I nx
L te x nx
n
pt
p
n p p
n p
n
pt
ν ν
ν
ν
ν
;
exp
exp ,
;
( )= ( )( )( ) −( )
( )=
+
+
n p
I nx
I nx
L t e x nx
n
p
n p p
n p
n
pt
+
( )( )
( ) ( )
( )=
+ + −( )
+
ν
ν
ν
ν
1 1
2
exp
exp ,
;
+






( )( )
( ) ( )+ + +
+ + −( )
+p
I nx
I nx n p
nx
n p
Ipn p p
n p
2
2 2 2ν
ν
ν
exp
exp
ν
ν
ν+ + −( )
+
( )( )
( ) ( )
1 1
nx
I nx
p
n p p
n p
exp
exp .
I z
I z
M p z p
n p
p
n p p
n p
ν
ν
ν
exp
, exp exp
+
+
( )( )
( )
≤ ( ) ( )−( )− +( )





1
2 
.
7Lemma 2.5 For all v ∈ R0 and p ∈ R+ there exists a positive constant M (v, p) such that 
for each n ∈ N we have
Proof. Pick v ∈ R0 and p ∈ R+. By definition (2) and Lemma 2.3 we get
for x ∈ R0 and n ∈ N.
Substituting nx = z and applying Lemma 2.4 we get
because 
so we have the following estimation:
for n ∈ N. From these inequalities and definition (3) we obtain (9).
An obvious consequence of the above lemma and definition (3) is
Theorem 2.1 For all v ∈ R0 and p ∈ R+ there exists a positive constant M (v, p) such that 
for each n ∈ N and f ∈ Ep we have:
Now we present the crucial lemma for the approximating theorems in the next section.
Lemma 2.6 For all v ∈ R0  and p ∈ R+ there exists a positive constant M (v, p)  such that 
for each n ∈ N and x ∈ R0 we have:
L w M pn p p
ν ν1 9; , ( )⋅( ) ≤ ( )
w x L w t x e
I nx
I nxp n p
px p
n p
p
n p( ) ( )( )= −( ) ( )( )( )
−
+
+ν ν ν
ν
1 ; exp
exp
,
w x L w t x M p zp n p
z p
n
p
n p
p
n p( ) ( )( )≤ ( ) −( ) −( ) ( )+ +ν νν1 ; , exp exp exp exp −( )− +( )





=
( ) − +( )( ) − ++ +
1
2
1
2
p
n p
M p zpn p
p
n
p
nν ν, exp exp exp p M p( )−( )( )≤ ( )1 ν,
exp
!
,
p
n p
k k
kk
p
n p k
p
n p
p
n+ =
∞
=
∞
( )− =
+





 < +





 =∑∑1 1
11
exp ( )
p
n p
p
n+( )− <1 10
L f M p fn p p
ν ν;. , .( ) ≤ ( )
w x L
t x
w t
x M p
x x
np n p
( )
−( )
( )








≤ ( )
+( )ν ν
2
1
11; , ( )
8Proof. Let us fix v ∈ R0 and p ∈ R+.
By (3), linearity of the operator  Ln
ν and Lemma 2.3 it follows that:
Now applying Lemmas 2.1, 2.4, estimation (10) we get (11).
The definition of the operator  Ln m,
,ν µ implies
for all functions of the form f(x, y) = f1(x)f2(y) where f1 ∈ Ep and f2 ∈ Eq, p, q ∈ R+.
In particular we get
w x L
t x
w t
x e L t e x xL te xp n
p
px
n
pt
n
pt( )
−( )
( )








= ( )−−ν ν ν
2
2 2; ; ;( )+ ( ) =
+






( )( )
( )
− + +
x L e x
e nx
n p
I nx
I nx
n
pt
px
p
n p
2
2
2
ν
ν
ν
;
exp
exp
exp
exp
p
n p
p
n p p
n p
nx
n p
I nx
I nx
2
2
1 12−( )
+
+ + −( )
+( )+
+( )
( )( )
( )
ν ν
ν
ν( )
−
+
( )( )
( ) ( )+
+ + −( )
+
2 2 1 1 2nx
n p
I nx
I nx
x
I nxpn p p
n p
ν
ν
ν ν
exp
exp
exp
p
n p p
n p
px p
n p
I nx
e n
n p
x
I
+ −
+
− −
+
+
( )( )
( ) ( ) =
+





 ( )
ν
ν
ν ν
exp
exp
2
2
2 2 12
nx
I nx
x
n
I nx
I nx
p
n p p
n p
p
n pexp
exp
exp+
+
+ +( )( )
( ) ( )+
( )( )
( )ν
ν
ν
exp
exp
exp
exp
p
n p
p
n p p
n
p
n p
x
I nx
I nx
x
I nx
+
+ + +
( )
−
( )( )
( ) ( )+2
2
1
2
ν
ν
ν ( )( )
( )
+ 





( )( )
( )
+
I nx
x p
n
I nx
I nx
x p
n
I nx
p
n p
p
n
ν
ν
ν
ν
2
2
2 2
exp
exp + + +
− −
( )( )
( )
−
( )( )
( ) ( ) ≤
p
p
n p p
n
px
I nx
x p
n
I nx
I nx
e
ν
ν
ν
2 2
1 exp
exp
exp
p
n p
p
n p
p
n p
p
n p
x
I nx
I nx
ν ν
ν
+
+ + +
+ +
( ) ( ) ( )( )( )( )
−2
2
2
1
exp exp
exp
exp
p
n p
p
n pI nx
I nx
I nx
I nx
x
+
+ +
+
+( ) ( )( )( )
( )
( )
+




−
ν
ν
ν
ν
1
1
1
2 1
exp
exp exp
exp
exp
p
n p
p
n p
p
n p
p
n pI nx
I nx
I nx
I n
+ + +
+
+( ) ( )( )
( )( )
( )( )ν
ν
ν
ν
1
x
x
n
I nx
I nx
I nx
I nx
x
p
n p
p
n p
( )
+ ( ) ( )( )( )
( )
( )
++
+ +
+
+2 1
1
1
exp
expν
ν
ν
ν
2
2
2
12
1
p
n
I nx
I nx
x p
n
I nxpn p
p
n p





( )( )
( )
+ −
( )+ + +ν
ν
νexp exp exp
p
n p
p
n p
p
n p
I nx
I nx
I nx
+
+
+( )( )
( )( )
( )( )
( )



ν
ν
νexp
exp
.
L f x y L f x L f yn m n m,
, ; , ; ; ( )ν µ ν µ( )= ( ) ( )1 2 12
L x y
L w x y L w x L w y
n m
n m p q n p m q
,
,
,
,
,
; , ,
; , ; ; .
ν µ
ν µ ν µ
1 1
1 1 1
( )=
( )= ( ) ( )
9From the above facts and Lemma 2.5 we derive
Lemma 2.8 For all v, μ ∈ R0 and p, q ∈ R+ there exists a positive constant M(v, μ, p, q) 
such that for each n, m ∈ N we have:
Lemma 2.9 For all v, μ ∈ R0 and p, q ∈ R+ there exists a positive constant M(v, μ, p, q) 
such that for each n, m ∈ N we have:
Proof. Applying definition (7), linearity of the operator and connection (12) we get
Hence the operator  Ln m,
,ν µ  transforms the space Ep,q into Ep,q .
3. Approximation theorems
The following theorem estimates a weighted error of approximation for functions 
belonging to the space 
Theorem 3.1 For all v, μ ∈ R0, p, q ∈ R+ and for each function g Ep q∈ ,
1  there exists 
a positive constant M(v, μ, p, q)  such that for all n, m ∈ N and x y R,( )∈ 02  we have:
The proofs of the above and the next theorems are analogous to the proofs of Theorems 
3.1, 3.2 ([2]) so we omit it.
The following theorem gives a degree of approximation of functions by operators  Ln m,
,ν µ .
Theorem 3.2 For all v, μ ∈ R0, p, q ∈ R+ and for each function f ∈ Ep,q there exists a 
positive constant M(v, μ, p, q) such that for all n, m ∈ N and x y R,( )∈ 02  we have
Theorem 3.2 implies the following corollaries.
L w M p qn m p q p q,
,
,
,
; , , , .ν µ ν µ1 ⋅( ) ≤ ( )
L f M p q fn m p q p q,
,
, ,
; , , , .ν µ ν µ⋅( ) ≤ ( )
w x y L f t s x y w x y L f t s x yp q n m p q n m, ,
,
, ,
,, , ; , , , ; ,( ) ( )( ) ≤ ( ) ( )( )=ν µ ν µ
w x y L w t s f t s
w t s
x y
f
p q n m p q
p q
p
, ,
,
,
,
,
, , ,
,
; ,( ) ( ) ( ) ( )





≤
ν µ 1
q p q n p m q p q
w x w y L w t x L w s y M p M q f( ) ( ) ( )( ) ( )( )≤ ( ) ( )ν µ ν µ1 1; ; , ,
,
E f E f Ep q p q p q, , ,: .
1 = ∈ ′∈{ }
w x y L g x y g x y
M p q g x
n
g
p q n m
x p q y p q
, ,
,
, ,
, ; , ,
, , ,
( ) ( )− ( ) ≤
( ) ′ + + ′
ν µ
ν µ
1 y
m
+





1
.
w x y L f x y f x y M p q f E x
n
y
mp q n m p q, ,
,
,, ; , , , , , , ; ,( ) ( )− ( ) ≤ ( )
+ +ν µ ν µ ω
1 1




.
10
Corollary 3.3 If v, μ ∈ R0, p, q ∈ R+ and f ∈ Ep,q then for all x y R,( )∈ 0
2
Moreover, the above convergence is uniform on every set [x1, x2] × [y1, y2] with 0 ≤ x1 < x2, 
0 ≤ y1 < y2.
Corollary 3.4 For all α, β ∈ (0,1], v, μ ∈ R0, p, q ∈ R+ and for each f ∈ Lip(Ep,q, α, β) 
there 
exists a positive constant M(v, μ, p, q) such that for all n, m ∈ N and x y R,( )∈ 02  we have:
R e f e r e n c e s
[1]  Herzog M., Approximation of functions from exponential weight spaces by operators 
of Szasz-Mirakyan type, Comment. Math. 43, 1, 2003, 77-94.
[2]  Herzog M., Approximation of functions of two variables by modified Szasz-Mirakyan 
operators, Comment. Math. 52, 1, 2012, 3-9.
[3]  Rempulska L. , Skorupka M., On some operators in weighted spaces of functions 
of two variables, Ricerche Mat. 48, 1, 1999, 1-20.
[4]  Wachnicki  E. , Approximation by bivariate Mazhar-Totik operators, Comment. Math. 
50, 2, 2010, 141-153.
[5]  Walczak  Z. ,  On certain modified Szasz-Mirakjan operators for functions of two 
variables, Demonstratio Math. 33, 1, 2000, 91-100.
[6]  Watson G.N. , Theory of Bessel functions, Cambridge Univ. Press, Cambridge 1966.
lim ; , , .
,
,
,
n m n m
L f x y f x y
→∞
( )= ( )ν µ
w x y L f x y f x y M p q x
n
y
mp q n m, ,
,, ; , , , , ,( ) ( )− ( ) ≤ ( ) +




 +
+ν µ
α
ν µ
1 1













( )= ∈ ( )=
β
α β ω
,
, , : , ; ,, , ,where Lip E f E f E t s Op q p q p q t s
α β+( ){ }.


Approximation of functions of two variables from exponential weight spaces

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Approximation of functions of two variables from exponential weight spaces

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