APPROXIMATION BY A GENERALIZED SZASZ TYPE OPERATOR FOR FUNCTIONS OF TWO VARIABLES

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 15 (2014), No 2, pp. 345-359 DOI: 10.18514/MMN.2014.666

Approximation by a generalized Szász type

operator for functions of two variables

Nursel Çetin, Sevilay Kirci Serenbay, and Çi§dem

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 15 (2014), No. 2, pp. 345–359

APPROXIMATION BY A GENERALIZED SZ ´ASZ TYPE

OPERATOR FOR FUNCTIONS OF TWO VARIABLES

NURSEL C¸ ET˙IN, SEVILAY KIRCI SERENBAY, AND C¸ I ˘GDEM ATAKUT Received 05 December, 2012

Abstract. In the present paper, we define a new Sz´asz-Mirakjan type operator in exponential weighted spaces for functions of two variables having exponential growth at infinity using a method given by Jakimovski-Leviatan. This operator is a generalization of two variables of an operator defined by A. Ciupa [1]. In this study, we investigate approximation properties and also estimate the rate of convergence for this new operator.

2010 Mathematics Subject Classification: 41A25;41A36

Keywords: linear positive operator, Jakimovski-Leviatan operator, weighted space, modulus of continuity, rate of convergence

1. INTRODUCTION

For a real function of real variable f W Œ0; 1/ ! R, the Sz´asz-Mirakjan operators are defined in [2] as Sn.fI x/ D e nx 1 X j D0 .nx/j j Š f . j n/ , x2 Œ0; 1/ ;

where the convergence of Sn.fI x/ to f .x/ under the exponential growth condition

on f that is jf .x/j  CeBx; for all x2 Œ0; 1/ ; with C; B > 0 was proved. Then, various modifications and further properties of the Sz´asz-Mirakjan operators have been studied intensively by many authors (e.g. [1,3–9]).

In [4], A. Jakimovski and D. Leviatan investigated approximation properties of a generalization of the Sz´asz-Mirakjan operators which are stated as follows:

Let g.´/D P1

nD0

an´nbe an analytic function in the diskj´j < R; R > 1 and

sup-pose g.1/¤ 0: Define the Appell polynomials pk.x/D pk.x; g/ .k 0/ by

g.u/eux_D 1 X kD0 pk.x/uk: c

For each function f defined in Œ0;1/ ; they considered the operators Lndefined by Ln.fI x/ D e nx g.1/ 1 X kD0 pk.nx/f  k n  ; n > 0

and also the authors obtained several approximation properties of these operators. A. Ciupa [1] introduced a Sz´asz-Mirakjan type operator that is a generalization of the operator defined by M. Lesniewicz and L. Rempulska [5] using the method given by Jakimovski-Leviatan. A. Ciupa studied the properties of approximation for functions of one variable in the space of continuous functions having an exponential growth at infinity.

In this paper, inspired by [1], for each function f defined in Œ0;1/  Œ0; 1/ ; we define the operators Ln;mby

Ln;m.fI x; y/ D e nxe my .g .1//2 1 X kD0 1 X j D0 pk.nx/ pj.my/ f  k n; j m  where g .u1/ eu1xg .u2/ eu2yD 1 X kD0 pk.x/ uk1 1 X j D0 pj.y/ uj₂:

Now, we consider the function g .x/_D P1

kD0

x2kC1

.2k_{C 1/Š} D sinh x where sinh x is the hyperbolic function of x and let pkbe the polynomials generated by relation

sinh u1sinh .u1x/ sinh u2sinh .u2y/D 1 X kD0 1 X j D0 p2k.x/ p2j.y/ u2k1 u 2j 2 :

Using the following equalities sinh u1sinh .u1x/D 1 2 1 X kD0 .1_{C x/}2k .1 x/2k .2k/Š u 2k 1 sinh u2sinh .u2y/D 1 2 1 X j D0 .1_{C y/}2j .1 y/2j .2j /Š u 2j 2 ; we have p2k.x/D .1C x/2k .1 x/2k 2 .2k/Š , p2j.y/D .1C y/2j .1 y/2j 2 .2j /Š : Let C R₁2
be the set of all real-valued continuous functions of two variables on R₁2WD f.x; y/ W x  1; y  1g :

For p; q > 0 and .x; y/2 R12, we define

wp;q.x; y/D wp.x/ wq.y/D e pxe qy

Cp;qD˚f 2 C R21
W wp;qf is uniformly continuous and bounded on R12

kf kp;qD sup

.x;y/2R2 1

wp.x/ wq.y/jf .x; y/j

and also for h; k 0, ı  0 , f 2 Cp;q; the first order modulus of continuity given by

! f; Cp;qI ı
D sup 0h;kı
h;kf _p;q where
h;kf .x; y/D f .x C h; y C k/ f .x; y/ :

In this study, in the space Cp;q; p; q > 0; we introduce the following positive linear

operators Pn;m.fI x; y/ D

1

.sinh 1/2sinh .nx/ sinh .my/

1 X kD0 1 X j D0 p2k.nx/ p2j.my/ f  2k n ; 2j m  (1) n; m_{2 N , .x; y/ 2 R}2₁and investigate the theorems on convergence of Pn;m.fI x; y/

operators to functions of two variables. We also estimate the rate of convergence for this new operator by using the modulus of continuity.

2. AUXILIARYRESULTS

In this section, we will give some useful results in order to study the convergence of the sequence Pn;mf
to the function f 2 Cp;q:

Lemma 1. If.x; y/_{2 R}2₁andn; m_{2 N; we have} Pn;m e0;0I x; y
D 1 Pn;m e1;0I x; y
D 1 ncoth 1C x coth .nx/ Pn;m e0;1I x; y
D 1

mcoth 1C y coth .my/ Pn;m e1;02 C e0;12 I x; y
D x2C y2
C  1 n2C 1 m2  .1_{C coth 1/} C .1 C 2 coth 1/x ncoth .nx/C y mcoth .my/  whereei;j.t1; t2/D t1it j

Lemma 2. If.x; y/2 R21; p; q > 0 and n; m2 N; then we have

Pn;m ept1I x; y
D

1

sinh 1 sinh .nx/sinh  ep=nsinh  nxep=n Pn;m eqt2I x; y
D 1

sinh 1 sinh .my/sinh  eq=msinh  myeq=m Pn;m t1ept1I x; y
D ep=n n 1 sinh 1 sinh .nx/ n cosh  ep=n  sinh  nxep=n  C nx sinhep=ncoshnxep=no

Pn;m t2eqt2I x; y
D

eq=m m

1 sinh 1 sinh .my/

n cosh  eq=msinh  myeq=m C my sinheq=mcoshmyeq=mo

Pn;m t12ept1I x; y
D 1 sinh 1 sinh .nx/ ( e2p=n n2 sinh  ep=nsinh  nxep=n C2x n e 2p=n cosh  ep=ncosh  nxep=n_{C x}2e2p=nsinh  ep=nsinh  nxep=n C 1 n2e p=n cosh  ep=nsinh.nxep=n/_Cx ne p=n sinh  ep=ncosh  nxep=n  Pn;m t22eqt2I x; y
D 1 sinh 1 sinh .my/

( e2q=m m2 sinh  eq=msinh  myeq=m C2y me

2q=m_cosh_eq=m_cosh_myeq=m

C y2e2q=msinheq=msinhmyeq=m C 1 m2e q=m cosh  eq=msinh.myeq=m/_Cy me q=m sinh  eq=mcosh  myeq=m  : Lemma 3. For all.x; y/_{2 R}2₁andn; m_{2 N; we have}

Pn;m  .t1 x/2ept1I x; y  D 1 sinh 1 sinh nx  x2sinh  ep=nsinh  nxep=n hep=n 1i2 C sinhnxep=n " e2p=n n2 sinh  ep=n_Ce p=n n2 cosh  ep=n 2x n e p=n cosh  ep=n  C coshnxep=n 2x n e 2p=n cosh  ep=n

C x ne

p=n_sinh_ep=ni _2x2_ep=n_sinh_ep=n_e nxep=no Pn;m  .t2 y/2eqt2I x; y  D 1 sinh 1 sinh my  y2sinh  eq=msinh  myeq=m heq=m 1i 2 C sinhmyeq=m " e2q=m m2 sinh  eq=m_Ce q=m m2 cosh  eq=m 2y me q=m_cosh_eq=m  C coshmyeq=m 2y me 2q=m_cosh_eq=m C y me q=m sinh  eq=m i 2y2eq=msinh  eq=m  e myeq=m o : Lemma 4. For all.x; y/_{2 R}2₁andn; m_{2 N; we have}

Pn;m  .t1 x/2I x; y  3 .xC 1/ n Pn;m  .t2 y/2I x; y  3 .y_mC 1/: Proof. By Lemma1, we get

Pn;m  .t1 x/2I x; y  D .coth .nx/ 1/ 2x 1 ncoth 1 x  Cx ncoth nx C 1 n2.1C coth 1/ :

Thus for .x; y/_{2 R1}2; we can write Pn;m  .t1 x/2I x; y  x 1 n C 2x n C 3 n2  3 .xC 1/ n : Similarly, we can easily obtain

Pn;m  .t2 y/2I x; y  3 .yC 1/ m :  Lemma 5. Letp; q > 0 , r > p; s > q and let n0D n0.p; r/ ; m0D m0.q; s/ be

fixed natural numbers such thatn0> p= .ln r ln p/ and m0> q= .ln s ln q/. Then

there exist positive constantsCp;r andCq;sdepending only onp; r and q; s such that

wr.x/ Pn;m  .t1 x/2ept1I x; y   Cp;r sinh  ep=n sinh 1 x_{C 2} n

ws.y/ Pn;m  .t2 y/2eqt2I x; y   Cq;s sinh  eq=m  sinh 1 y_{C 2} m for all.x; y/_{2 R}2₁andn_{ n}0; m m0:

Proof. Firstly, for m; n2 N; we consider the sequence of real numbers .pn/ and

.qm/ ; pnD n  ep=n_{C 1} (2.1) qmD m  eq=m_{C 1} (2.2) which are decreasing and lim

n!1pnD p; limm!1qmD q. Thus

p < pn< pep=n pep (2.3)

q < qm< qeq=m qeq (2.4)

Since n0> p= .ln r ln p/ ; we have ep=n0< eln.r=p/D r=p and r > pep=n0> pn0>

pn for n n0: Also, because m0> q= .ln s ln q/ ; we get eq=m0 < eln.s=q/D s=q

and s > qeq=m0> qm0> qmfor m m0:

Applying2.1, we obtain sinh



nxep=n.sinh nx/ 1_{ 2e}pnx

coshnxep=n.sinh nx/ 1 epnx (2.5) x2.sinh nx/ 1x

n: Also using2.2, we get

sinh 

myeq=m.sinh my/ 1_{ 2e}qmy

coshmyeq=m.sinh my/ 1_{ e}qmy (2.6) y2.sinh my/ 1 y

m:

By writing the last inequalities in Lemma2, we get respectively Pn;m ept1I x; y
 sinh  ep=n sinh 1 2e pnx Pn;m eqt2I x; y
 sinh  eq=m sinh 1 2e qmy

and Pn;m ept1I x; y
_r sup sinh  ep=n sinh 1 2e .pn r/x P_n;m eqt2I x; y
_s sup sinh  eq=m sinh 1 2e .qm s/y_:

Taking into account Lemma3and2.5, we obtain Pn;m  .t1 x/2ept1I x; y   1 sinh 1  2x2epnxp 2 n n2sinh  ep=nC2x n sinh  ep=nepn nxep=n C 2 n2e pnx_e2p=n sinh  ep=n_C 2 n2e pnx_ep=n cosh  ep=n 4x n e pnx_ep=n cosh  ep=n_C2x n e pnx_e2p=n cosh  ep=n Cx ne pnx_ep=n sinh  ep=no  sinh  ep=n sinh 1  2x2p 2 n n2e pnx C2x n e p=n C 2 n2e pnx e2p=n C_n22e pnx_ep=n_coth_ep=n C2x_n epnxe2p=ncothep=n Cx ne pnx_ep=no Since coth 

ep=n_{ coth 1 < 2 and e}p=n< ep; we can write Pn;m  .t1 x/2ept1I x; y   sinh  ep=n sinh 1  2x2 n2 p 2 nepnxC 2x n e p C 2 n2e pnx_e2p C 4 n2e pnx_ep_C4x n e pnx_e2p_Cx ne pnx_ep  : Say wr.x/D e rx: Thus, we get

wr.x/ Pn;m  .t1 x/2ept1I x; y   sinh  ep=n sinh 1  2x n e .pn r/x x np 2 nC ep 2 C 2e 2p  C2x n e p rx C 2 n2e .pn r/x _e2p C 2ep
 :

Now, by using2.1and inequalities x np 2 n< x np 2_e2p=n_{< xp}2_e2p_; it follows wr.x/ Pn;m  .t1 x/2ept1I x; y   sinhep=n sinh 1  2x n e .pn r/x  xp2e2p_Ce p 2 C 2e 2p  C2x n e p C 2 n2e .pn r/x _e2p C 2ep
 :

Also, we have r pn r pn0> 0 and xe .r pn/x xe .r pn0/x  1= .r pn0/

for n n0: Applying e.pn r/x< 1; we obtain

wr.x/ Pn;m  .t1 x/2ept1I x; y   sinh  ep=n sinh 1  2 n 1 r pn0  xp2e2p_{C 2e}2p_Ce p 2  C2x n e p C 2 n2 e 2p C 2ep
  sinh  ep=n sinh 1  2x n e2p r pn0 p2_C2 n 1 r pn0  2e2p_Ce p 2  C2x n e p C 2 n2 e 2p C 2ep
  Cp;r sinh  ep=n sinh 1 xC 2 n :

Similarly as above, applying 2.6 to Lemma 3, by simple calculations we easily obtain the required inequality

ws.y/ Pn;m  .t2 y/2eqt2I x; y   Cq;s sinheq=m sinh 1 yC 2 m :  Lemma 6. Ifp; q > 0 , r > p; s > q and n0D n0.p; r/ ; m0D m0.q; s/ be fixed

natural numbers such that n0> p= .ln r ln p/ ; m0 > q= .ln s ln q/ and if f 2

Cp;q; then we have

kPn;m.fI x; y/k_r;s 2 kf kp;q

sinh.ep/ sinh.eq/ .sinh 1/2 :

Proof. By (1), we can write e rxe sy_jPn;m.fI x; y/j

D e

rx_e sy

.sinh 1/2sinh .nx/ sinh .my/ ˇ ˇ ˇ ˇ ˇ ˇ 1 X kD0 1 X j D0 p2k.nx/ p2j.my/ e 2kp n e 2j q m
f  2k_n ;2j m  e2kpn _e 2j q m ˇ ˇ ˇ ˇ : Sincekf kp;q D sup .x;y/2R21

e pxe qyjf .x; y/j ; it follows that e rxe sy_jPn;m.fI x; y/j

 e

rx_e sy

.sinh 1/2sinh .nx/ sinh .my/kf kp;q

1 X kD0 1 X j D0 p2k.nx/ p2j.my/ e 2kp n e 2j q m D e rxe sy_{kf kp;q}Pn;m ept1I x; y
Pn;m eqt2I x; y
D e rxe sy_{kf kp;q}

sinhep=nsinhnxep=n sinh 1 sinh .nx/

sinheq=msinhmyeq=m sinh 1 sinh .my/ : Using the notations in Lemma5, the inequalities (2.5) and (2.6), we obtain e rxe syjPn;m.fI x; y/j  kf kp;qe rxe sy sinh  ep=n sinh 1 2e pnx sinh  eq=m sinh 1 2e qmy D 4 kf kp;qe x.r pn/e y.s qm/ sinh  ep=n  sinh  eq=m  .sinh 1/2  4 kf kp;q sinh .ep/ sinh .eq/ .sinh 1/2 :  3. APPROXIMATION BYPn;m OPERATORS

In this section, we give theorems on the degree of approximation of functions of two variables by these operators.

Theorem 1. Letp; q > 0 , r > p; s > q and n0D n0.p; r/ ; m0D m0.q; s/ be

fixed natural numbers such thatn0> p= .ln r ln p/ and m0> q= .ln s ln q/. If f 2

C_p;q1 ; where C_p;q1 _{D˚f 2 C}p;q W fx; fy2 Cp;q , then there exists a positive constant

wr;s.x; y/jPn;m.fI x; y/ f .x; y/j  Mp;q;r;s ( @f @x _p;q r x C 2 n C @f @y _p;q r y C 2 m )

Proof. Let .x; y/ be a fixed point in R2₁: For f _{2 Cp;q}1 and .t1; t2/2 R2₁we have

f .t1; t2/ f .x; y/D t1 Z x @f @u.u; t2/ duC t2 Z y @f @v.x; v/ dv: Using Pn;m.1I x; y/ D 1; it results that

Pn;m.f .t1; t2/I x; y/ f .x; y/ D Pn;m 0 @ t1 Z x @f @u.u; t2/ duI x; y 1 AC Pn;m 0 @ t2 Z y @f @v.x; v/ dvI x; y 1 A: For r > p; s > q and m; n2 N; we have

wr;s.x; y/jPn;m.f .t1; t2/I x; y/ f .x; y/j  wr;s.x; y/ Pn;m 0 @ ˇ ˇ ˇ ˇ ˇ ˇ t1 Z x @f @u.u; t2/ du ˇ ˇ ˇ ˇ ˇ ˇ I x; y 1 A C wr;s.x; y/ Pn;m 0 @ ˇ ˇ ˇ ˇ ˇ ˇ t2 Z y @f @v .x; v/ dv ˇ ˇ ˇ ˇ ˇ ˇ I x; y 1 A: By using the following inequalities

ˇ ˇ ˇ ˇ ˇ ˇ t1 Z x @f @u.u; t2/ du ˇ ˇ ˇ ˇ ˇ ˇ  @f @x _p;q ˇ ˇ ˇ ˇ ˇ ˇ t1 Z x 1 wp;q.u; t2/ du ˇ ˇ ˇ ˇ ˇ ˇ  @f @x _p;q 1 wq.t2/  ₁ wp.t1/C 1 wp.x/  jt1 xj ; ˇ ˇ ˇ ˇ ˇ ˇ t2 Z y @f @v .x; v/ dv ˇ ˇ ˇ ˇ ˇ ˇ  @f @y _p;q ˇ ˇ ˇ ˇ ˇ ˇ t2 Z y 1 wp;q.x; v/ dv ˇ ˇ ˇ ˇ ˇ ˇ  @f @y _p;q 1 wp.x/  1 wq.t2/C 1 wq.y/  jt2 yj

and H¨older inequality, we can write wr;s.x; y/ Pn;m 0 @ ˇ ˇ ˇ ˇ ˇ ˇ t1 Z x @f @u.u; t2/ du ˇ ˇ ˇ ˇ ˇ ˇ I x; y 1 A  @f @x _p;q wr;s.x; y/ Pn;m  ₁ wq.t2/I x; y   Pn;m jt 1 xj wp.t1/I x; y  C 1 wp.x/ Pn;m.jt1 xj I x; y/   @f @x _p;q _h wr.x/Pn;m  .t1 x/2ept1I x; y i1=2 wr.x/Pn;m ept1I x; y
1=2 C e.p r/xhPn;m  .t1 x/2I x; y i1=2 ws.y/Pn;m eqt2I x; y
; wr;s.x; y/ Pn;m 0 @ ˇ ˇ ˇ ˇ ˇ ˇ t2 Z y @f @v.x; v/ dv ˇ ˇ ˇ ˇ ˇ ˇ I x; y 1 A  @f @y _p;q wr.x/ wp.x/ Pn;m.1I x; y/ ws.y/  Pn;m jt2 y_j wq.t2/I x; y  C_w 1 q.y/ Pn;m.jt2 yj I x; y/   @f @y _p;q  h ws.y/ Pn;m  .t2 y/2eqt2I x; y i1=2
ws.y/ Pn;m eqt2I x; y
1=2 ChPn;m  .t2 y/2I x; y i1=2 : Applying the inequalities2.5and2.6to Lemma2, we obtain

wr.x/Pn;m ept1I x; y
 e rx sinh  ep=n sinh 1 2e pnx D 2 sinh  ep=n sinh 1 e x.r pn/  2 sinhep=n sinh 1 and ws.y/Pn;m eqt2I x; y
 e sy sinh  eq=m sinh 1 2e qmy

D 2 sinh  eq=m  sinh 1 e y.s qm/  2 sinh  eq=m sinh 1 : By these inequalities, Lemma5and Lemma4, we get to

wr;s.x; y/ Pn;m 0 @ ˇ ˇ ˇ ˇ ˇ ˇ t1 Z x @f @u.u; t2/ du ˇ ˇ ˇ ˇ ˇ ˇ I x; y 1 A  2 @f @x _p;q sinh  ep=nsinh  eq=m .sinh 1/2 r 2Cp;r x_{C 2} n C 2 @f @x _p;q sinh  eq=m sinh 1 e x.r p/ r 3 .x_{C 1/} n  @f @x _p;q Mp;rr x C 2 n ; and wr;s.x; y/ Pn;m 0 @ ˇ ˇ ˇ ˇ ˇ ˇ t2 Z y @f @v.x; v/ dv ˇ ˇ ˇ ˇ ˇ ˇ I x; y 1 A  @f @y _p;q sinh  eq=m sinh 1 r 2Cq;s y_{C 2} m C @f @y _p;q r 3.y_{C 1/} m  @f @y _p;q Mq;sr y C 2 m

for all m m0and n n0: This proves the theorem. 

Theorem 2. Suppose thatf 2 Cp;q andp; r; q; s; n0; m0 satisfy the conditions of

Theorem1. Then there exists positive constant M_{D M}p;q;r;s depending only on

p; q; r; s such that wr;s.x; y/jPn;m.fI x; y/ f .x; y/j  M! f; Cp;qI x C 2 n 1=2 ; y C 2 m 1=2! for all.x; y/2 R21andm m0; n n0:

Proof. Similarly as in [5]; we consider the Steklov means for f 2 Cp;q fh;ı.x; y/D 1 hı h Z 0 ı Z 0 f .x_{C u; y C v/ dudv ; h; ı > 0; .x; y/ 2 R1}2: We have fh;ı.x; y/ f .x; y/D 1 hı h Z 0 ı Z 0
u;vf .x; y/ dudv; @fh;ı @x .x; y/D 1 hı ı Z 0 .f .x_{C h; y C v/ f .x; y C v// dv;} @fh;ı @y .x; y/D 1 hı h Z 0 .f .x_{C u; y C ı/ f .x C u; y// du;} which implies fh;ı2 Cp;q1 .h; ı > 0/ and

fh;ı f _p;q  w f; Cp;qI h; ı
; @fh;ı @x _p;q  sup .x;y/2R2 1 wp;q.x; y/ 1 hı ı Z 0 ˇ ˇ
h;vf .x; y/ ˇ ˇC j
0;vf .x; y/j
dv  2 hw f; Cp;qI h; ı
and @fh;ı @y _p;q 2 ıw f; Cp;qI h; ı
for h; ı > 0:

For every fixed .x; y/2 R12; r > p; s > q and n; m2 N; h; ı > 0 we have

wr;s.x; y/jPn;m.fI x; y/ f .x; y/j  wr;s.x; y/ ˚ˇ ˇP_n;m f f_h;ıI x; y
ˇ ˇ CˇˇPn;m fh;ıI x; y
fh;ı.x; y/ ˇ ˇC ˇ ˇfh;ı.x; y/ f .x; y/ ˇ ˇ : By Lemma5, one obtains

wr;s.x; y/

ˇ

ˇPn;m f fh;ıI x; y
ˇˇ 4w f; Cp;qI h; ı

for all m m0and n n0: Since Theorem1, we can write

wr;s.x; y/ ˇ ˇPn;m fh;ıI x; y
fh;ı.x; y/ ˇ ˇ

 Mp;q;r;sw f; Cp;qI h; ı
( 1 h r x C 2 n C 1 ı r y C 2 m ) for m m0, n n0: Therefore wr;s.x; y/jPn;m.fI x; y/ f .x; y/j  4w f; Cp;qI h; ı
C Mp;q;r;sw f; Cp;qI h; ı
( 1 h r x C 2 n C 1 ı r y C 2 m ) C wr;s.x; y/ ˇ ˇfh;ı.x; y/ f .x; y/ ˇ ˇ  w f; Cp;qI h; ı
5C Mp;q;r;s ( 1 h r x C 2 n C 1 ı r y C 2 m )! for all h; ı > 0 and m m0, n n0: Setting hD

q xC2 n ; ıD q yC2 m we obtain the desired result. 

Corollary 1. Iff _{2 C}p;q; then for all .x; y/2 R2₁

lim

m;n!1Pn;m.fI x; y/ D f .x; y/ :

Also, the convergence is uniform on every rectangle 1 x  a; 1  y  b: REFERENCES

[1] A. Ciupa, “Approximation by a generalized Szasz type operator,” J. Comput. Anal. Appl., vol. 5, no. 4, pp. 413–424, 2003.

[2] J. Favard, “Sur les multiplicateurs d’interpolation,” J. Math. Pures Appl., vol. 9, no. 23, pp. 219– 247, 1944.

[3] Z. Finta, N. K. Govil, and V. Gupta, “Some results on modified Sz´asz-Mirakjan operators,” J. Math. Anal. Appl., vol. 327, no. 2, pp. 1284–1296, 2007.

[4] A. Jakimovski and D. Leviatan, “Generalized Szasz operators for the approximation in the finite interval.” Mathematica, Cluj, vol. 11, pp. 97–103, 1969.

[5] M. Le´sniewicz and L. Rempulska, “Approximation by some operators of the Szasz-Mirakjan type in exponential weight spaces,” Glas. Mat., III. Ser., vol. 32, no. 1, pp. 57–69, 1997.

[6] R. N. Mohapatra and Z. Walczak, “Remarks on a class of Sz´asz-Mirakyan type operators,” East J. Approx., vol. 15, no. 2, pp. 197–206, 2009.

[7] C. Mortici, “An extension of the Sz´asz-Mirakjan operators,” An. S¸tiint¸. Univ. “Ovidius” Constant¸a, Ser. Mat., vol. 17, no. 1, pp. 137–144, 2009.

[8] L. Rempulska and S. Graczyk, “On certain class of Sz´asz-Mirakyan operators in exponential weight spaces,” Int. J. Pure Appl. Math., vol. 60, no. 3, pp. 259–267, 2010.

[9] Z. Walczak, “On the convergence of the modified Szasz-Mirakyan operators,” Yokohama Math. J., vol. 51, no. 1, pp. 11–18, 2004.

Authors’ addresses Nursel C¸ etin

Ankara University, Faculty of Science, Department of Mathematics, Tandogan 06100 Ankara, Tur-key

E-mail address: ncetin@ankara.edu.tr Sevilay Kirci Serenbay

Bas¸kent University, Department of Mathematics Education, 06530 Ankara, Turkey E-mail address: kirci@baskent.edu.tr

C¸ i˘gdem Atakut

Ankara University, Faculty of Science, Department of Mathematics, Tandogan 06100 Ankara, Tur-key