Thai Journal of Mathematics [616078]

Thai Journal of Mathematics
Volume 2 (2004) Number 2 : 203{216
Approximation by Generalized Baskakov
Operators for Functions of One and
Two Variables in Exponential and
Polynomial Weight Spaces
Abdul Wa¯ and Salma Khatoon
Abstract : In the present paper our aim is to investigate the convergence, degree
of approximation and direct theorems for functions of one and two variables by
generalized Baskakov operators in exponential and polynomial weight spaces.
Keywords : Exponential weight spaces, polynomial weight spaces, convergence,
degree of approximation, moduli of smoothness, Lipschitz class.
2000 Mathematics Subject Classi¯cation : 41A36, 41A25, 41A63.
1 Introduction
There is a large number of literature available on approximation of functions of
one variable. But the corresponding problem for functions of several variables had
received less attention. Bernstein operator for two variables was ¯rst introduced
by A. Dhingas [5] and it was also mentioned by G.G. Lorentz [9].
Later on, some investigation in this direction was done by D.D. Stancu [12, 13,
14]. Recently a series of research papers [6, 7, 8, 11, 15] was published which are
devoted to various modi¯ed Sz¶ asz-Mirakjan and Baskakov operators for functions
of one and two variables. In these papers authors studied the convergence, degree
of approximation, Voronovskaja-type theorems and convergence of derivatives of
these operators.
In this paper, we will introduce the generalized Baskakov operators [10] in the
space of continuous functions of two variables having exponential and polynomial
growth. We also investigate the degree of approximation of functions of one and
two variables.

204 Thai J. Math. 2(2004)/ Abdul Wa¯ and Salma Khatoon
2 Approximation of functions of one variable
2.1 Notations
Let C¡
R0= [0;+1)¢
be the set of all real-valued functions continuous on x2R0.
Following [3], for q >0 we de¯ne weights
vq(x) =e¡qx; x 2R0; (2.1)
spaces Cq=©
f2C(R0) :vqfis uniformly continuous and bounded on R0ă
, with
norm
kfkq= sup
x2R0vq(x)jf(x)j:
Assuming further for h; ±¸0;0< ®·1 and f2Cq
¢hf(x) =f(x+h)¡f(x);!(f; Cq;±) = sup
0·h·±k¢hfkq;
Lip(Cq; ®) =n
f2Cq:!(f; Cq;±) =O(±®)!0 +o
(2.2)
and
Cm
q=n
f2Cq:f(k)2Cq; k= 1;2; : : : ; mo
;
for ¯xed qandm2N.
Mihesan [10] introduced the following generalized Baskakov operators with
non-negative constant a¸0 independent of n
Ba
n(f;x) =1X
k=0pn;k(x; a)f(k=n); x2R0; k= 0;1;2; : : : ; n = 1;2; : : : ; (2.3)
where
pn;k(x; a) =e¡ax
1+xpk(n; a)
k!xk
(1 +x)n+k(2.4)
and
pk(n; a) =kX
i=0(k
i)(n)iak¡i
with ( n)0= 1;(n)i=n(n+ 1): : :(n+i¡1);fori¸1;de¯ned for f2C(R0), the
space of functions continuous on R0.
It is clear from the de¯nition of Ba
n(f;x),vq(x) and CqthatBa
n(f;x) is a linear
positive operator from CqintoCqand
1X
k=0pn;k(x; a) = 1 :

Approximation by Generalized Baskakov Operators for . . . 205
2.2 Auxiliary results
To prove the main theorems we need following lemmas [1, 2, 10].
Lemma 2.1 Fora; x¸0,n= 1;2; :::;we have
Ba
n(1;x) = 1; Ba
n(t;x) =x+ax
n(1 +x); (2.5)
Ba
n¡
(t¡x)2;x¢
=x(1 +x)
n+1
n2ax
1 +x(a+ 1)x+ 1
1 +x: (2.6)
Lemma 2.2 ForBa
n(f;x)de¯ned by (2.3), we have,
Ba
n(eqt;x) =eax
1+x(eq=n¡1)(1 +x¡xeq=n)¡n;
Ba
n¡
(t¡x)2eqt;x¢
=eax
1+x(eq=n¡1)(1 +x¡xeq=n)¡n¡2x(1 +x)
n
£nh
eq=n+x(1 +x)
nn2(eq=n¡1)2i
+haeq=n
n(1 +axeq=n
1 +x)
£(1¡xeq=n
1 +x)2+ 2axeq=n(eq=n¡1)(1¡xeq=n
1 +x)io
;
forx <(eq=n¡1)¡1< n=q .
If in addition, we assume that x·´pn, where, ´=1
3min(q¡2;1)andn¸2q,
we have
Ba
n(eqt;x)·ea+1eqx; (2.7)
Ba
n¡
(t¡x)2eqt;x¢
·M1x(1 +x)
neqx: (2.8)
Lemma 2.3 Forq¸0andn¸2qthere exists a constant Mq(a)such that for all
n2N,x2R0,
Ba
n¡
(t¡x)2;x¢
·Mq(a)x(1 +x)
n:
This lemma can be easily proved using (2.6).
Lemma 2.4 Ifq > 0,x·´pn, where ´=1
3min(q¡2;1)andn¸2qthen
following holds for all n2N,x2R0,
kBa
n(1
vq(t);x)kq·ea+1: (2.9)

206 Thai J. Math. 2(2004)/ Abdul Wa¯ and Salma Khatoon
Proof. Using Lemma(2.2), we obtain,
vq(x)jBa
n(1
vq(t);x)j=e¡qxjBa
n(eqt;x)j
·e¡qxea+1eqx
·ea+1;
which readily implies to (2.9). ¤
With the application of Lemma (2.4), we can prove following lemma [2].
Lemma 2.5 Forkfkq<1, we have,
kBa
n(f;x)kq· kfkqea+1; (2.10)
forx·´pn, where ´=1
3min(q¡2;1)andn¸2q.
2.3 Approximation theorems
In this section we will investigate the degree of approximation for functions of one
variable by Ba
n(f;x) in exponential weight spaces.
Theorem 2.6 Letg2C1
qwith some q >0,x2R0, satisfying x <(eq=n¡1)¡1<
n=qforn2N. Then there exists a constant Mq(a)depending only on q and a
such that
vq(x)jBa
n(g;x)¡g(x)j · kg0kqMq(a)nx(1 +x)
no1=2
; (2.11)
forx·´pn, where ´=1
3min(q¡2;1)andn¸2q.
Proof. For a ¯x x2R0,g2C1
qandt2R0, we have
g(t)¡g(x) =Zt
xg0(u)du:
It follows from (2.5) that for every n2N
jBa
n(g(t);x)¡g(x)j=Ba
n(jZt
xg0(u)duj;x): (2.12)
But from [8], we have
Zt
xg0(u)du· kg0kqjZt
x1
vq(t)duj · kg0kq(eqt+eqx)jt¡xj;
with which (2.12) reduces to
vq(x)jBa
n(g;x)¡g(x)j · kg0kqn
e¡qxBa
n(eqtjt¡xj;x) +Ba
n(jt¡xj;x)o
:(2.13)

Approximation by Generalized Baskakov Operators for . . . 207
By the HÄ older inequality, (2.5), (2.7), (2.8) and Lemma (2.3), we get
Ba
n(jt¡xj;x)·n
Ba
n((t¡x)2;x)o1=2
fBa
n(1;x)g1=2·Mq(a)nx(1 +x)
no1=2
(2.14)
and
e¡qxBa
n(eqtjt¡xj;x)·n
e¡qxBa
n((t¡x)2eqt;x)o1=2n
e¡qxBa
n(eqt;x)o1=2
·Mq(a)nx(1 +x)
no1=2
: (2.15)
Using (2.14) and (2.15) in (2.13), we get
vq(x)jBa
n(g;x)¡g(x)j · kg0kqMq(a)nx(1 +x)
no1=2
;
forx·´pn, where ´=1
3min(q¡2;1) and n¸2q. ¤
Next, we prove the following theorem.
Theorem 2.7 Suppose that f2Cq,q >0,x2R0, satisfying x <(eq=n¡1)¡1<
n=qforn2N. then there exists a constant M0
q(a)depending only on q and a such
that
vq(x)jBa
n(f;x)¡f(x)j ·M0
q(a)!³
f; Cq;nx(1 +x)
no1=2´
;
forx·´pn, where ´=1
3min(q¡2;1)andn¸2q.
Proof. Letf±be the Steklov mean of f2Cq, i.e.,
f±(x) =1
±Z±
0f(x+u)du; x ¸0; ± > 0:
Forx¸0,± >0, we have
f±(x)¡f(x) =1
±Z±
0[f(x+u)¡f(u)]du;
f0
±(x) =1
±[f(x+±)¡f(x)];
which implies f±(x)2C1
qand
kf±¡fkq·!(f; Cq;±) (2.16)
kf0
±kq·±¡1!(f; Cq;±): (2.17)
Thus, for every n2N,x¸0 and ± >0, we obtain
vq(x)jBa
n(f;x)¡f(x)j ·vq(x)jBa
n(f±¡f;x)j+vq(x)jBa
n(f±;x)¡f±(x)j+vq(x)jf±¡fj:

208 Thai J. Math. 2(2004)/ Abdul Wa¯ and Salma Khatoon
Using (2.10), (2.16), (2.17) and Theorem (2.6) in above, we yield
vq(x)jBa
n(f;x)¡f(x)j ·ea+1!(f; Cq;±)
+±¡1Mq(a)!(f; Cq;±)nx(1 +x)
no1=2
+!(f; Cq;±)
·M0
q!(f; Cq;nx(1 +x)
no1=2
):
¤
From Theorem (2.7), we have following two corollaries giving convergence and
direct theorems for generalized Baskakov operators in one variable.
Corollary 2.8 Iff2Cqwith some q >0, then
Ltn!1Ba
n(f;x) =f(x)
for each x2R0. This statement holds uniformly on every interval [0; x1]; x1>0.
Corollary 2.9 Iff2Lip(Cq; ®)with some q >0and0< ®·1, then there
exists a positive constant M0
q(a)depending only on q and a such that
vq(x)jBa
n(f;x)¡f(x)j ·M0
q(a)nx(1 +x)
no®=2
;
forx·´pn, where ´=1
3min(q¡2;1)andn¸2q.
3 Approximation of functions of two variables
3.1 Notations and auxiliary results
LetC(R2
0= [0;1)£[0;1)) be the set of all real-valued functions of one and two
variables continuous on R2
0. Using (2.1), let us de¯ne for q1; q2>0 and ( x; y)2R2
0
vq1;q2(x; y) =vq1(x)vq2(y) =e¡q1xe¡q2y; (3.1)
Cq1;q2=n
f2C(R2
0) :vq1;q2fis uniformly continuous and bounded on R2
0o
,
kfkq1;q2= sup
(x;y)2R2
0vq1;q2(x; y)jf(x; y)j
and
!(f; Cq1;q2;s; t) = sup
0·u·s;0·v·tk¢u;vf(:; :)kq1;q2;
where,
¢u;vf(x; y) =f(x+u; y+v)¡f(x; y):

Approximation by Generalized Baskakov Operators for . . . 209
Also, let for some q1; q2>0;0< ®; ¯ ·1
Lip(Cq1;q2;®; ¯) =n
f2Cq1;q2:!(f; Cq1;q2;s; t) =O(s®+t¯) ass; t!0 +o
and
C1
q1;q2=n
f2Cq1;q2:@f
@x;@f
@y2Cq1;q2o
:
Now, we de¯ne the generalized Baskakov operators for function of two variables
in the space Cq1;q2; q1; q2>0 with non-negative constants a; b¸0 independent
of n as
Ba;b
m;n(f;x; y) =1X
j=01X
k=0pm;j(x; a)pn;k(y; b)f(j=m; k=n ); m; n 2N;(x; y)2R2
0;
(3.2)
where, pm;j(x; a); pn;k(y; b) are de¯ned by (2.4).
It can be easily veri¯ed that Ba;b
m;n(f:x; y) is a linear positive operator from
Cq1;q2intoCq1;q2, provided m, n are large enough and also
1X
j=01X
k=0pm;j(x; a)pn;k(y; b) = 1 : (3.3)
Analogously as in (2.5), we have
Ba;b
m;n(1;x; y) = 1 ;for all m; n2N;(x; y)2R2
0: (3.4)
Iff(x; y) =f1(x)f2(y) and f12Cq1; f22Cq2, then for all m; n2N;(x; y)2
R2
0
Ba;b
m;n(f;x; y) =Ba
m(f1(t);x)Bb
n(f2(z);y): (3.5)
Using Lemma (2.4), Lemma (2.5) and equations (3.1) – (3.5), we obtain fol-
lowing lemmas :
Lemma 3.1 Letq1; q2>0. Then with
x·´1pm; where; ´ 1=1
3min(q¡2
1;1)and m ¸2q1;
y·´2pn; where; ´ 2=1
3min(q¡2
2;1)and n ¸2q2;
following holds for all m; n2N;(x; y)2R2
0
kBa;b
m;n(1
vq1;q2(t; z);x; y)kq·ea+b+2:
Lemma 3.2 Iff2Cq1;q2; q1; q2>0, then for
x·´1pm; where; ´ 1=1
3min(q¡2
1;1)and m ¸2q1;
y·´2pn; where; ´ 2=1
3min(q¡2
2;1)and n ¸2q2;
we have,
kBa;b
m;n(f;x; y)kq1;q2· kfkq1;q2ea+b+2:

210 Thai J. Math. 2(2004)/ Abdul Wa¯ and Salma Khatoon
3.2 Approximation theorems
In this section we shall state and prove analogues of Theorems 2.6 and 2.7.
Theorem 3.3 Letg2C1
q1;q2with some q1; q2>0,(x; y)2R2
0satisfying x <
(eq1=m¡1)¡1< m=q 1andy <(eq2=n¡1)¡1< n=q 2form; n2N. Then there
exists a constant Mq1;q2(a; b)depending only on q1; q2, a and b such that
vq1;q2(x; y)jBa;b
m;n(g;x; y)¡g(x; y)j ·Mq1;q2(a; b)h
k@g
@ukq1;q2nx(1 +x)
mo1=2
+k@g
@vkq1;q2ny(1 +y)
no1=2i
; (3.6)
forx·´1pm;where ´1=1
3min(q¡2
1;1)andm¸2q1;y·´2pn;where ´2=
1
3min(q¡2
2;1)andn¸2q2:
Proof. Let (x; y)2R2
0be a ¯xed point, we can write for ( t; z)2R2
0
g(t; z)¡g(x; y) =Zt
x@g
@u(u; z)du+Zz
y@g
@v(x; v)dv;
thus, for m; n2N, we have
vq1;q2(x; y)jBa;b
m;n(g;x; y)¡g(x; y)j ·vq1;q2(x; y)Ba;b
m;n³
jZt
x@g
@u(u; z)duj;x; y´
+vq1;q2(x; y)Ba;b
m;n³
jZz
y@g
@v(x; v)dvj;x; y´
(3.7)
Now, as in the proof of the Theorem 2.6, we have
jZt
x@g
@u(u; z)duj · k@g
@ukq1;q2jZt
x1
vq1;q2(u; z)duj
· k@g
@ukq1;q2eq2z[eq1t+eq1x]jt¡xj (3.8)
and
jZz
y@g
@v(x; v)dvj · k@g
@vkq1;q2jZz
y1
vq1;q2(x; v)dvj
· k@g
@vkq1;q2eq1x[eq2z+eq2y]jz¡yj: (3.9)
Using (3.8), ¯rst term on the right hand side of (3.7) reduces to
vq1;q2(x; y)Ba;b
m;n(jZt
x@g
@u(u; z)duj;x; y)

Approximation by Generalized Baskakov Operators for . . . 211
· k@g
@ukq1;q2fe¡q1xBa
m(eq1tjt¡xj;x)Ba
m(eq1tjt¡xj;x)g £eq2ybb
n(eq2z;y) (3.10)
and by the HÄ older inequality, (2.5), (2.7), (2.8) and Lemma 2.3, we get
e¡q1xBa
m(eq1tjt¡xj;x)· fe¡q1xBa
m(eq1t(t¡x)2;x)g1
2fe¡q1xBa
m(eq1t;x)g1
2
·Mq1(a)nx(1 +x)
no1
2; (3.11)
Ba
m(jt¡xj;x)· fBa
m((t¡x)2;x)g1
2fBa
m(1;x)g1
2
·Mq1(a)nx(1 +x)
no1
2(3.12)
and
e¡q2yBb
n(eq2z;y)·eb+1: (3.13)
Substituting values from (3.11), (3.12) and (3.13), (3.10) takes the form
vq1;q2(x; y)Ba;b
m;n(jZt
x@g
@u(u; z)duj;x; y)· k@g
@ukq1;q2Mq1;q2(a; b)nx(1 +x)
no1=2
:
(3.14)
Similarly using (3.9), second term on the right hand side of (3.7) changes to
vq1;q2(x; y)Ba;b
m;n(jZz
y@g
@v(x; v)dvj;x; y)· k@g
@vkq1;q2fe¡q2yBb
n(eq2zjz¡yj;y)
+Bb
n(jz¡yj;y)g
· k@g
@vkq1;q2Mq1;q2(a; b)fy(1 +y)
ng1=2:
(3.15)
Hence, with the use of (3.14) and (3.15), (3.7) reduces to (3.6). ¤
Theorem 3.4 Suppose that f2Cq1;q2,q1; q2>0,(x; y)2R2
0, satisfying the
conditions of Theorem (3.3). Then there exists a constant Mq1;q2(a; b)depending
only on q1; q2, a and b such that
vq1;q2(x; y)jBa;b
m;n(f;x; y)¡f(x; y)j ·Mq1;q2(a; b)!(f; Cq1;q2;fx(1 +x)
mg1=2;fy(1 +y)
ng1=2);
(3.16)
forx·´1pmwhere ´1=1
3min(q¡2
1;1)andm¸2q1;y·´2pnwhere ´2=
1
3min(q¡2
2;1)andn¸2q2:
Proof. Again, as in the proof of Theorem 2.7, we consider the Steklov mean for
f2Cq1;q2,
f±1;±2(x; y) =1
±1±2Z±1
0Z±2
0f(x+u; y+v)dudv; (x; y)2R2
0; ±1; ±2>0:

212 Thai J. Math. 2(2004)/ Abdul Wa¯ and Salma Khatoon
For (x; y)2R2
0; ±1; ±2>0, we have
f±1;±2(x; y)¡f(x; y) =1
±1±2Z±1
0Z±2
0¢u;vf(x; y)dudv;
@f±1;±2(x; y)
@x=1
±1±2Z±2
0[f(x+±1; y+v)¡f(x; y+v)]dv;
@f±1;±2(x; y)
@y=1
±1±2Z±1
0[f(x+u; y+±2)¡f(x+u; y)]du;
which implies f±1;±22C1
q1;q2(±1; ±2>0) and
kf±1;±2¡fkq1;q2·!(f; Cq1;q2;±1; ±2); (3.17)
k@f±1;±2
@xkq1;q2·sup
(x;y)2R2
0vq1;q21
±1±2Z±2
0(j¢±1;vf(x; y)j+j¢0;vf(x; y)j)dv
·2±¡1
1!(f; Cq1;q2;±1; ±2) (3.18)
k@f±1;±2
@ykq1;q2·sup
(x;y)2R2
0vq1;q21
±1±2Z±1
0(j¢u;±2f(x; y)j+j¢u;0f(x; y)j)du
·2±¡1
2!(f; Cq1;q2;±1; ±2): (3.19)
For each ¯xed ( x; y)2R2
0,q1; q2>0 and for all m; n2N,±1; ±2>0, we have
vq1;q2(x; y)jBa;b
m;n(f;x; y)¡f(x; y)j ·vq1;q2(x; y)fjBa;b
m;n(f¡f±1;±2;x; y)
+jBa;b
m;n(f±1;±2;x; y)¡f±1;±2j+jf±1;±2¡fjg:
(3.20)
Using Lemma 3.2, Theorem 3.3, (3.17), (3.18) and (3.19) in the above expression,
we arrive at (3.16). ¤
The immediate consequences of the Theorem (3.4) are following two corollaries
establishing convergence and direct theorems for functions of two variables.
Corollary 3.5 Iff2Cq1;q2with some q1; q2>0, then
Lt
m;n!1Ba;b
m;n(f;x; y) =f(x; y);
for all (x; y)2R2
0. Moreover, this statement holds uniformly on every rectangle
0·x·x1,0·y·y1,x1>0,y1>0.
Corollary 3.6 Iff2Lip(Cq1;q2;®; ¯)with some q1; q2>0and0< ®,¯·1,
then there exists a constant Mq1;q2(a; b)depending only on q1,q2, a and b such
that
vq1;q2(x; y)jBa;b
m;n(f;x; y)¡f(x; y)j ·Mq1;q2(a; b)[fx(1 +x)
mg®=2+fy(1 +y)
ng¯=2];
forx·´1pmwhere ´1=1
3min(q¡2
1;1)andm¸2q1;y·´2pnwhere ´2=
1
3min(q¡2
2;1)andn¸2q2:

Approximation by Generalized Baskakov Operators for . . . 213
4 Approximation in polynomial weight spaces
Following [4] we de¯ne weights, wp(x),p2W=N[ f0gas
w0(x) = 1 ; wp(x) = (1 + xp)¡1; p > 0
spaces Cp=n
f2C(R0) :wpfis uniformly continuous and bounded on R0o
with
kfkp= sup
x2R0wp(x)jf(x)j:
Assume further for h; ±¸0;0< ®·1 and for f2Cp
¢hf(x) =f(x+h)¡f(x);!(f; Cp;±) = sup
0·h·±k¢hfkp;
Lip(Cp; ®) =ff2Cp:!(f; Cp;±) =O(±®)!0+g
and
Cm
p=n
f2Cp:f(k)2Cp; k= 1;2; :::; mo
;
for ¯xed pandm2N.
It is clear from the de¯nition of Ba
n(f;x),wp(x) and Cpthat Ba
n(f;x) is a
linear positive operator from CpintoCpand
1X
k=0pn;k(x; a) = 1 :
As in section 2.3, we can prove following theorems on degree of approximation
of function of one variable by Ba
n(f;x) in polynomial weight spaces.
Theorem 4.1 Letg2C1
pwith some p2W, then there exists a constant Mp(a)
depending only on p and a such that
wp(x)jBa
n(g;x)¡g(x)j · kg0kqMp(a)nx(1 +x)
no1=2
; (4.1)
for all x2R0,n2N.
Theorem 4.2 Suppose that f2Cp,p2W.Then there exists a constant Mp(a)
depending only on p and a such that
wp(x)jBa
n(f;x)¡f(x)j ·Mp(a)!³
f; Cp;fx(1 +x)
ng1=2´
;
for all x2R0,n2N.

214 Thai J. Math. 2(2004)/ Abdul Wa¯ and Salma Khatoon
Two immediate consequences of Theorem 4.2 are corollaries analogous to
Corollary 2.8 and 2.9.
For functions of two variables, we consider the weights wp1;p2(x; y),p1; p22
W=N[ f0g,(x; y)2R2
0as
wp1;p2(x; y) =wp1(x)wp2(y);
spaces
Cp1;p2=n
f2C(R2
0) :wp1;p2fis uniformly continuous and bounded on R2
0o
;
norm
kfkp1;p2= sup
(x;y)2R2
0wp1;p2(x; y)jf(x; y)j
and
!(f; Cp1;p2;s; t) = sup
0·u·s;0·v·tk¢u;vf(:; :)kp1;p2;
where,
¢u;vf(x; y) =f(x+u; y+v)¡f(x; y):
Also, let for some p1; p22W;0< ®; ¯ ·1
Lip(Cp1;p2;®; ¯) =n
f2Cp1;p2:!(f; Cp1;p2;s; t) =O(s®+t¯)as s; t !0 +o
and
C1
p1;p2=n
f2Cp1;p2:@f
@x;@f
@y2Cp1;p2o
:
It can be easily veri¯ed that Ba;b
m;n(f:x; y) is a linear positive operator from
Cp1;p2intoCp1;p2, provided m, n are large enough and also
1X
j=01X
k=0pm;j(x; a)pn;k(y; b) = 1 :
Now we shall give here following two theorems regarding degree of approxi-
mation of functions of two variables by the operator (3.2) in polynomial weight
spaces.
Theorem 4.3 Letg2C1
p1;p2with some p1; p22W, then there exists a constant
Mp1;p2(a; b)depending only on p1; p2, a and b such that
wp1;p2(x; y)jBa;b
m;n(g;x; y)¡g(x; y)j ·Mp1;p2(a; b)h
k@g
@ukp1;p2nx(1 +x)
mo1=2
+k@g
@vkp1;p2fy(1 +y)
ng1=2i
(4.2)
for all (x; y)2R2
0andm; n2N.

Approximation by Generalized Baskakov Operators for . . . 215
Theorem 4.4 Suppose that f2Cp1;p2,p1; p22W. Then there exists a constant
Mp1;p2(a; b)depending only on p1; p2, a and b such that
wp1;p2(x; y)jBa;b
m;n(f;x; y)¡f(x; y)j
·Mp1;p2(a; b)!(f; Cp1;p2;fx(1 +x)
mg1=2;fy(1 +y)
ng1=2);
for all (x; y)2R2
0andm; n2N.
Corollaries giving convergence and direct theorems for functions of two vari-
ables in polynomial weight spaces (corresponding to Corollary 3.5 and 3.6) can
also be obtained from Theorem 4.4.
References
[1]Abdul Wa¯ and Salma Khatoon, On the order of approximation of functions
by generalized Baskakov operators, Indian J. Pure Appl. Math. 35(3)(2004),
347{358.
[2]Abdul Wa¯ and Salma Khatoon, Inverse theorem for Baskakov operator , com-
municated.
[3]M. Becker, D. Kucharski, R. J. Nessel, Global approximation theorems for
the Sz¶ asz-Mirkjan operators in exponential weight spaces , Linear Spaces and
Approximation (Proc. Conf. Oberwolfach, 1977), BirkhÄ auser Verlag, Basel,
ISNM, 40(1978), 319{333.
[4]M. Becker, Global Approximation theorems for Sz¶ asz-Mirkjan and Baskakov
operators in polynomial weight spaces, Indiana Univ. Math. J., 27(1)(1978),
127{142.
[5]A. Dhingas, ÄUber einige IdentitÄ aten vom Bernstein Types, Norske Vid. Selsk.
Trodheim 24(1951), 96{97.
[6]B. Firlej and L. Rempulska, Approximation of functions by some operators of
the Sz¶ asz-Mirkjan-type, Fasc. Math. 27(1997), 15{27.
[7]M. Gurdek, L. Rempulska and M. Skorupka, The Baskakov operators for func-
tions of two variables, Collect. Math. 50(3) (1999), 289{302.
[8]M. Lesniewicz and L. Rempulska, Approximation by some operators of
the Szasz-Mirkjan type in exponential weight spaces, Glasnik Matematicki
52(1997), 57{69.
[9]G. G. Lorentz, Bernstein Polynomials, Univ. of Toronto Press, Toronto (1953).
[10]V. Mihesan, Uniform approximation with positive operators genenrated by
generalized Baskakov method, Automat. Comput. Appl. Math. 7(1)(1998), 38{
97.

216 Thai J. Math. 2(2004)/ Abdul Wa¯ and Salma Khatoon
[11]M. Skorupka, On approximationof functions of two variables by some linear
positive operators, Le Matematiche 50(2)(1995), 323{336.
[12]D. D. Stancu, On certain polynomials of Bernstein type, Studii Cercet. St.
Matem. (Iasi) 11(1960), 221-233.
[13]D. D. Stancu, On certain polynomials of two variables of Bernstein type and
some applications of them, Dokl. Akad. Nauk SSSR 134(1)(1960), 48{51.
[14]D. D. Stancu, De lapproximation, par des polyn^ omes du type Bernstein, des
functions de deux variable, Comm. Acad. R.P. Rom^ ³ne 9(1959), 773{777.
[15]G. You and P. Xaun, Weighted approximation by multidimensional Baskakov
operators, J. Math. Res. Exposition 20(1) (2000), 43{50.
(Received 10 January 2004)
Abdul Wa¯ and Salma Khatoon
Department of Mathematics
Faculty of Natural Sciences
Amia Millia Islamia, New Delhi-110025, INDIA
e-mail : skhareem@hotmail.com