IS S N 1 3 0 3 –5 9 9 1
APPROXIMATION BY THE BIVARIATE COMPLEX BASKAKOV-STANCU OPERATORS
·
ISP·IR NURHAYAT AND MANAV NES·IBE
Abstract. In this paper we study the approximation properties of the Stancu type bivariate generalization of the complex Baskakov operators. We obtain a Voronovskaja type result with quantitative estimates for bivariate complex Baskakov-Stancu operators attached to analytic functions having suitable ex-ponential growth on compact polydisks. Also we give the exact order of ap-proximation.
1. Introduction
In the present paper we deal with the following type complex Baskakov operators relating to divided di¤erence of an analytic function f: For a complex valued function f de…ned on [R; 1) [ DR with DR = fz 2 C : jzj < Rg ; the complex
Baskakov operators de…ned by Wn(f )(z) = 1 X j=0 n(n + 1):::(n + j 1) nj [0; 1=n; :::; j=n; f ] z j (1.1)
were studied in [8. pp.124-134]. Here the function f : [R; 1) [ DR! C is analytic
in DR and all its derivatives bounded on [0; 1) by the same constant and also
has an expontential growth condition for all z 2 DR and for j = 0; one takes
n(n + 1):::(n + j 1) = 1. In [8], the Voronovskaja type results with a quantitative estimate. The exact order of simultaneous approximation for these operators were given. Considering the real Baskakov operators de…ned in [4], the classical complex Baskakov operator is de…ned by
Bn(f ) (z) = (1 + z) n 1 X k=0 n + k 1 k z 1 + z k f k n ; z 2 C:
Received by the editors July 24, 2014, Accepted: Sept. 26, 2014. 2000 Mathematics Subject Classi…cation. 47A58.
Key words and phrases. Bivariate complex Baskakov-Stancu operators, rate of convergence, exact order.
c 2 0 1 4 A n ka ra U n ive rsity
If z = x; x 2 R and x 0 then Wn(f )(z) = Bn(f ) (z) : But if x < 0 then Wn(f )(z)
may be di¤erent from Bn(f ) (z) [8. page 124]. Therefore Wn(f )(z) and Bn(f ) (z)
do not necessarily coincide for all z 2 C: In [8], the approximation properties of these operators were studied separately, under di¤erent hypothesis on f and z 2 C: Furthermore, bivariate form of the operators Wn(f ) given by (1.1) was introduced
and the results in univariate case were extended to the bivariate case for the analytic functions on polydisks [8.pp.172-180].
The Stancu type generalization of the complex Baskakov operators were studied by Gal et.all. [6] which are de…ned as follows
Wn; (f )(z) = 1 X j=0 n(n + 1):::(n + j 1) (n + )j n + ; + 1 n + ; :::; + j n + ; f z j ; z 2 C (1.2) where 0 and ; are real numbers with independent of n, [x0; x1; :::; xn; f ]
denotes the divided di¤erence of the function f on the distinct points x0; x1; :::; xn.
In [6], Voronoskaja type result with quantitative estimates and convergence results for the operators (1.2) attached the analytic functions on compact disks were ob-tained. The similar results for complex Bernstein-Stancu polynomials in [7],[9], and for complex Durrmeyer-Stancu and genuine Durrmeyer-Stancu operators in [13] and [10] were obtained. In case of real variables, some approximation properties of the Baskakov and Baskakov-Stancu operators were investigated in [1], [5],[11],[12] and [14].
The aim of the present paper is to investigate the approximation properties of bivariate complex Baskakov-Stancu operators of tensor product kind. We extend the approximation results from the univariate case, obtained in [6] for the complex Baskakov-Stancu operators, to the bivariate case.
First we present a few concepts in the bivariate case which are natural extensions of the usual concepts in the univariate case. Let DRj := fzj 2 C : jzjj < Rj; j = 1; 2g and DR1 DR2 denotes an open polydisk (of center 0 and radius R) where R = (R1; R2) and jz1j r1; jz2j r2; r1 < R1 with r2 < R2. Let also DR1 DR2 =
(z1; z2) 2 C2: jzjj Rj; j = 1; 2 denotes the closed polydisk.
We de…ned the bivariate complex Baskakov-Stancu operators as follows Wn;m; ; ; (f )(z1; z2) = 1 X =0 1 X =0 n(n + 1):::(n + 1) (n + ) m(m + 1):::(m + 1) (m + ) " n + ; + 1 n + ; :::; + n + ; m + ; + 1 m + ; :::; + m + ; f (:; :) z2 # z1 z1z2 (1.3) where f : [R1; +1) [ DR1 [R2; +1) [ DR2 ! C is analytic in DR1 DR2and f has all partial derivatives bounded on [0; +1) [0; +1), by the same constant, and satis…es an exponential growth condition, namely jf(z1; z2)j M eA1jz1j+A2jz2j;
for all z1 2 DR1, z2 2 DR2 and for = 0; = 0; n(n + 1):::(n + 1) = 1; m(m + 1):::(m + 1) = 1:
In this paper, we would like to obtain the exact order of approximation for the operators given by (1.3) on compact polydiscs. First we give the order of approximation and the Voronovskaja type theorems with quantitative estimate for the operators W ; ; ;
n;m (f ) de…ned by (1.3). These results allow us to obtain the
exact order in approximation by the operators W ; ; ; n;m (f ):
2. Auxiliary Results
In order to establish the next results, we need the following auxiliary lemmas. Lemma 2.1. ([6] Lemma1) For all n; k 2 N [ f0g, 0 , z 2 C, let us de…ne
Vn; (ek; z) = 1 X =0 n (n + 1) ::: (n + 1) (n + ) n + ; + 1 n + ; :::; + n + ; ek z ; where ek(z) = zk. Then Vn; (e0; z) = 1 and we have the following recurrence
relation: V ; n (ek+1; z) = z (1 + z) n + V ; n (ek; z) 0+ nz + n + V ; n (ek; z) : As a result, Vn; (e1; z) = nz + n + ; V ; n (e2; z) = n (n + 1) z2 (n + )2 + nz (1 + 2 ) z2 (n + )2 + 2 (n + )2. Throughout the paper we use the two dimensional test functions ei;j: [0; +1)
[0; +1) ! R; ei;j(x1; x2) = ei(x1)ej(x2) with ei(x1) = xi1 and ej(x2) = xj2 for
i; j 2 f0; 1; 2g.
Lemma 2.2. ([6] Lemma 3) For all n; k 2 N [ f0g, 0 , z 2 C and jzj r, r 1 then we have
Vn; (ek; z) (k + 1)!rk.
Lemma 2.3. For all n; m 2 N [ f0g we have Wn;m; ; ; (e0;0)(x1; x2) = 1 Wn;m; ; ; (e1;0)(x1; x2) = nx1+ n + Wn;m; ; ; (e0;1)(x1; x2) = mx2+ m + Wn;m; ; ; (e2;0)(x1; x2) = n (n + 1) x2 1 (n + )2 + nx1(1 + 2 ) (n + )2 + 2 (n + )2 Wn;m; ; ; (e0;2)(x1; x2) = m (m + 1) x2 2 (m + )2 + mx2(1 + 2 ) (m + )2 + 2 (m + )2
where ei;j(i; j = 0; 1; 2) are the test functions.
Proof. Considering Lemma 1 and using Barbosu method in [2], [3], it can be easily proved. So we omit the details of proof.
Lemma 2.4. Let f : [0; +1) [0; +1) ! R has all the partial derivatives bounded by the same constant in [0; +1) [0; +1),then W ; ; ;
n;m (f ) uniformly converges
to f on [0; r1] [0; r2], for r1; r2> 0:
Proof. Considering Lemma 3 we obtain lim n;m!1 W ; ; ; n;m (e0;0) e0;0 r 1;r2 = 0; lim n;m!1 W ; ; ; n;m (e1;0) e1;0 r 1;r2 = 0; lim n;m!1 W ; ; ; n;m (e0;1) e0;1 r1;r2 = 0; lim n;m!1 W ; ; ; n;m (e2;0+ e0;2) (e2;0+ e0;2) r 1;r2 = 0: Hence by Volkov’s theorem in [15], we reach the desired result.
Lemma 2.5. For all ; 2 N [ f0g ; 0 ; 0 and jz1j r1; jz2j r2
and r1; r2 1 we have
Wn;m; ; ; (e ; )(z1; z2) ( + 1)! ( + 1)!r1r2.
Proof. Using the equality ek;j(z1; z2) = ek(z1) ej(z2) and by the de…nition of
W ; ; ;
n;m , from Lemma 2 we get the result.
3. Approximation by bivariate complex Baskakov-Stancu operators In this section we will give some convergence results with quantitative estimates for the operators W ; ; ;
n;m (f ).
Theorem 3.1. Let n0; m0 2 N and 3 n0 < 2R1 < 1; 3 m0 < 2R2 < 1
and ; 2 N [ f0g ; 0 ; 0 . Suppose that f : [R1; +1) [ DR1 [R2; +1) [ DR2 ! C has all the partial derivatives bounded by the same constant in [0; +1) [0; +1), analytic in DR1 DR2, that means f (z1; z2) =
P1
=0
P1
=0c ; z1z2,
for all jz1j R1; jz2j R2 and suppose that there exist M > 0 and Ai 2
1 Ri
; 1 , i = 1; 2, with the property that jc ; j M
A1A2
! ! , for all ; = 0; 1; 2; :::, ( which implies jf(z1; z2)j M eA1jz1j+A2jz2j, for all z1 2 DR1; z2 2 DR2 ). If 1 r1< min n0 2 ; 1 A1 ; 1 r2< min m0 2 ; 1
A2 then for all jz1j
r1; jz2j r2
and n > n0, m > m0 the sequence of the operators Wn;m; ; ; (f ) is uniformly
Proof. Using the results of Lemma 4 and page 159 in [6] we have Wn;m; ; ; (f )(z1; z2) 1 X =0 1 X =0 c ; Wn;m; ; ; (e ; )(z1; z2) M 1 X =0 1 X =0 ( + 1) ( + 1) (r1A1) (r2A2) < 1
where the last series is convergent for all n; m 2 N, jz1j r1; jz2j r2; n n0;
m m0 with 1 r1< min n0 2 ; 1 A1 ; 1 r2< min m0 2 ; 1 A2 . On the other hand, from Lemma 3 we have
lim
n;m!1W ; ; ;
n;m (f )(x1; x2) = f (x1; x2)
for all (x1; x2) 2 [0; r1] [0; r2] and by the classical Vitali’s theorem we arrive at
W ; ; ;
n;m (f ) is uniformly converges to f on Dr1 Dr2for all n n0; m m0:
Now, we can give the following estimate in approximation for W ;
n;m(f ) to f:.
Theorem 3.2. Let 0 ; 0 . Suppose that the hypotheses are same on the function f and on the constants n0, m0, R1, R2, M , A1, A2in the statement
of Theorem 1. Then
(i): Suppose that 1 r1 < min
n0 2 ; 1 A1 ; 1 r2 < min m0 2 ; 1 A2 . Then for all jz1j r1; jz2j r2 and n > n0, m > m0 we have
Wn;m; ; ; (f )(z1; z2) f (z1; z2) + r1 n + Dr1;r2(f ) + Ar1(f ) n + + Br1(f ) n + + Cr1(f ) n + + + r2 m + Fr1;r2(f ) + Ar2(f ) m + + Br2(f ) m + + Cr2(f ) m +
where Dr1;r2(f ) = 1 X =1 1 X =0 jc ; j r1 1r2 < +1; Fr1;r2(f ) = 1 X =0 1 X =1 jc ; j r2 1r1( + 1)! < +1; Ar1(f ) = (1 + r1) 1 X =1 1 X =0 jc ; j ( + 1)!r1 1r2 < +1; Ar2(f ) = (1 + r2) 1 X =0 1 X =1 jc ; j ( + 1)!r2 1< +1; Br1(f ) = 1 X =1 1 X =0 jc ; j r2 < +1; Br2(f ) = 1 X =0 1 X =1 jc ; j r2 1< +1; Cr1(f ) = 1 X =1 1 X =0 jc ; j r2r1 < +1; Cr2(f ) = 1 X =0 1 X =1 jc ; j r2 < +1:
(ii): Let 1; 2 2 N be with 1+ 2 1 and 1 r1 < r1 < min
n0 2 ; 1 A1 ; 1 r2 < r2 < min m0 2 ; 1
A2 be arbitrary …xed. Then for all jz1j
r1; jz2j r2, n > n0 and m > m0 we have @ 1+ 2W ; ; ; n;m (f ) @z 1 1 @z22 (z1; z2) @ 1+ 2f @z 1 1 @z22 (z1; z2) Cr; ; ; 1;r2;n;m(f ) 1! (r1 r1) 1+1 2! (r2 r2) 2+1
where the constant Cr; ; ; 1;r2;n;m(f ) = + r1 n + 1 X =1 1 X =0 jc ; j (r1) 1 (r2) +Ar1(f ) n + + Br1(f ) n + + Cr1(f ) n + + + r2 m + 1 X =0 1 X =1 jc ; j (r2) 1 [(r1) ( + 1)!] + Ar2(f ) m + + Br2(f ) m + + Cr2(f ) m + and Ar1(f ), Ar2(f ), Br1(f ), Br2(f ), Cr1(f ), Cr2(f ) is given as the above.
Proof. (i) Denote e ; (z1; z2) = e (z1):e (z2) where e (t) = t . From Lemma 2 we
get Wn;m; ; ; (f )(z1; z2) f (z1; z2) 1 X =0 1 X =0 c ; Wn;m; ; ; (e ; )(z1; z2) 1 X =0 1 X =0 c ; e ; (z1; z2) 1 X =0 1 X =0 jc ; j Wn;m; ; ; (e ; )(z1; z2) e ; (z1; z2)
taking into account the estimate Vn; (ek)(z) ek(z) rk 1 + r n + + r (1 + r) n + k (k + 1)!r k 2+ k n + r k 1+ k n + r k
for k 2 N; jzj r; r 1;in the proof of Theorem1 in [6] for all jz1j r1 and
jz2j r2and using Lemma 1 we obtain
Wn;m; ; ; (e ; )(z1; z2) e ; (z1; z2) = Wn; (e )(z1):Wm; (e )(z2) z1z2 Wn; (e )(z1):Wm; (e )(z2) Wn; (e )(z1):z2 + Wn; (e )(z1):z2 z1z2 Wn; (e )(z1) : Wm; (e )(z2) z2 + jz2j : Wn; (e )(z1) z1 r1( + 1)! r2 1 + r2 m + +r2(1 + r2) m + ( + 1)!r 2 2 +m + r 1 2 +m + r2 +r2 r1 1 + r2 n + +r1(1 + r1) n + ( + 1)!r 2 1 +n + r 1 1 +n + r1
which from the conditions on the coe¢ ents c ; implies Wn;m; ; ; (f )(z1; z2) f (z1; z2) 1 X =0 1 X =0 jc ; j Wn;m; ; ; (e ; )(z1; z2) e ; (z1; z2) 1 X =0 1 X =0 jc ; j r1( + 1)! r 1 2 + r2 m + + r2(1 + r2) m + ( + 1)!r 2 2 +m + r 1 2 + m + r2 + r2 r 1 1 + r1 n + + r1(1 + r1) n + ( + 1)!r 2 1 +n + r1 1 + n + r1 = + r1 n + Dr1;r2(f ) + Ar1(f ) n + + Br1(f ) n + + Cr1(f ) n + + + r2 m + Fr1;r2(f ) + Ar2(f ) m + + Br2(f ) m + + Cr2(f ) m + which proves (i).
Here, the analyticity of f implies that the series Dr1;r2(f ), Fr1;r2(f ), Br1(f ), Br2(f ), Cr1(f ), Cr2(f ) are convergent and the convergency of Ar1(f ), Ar2(f ) follows from jc ; j M
A1A2 ! ! .
(ii) Now we give the rate of convergence in simultaneous approximation. Let 1 r1< r1< R1; 1 r2< r2< R2. By the Cauchy’s formula we get
@ 1+ 2W ; ; ; n;m (f ) @z 1 1 @z 2 2 (z1; z2) @ 1+ 2f @z 1 1 @z 2 2 (z1; z2) 1! 2! (2 i)2 Z ju2 z2j=r2 Z ju1 z1j=r1 Wn;m; ; ; (f )(u1; u2) f (u1; u2) ju1 z1j 1+1ju2 z2j 2+1 du1du2
passing to absolute value with jz1j r1; jz2j r2 and taking into account that
ju1 z1j = r1 r1; ju2 z2j = r2 r2, by applying the estimate in (i) we obtain
@ 1+ 2W ; ; ; n;m (f ) @z 1 1 @z22 (z1; z2) @ 1+ 2f @z 1 1 @z22 (z1; z2) Cr; ; ; 1;r2;n;m(f ) 1! (r1 r1) 1+1 2! (r2 r2) 2+1
The second result is about the Voronovskaja-type theorem for operator (1.3). This Voronovskaja-type formula will be the product of the parametric extensions generated by the Voronovskaja-type formula in univariate case in Theorem 2 [6]. Thus, for f (z1; z2) de…ning the parametric extensions of Voronoskaja formula by
z1L ; n (f )(z1; z2) : = Wn; (f )(:; z2)(z1) f (z1; z2) z1 n + @f @z1 (z1; z2) z1(1 + z1) 2n @2f @z2 1 (z1; z2); z2L ; m (f )(z1; z2) : = Wm; (f )(z1; :)(z2) f (z1; z2) z2 m + @f @z2 (z1; z2) z2(1 + z2) 2m @2f @z2 2 (z1; z2):
their product gives
z2L ; m (f )(z1; z2) z1L ; n (f )(z1; z2) = Wm; Wn; (f )(:; z2)(z1) f (z1; z2) z1 n + @f @z1 (z1; z2) z1(1 + z1) 2n @2f @z2 1 (z1; z2) Wn; (f )(:; z2)(z1) f (z1; z2) z1 n + @f @z1 (z1; z2) z1(1 + z1) 2n @2f @z2 1 (z1; z2) z2(1 + z2) 2m W ; n @2f @z2 2 (:; z2) (z1) @2f @z2 2 (z1; z2) z1 n + @2 @z2 2 @f @z1 (z1; z2) z1(1 + z1) 2n @2 @z2 2 @2f @z2 1 (z1; z2) : = E1 E2 E3:
By simple calculation we can write
z2L ; m (f )(z1; z2) z1L ; n (f )(z1; z2) = Wn;m; ; ; (f )(z1; z2) Wm; (f ) (z1; :) (z2) z1 n + W ; m @f @z1 (z1; :) (z2) z1(1 + z1) 2n W ; m @2f @z2 1 (z1; :) (z2) Wn; (f )(:; z2)(z1) + f (z1; z2) + z1 n + @f @z1 (z1; z2) + z1(1 + z1) 2n @2f @z2 1 (z1; z2) z2(1 + z2) 2m W ; n @2f @z2 2 (:; z2) (z1) + z2(1 + z2) 2m @2f @z2 2 (z1; z2) +z2(1 + z2) 2m z1 n + @2 @z2 2 @f @z1 (z1; z2) + z1(1 + z1) 2n z2(1 + z2) 2m @4f @z2 1@z22 (z1; z2)
from which can be derived the commutativity property
The Voronovskaja’s theorem can be stated as follows.
Theorem 3.3. Let 0 ; 0 . Suppose that the hypothesis on the function f and on the constants n0, m0, R1, R2, M , A1, A2 in the statement of
Theorem (1) hold and let 1 r1 < min
n0 2 ; 1 A1 , 1 r2 < min m0 2 ; 1 A2 be …xed. For all n > n0, m > m0 and jz1j r1, jz2j r2 we have the following
Voronovskaja-type result z2L ; m (f )(z1; z2) z1L ; n (f )(z1; z2) M1;r1;r2(f ) 1 n2 + 1 m2 + 6 X k=2 Mk;r1;r2(f ) " 1 (n + )2 + 1 (m + )2 # where M1;r1;r2(f ) : = 16M 1 X =2 1 X =0 (r1A1) (r2A2) ( 1)( 2)2( + 1) < +1; M2;r1;r2(f ) : = 2M 1 X =2 1 X =0 (r1A1) 2(r2A2) ( 1) 2 ( + 1) < +1; M3;r1;r2(f ) : = 2 M 1 X =2 1 X =0 (r1A1) 2(r2A2) 2( + 1) r1< +1; M4;r1;r2(f ) : = 2 2 + 2 M 1 X =2 1 X =0 (r1A1) 2(r2A2) 2( + 1) ( + 1) r12< +1; M5;r1;r2(f ) : = M 1 X =2 1 X =0 (r1A1) 2 ( 2)! (r2A2) ( + 1) r1< +1; M6;r1;r2(f ) : = 2 M 1 X =2 1 X =0 (r1A1) 2 ( 2)! (r2A2) ( + 1) r 2 1< +1:
Proof. By the hypothesis we can write f (z1; z2) =
P1 =0f (z2) z1, where f (z2) = P1 =0c ; z2. It follows @2f @z2 1 (z1; z2) = P1 =2f (z2) ( 1) z1 2and @2f @z2 2 (z1; z2) =
P1 =0 @2f @z2 2 (z2) z1, where @2f @z2 2 (z2) =P1=2c ; ( 1) z2 2, @2 @z2 2 @f @z1 (z1; z2) = @2 @z2 2 1 X =1 z1 1f (z2) ! = 1 X =1 z1 1@ 2f @z2 2 (z2) = 1 X =1 1 X =2 c ; z1 1 ( 1) z 2 2 @f @z1
(z1; z2) =P1=1 z1 1f (z2). This implies that Wn; (f )(:; z2)(z1) =P1=0f (z2) Wn; (e ) (z1)
and Wn; (f )(:; z2)(z1) f (z1; z2) z1 n + @f @z1 (z1; z2) z1(1 + z1) 2n @2f @z2 1 (z1; z2) = Wn(f ) (:; z2) (z1) f (z1; z2) z1(1 + z1) 2n @2f @z2 1 (z1; z2) +Wn; (f )(:; z2)(z1) Wn(f ) (:; z2) (z1) z1 n + @f @z1 (z1; z2) = 1 X =2 f (z2) Wn(e ) (z1) 1 X =0 f (z2) z1 z1(1 + z1) 2n 1 X =2 f (z2) ( 1) z1 2 + 1 X =2 f (z2) Wn; (e ) (z1) 1 X =2 f (z2) Wn(e ) (z1) z1 n + 1 X =1 z1 1f (z2) = 1 X =2 f (z2) Wn(e ) (z1) e (z1) z1(1 + z1) 2n ( 1) z 2 1 + 1 X =2 f (z2) Wn; (e ) (z1) Wn(e ) (z1) z1 n + z 1 1
Applying W ;
m to the last expression with respect to z2, we obtain
E1 = 1 X =2 Wm; (f ) (z2) Wn(e ) (z1) e (z1) z1(1 + z1) 2n ( 1) z 2 1 + 1 X =2 Wm; (f ) (z2) Wn; (e ) (z1) Wn(e ) (z1) z1 n + z 1 1 = 1 X =2 1 X =0 c ; Wm; (e ) (z2) ! Wn(e ) (z1) e (z1) z1(1 + z1) 2n ( 1) z 2 1 + 1 X =2 1 X =0 c ; Wm; (e ) (z2) ! Wn; (e ) (z1) Wn(e ) (z1) z1 n + z 1 1
Passing to absolute value with jz1j r1and jz2j r2 and taking into account the
estimates in proofs of Theorem 2.4.2 in [8. pp.175-176], it follows jE1j 1 X =2 1 X =0 jc ; j r2( + 1)! 16r1 !( 1)( 2)2 n2 + 1 X =2 1 X =0 jc ; j r2( + 1)! " ( 1) ! 2 2 (n + )2 r 2 1 + 2 2 ! (n + )2r 1 1 + 2( + 1)! (n + )2 2 2 + 2 r1+ ( 1) (n + )2 r 1 1 + ( 1) 2 (n + )2 r1 # 1 n2 1 X =2 1 X =0 16M (A1r1) (A2r2) ( 1)( 2)2( + 1) + 1 (n + )2 1 X =2 1 X =0 M(A1r1) 2 (A2r2) ! ( + 1) ( 1) ! 2 2 + 2 2 !r 1 + 2( + 1)! 2 2 + 2 r 2 1+ ( 1) r1+ ( 1) 2r12 Similarly, jE2j 1 n2 1 X =0 1 X =2 16M(r1A1) (r2A2) ! ( 1)( 2) 2 + 1 (n + )2 1 X =0 1 X =2 M(A1r1) 2 (r2A2) ! ! ( 1) ! 2 2 + 2 2( + 1)!r 1 + 2 2 + 2 !r 2 1+ ( 1) r1+ ( 1) 2r21
Then Wn; @ 2f @z2 2 (:; z2) (z1) = 1 X =0 @2f @z2 2 (:; z2) Wn; (e ) (z1) = 1 X =0 1 X =2 c ; ( 1) z1 2Wn; (e ) (z1) and Wn; @ 2f @z2 2 (:; z2) (z1) @2f @z2 2 (z1; z2) z1 n + @2 @z2 2 @f @z1 (z1; z2) z1(1 + z1) 2n @2 @z2 2 @2f @z2 1 (z1; z2) = 1 X =2 1 X =2 c ; ( 1) z2 2Wn; (e ) (z1) 1 X =2 1 X =2 c ; ( 1) z2 2(e ) (z1) z1 n + 1 X =1 1 X =2 c ; z1 1 ( 1) z 2 2 z1(1 + z1) 2n 1 X =2 1 X =2 c ; z1 2 ( 1) = 1 X =2 1 X =2 c ; ( 1) z2 2 Wn; (e ) (z1) (e ) (z1) z1 n + z 1 1 z1 1(1 + z1) ( 1) 2n
which again by Theorem 2.4.2 in [8], implies jE3j r2(1 + r2) 2m 16M n2 1 X =2 1 X =2 (r1A1) (r2A2) ! ( 1) ( 1)( 2) 2 +r2(1 + r2) 2m 1 (n + )2 1 X =2 1 X =2 M(r1A1) 2 (r2A2) ! ! ( 1) ( 1) ! 2 2 +2 2r1( + 1)! + 2 2 + 2 !r 2 1+ ( 1) r1+ ( 1) 2r21
Interchanging above the places of n and m; by reason of symmetry, we get a similar order of approximation for z1L
;
n (f )(z1; z2) z2L ;
m (f )(z1; z2) .
In conclusion if we use the commutativity property ofz2L ; m (f )(z1; z2) z1L ; n (f )(z1; z2), z2L ; m (f )(z1; z2) z1L ; n (f )(z1; z2) jE1j + jE2j + jE3j M1;r1;r2(f ) 1 n2 + 1 m2 + 6 X k=2 Mk;r1;r2(f ) " 1 (n + )2+ 1 (m + )2 #
where the series Mi;r1;r2(f ) ; i = 1; 2; 3; 4; 5; 6 given by the statement are convergent due to jc ; j M
A1A2 ! ! .
The following theorem will be useful to …nd exact order of approximation by W ;
n;n(f ).
Theorem 3.4. Let 0 ; 0 . Suppose that n0= m0and the hypothesis
on the function f and on the constants n0, m0, R1, R2, M , A1, A2in the statement
of Theorem 1 hold and let 1 r1 < min
n0 2 ; 1 A1 , 1 r2 < min m0 2 ; 1 A2
be …xed. Denoting kfkr1;r2 = sup fjf(z1; z2)j ; jz1j r1; jz2j r2g and f is not a solution of the complex partial di¤ erantial equation
( z1) @f @z1 (z1; z2) + z1(1 + z1) 2 @2f @z2 1 (z1; z2) + ( z2) @f @z2 (z1; z2) +z2(1 + z2) 2 @2f @z2 2 (z1; z2) = 0; jz1j R1; jz2j R2 (3.1)
then for all n > n0 we have
Wn;n; (f ) f r1;r2 Kr1;;r2;f n where Kr; 1;r2;f depends only on f , , , r1, r2.
Proof. For all jz1j r1, jz2j r2 and n 2 N , we can write
Wn;n; (f )(z1; z2) f (z1; z2) = 2 n z2(1 + z2) 4 @2f @z2 2 (z1; z2) + z1(1 + z1) 4 @2f @z2 1 (z1; z2) +n ( z2) 2 (n + ) @f @z2 (z1; z2) + n ( z1) 2 (n + ) @f @z1 (z1; z2) +2 n n2 4 z2 Ln; (f )(z1; z2) z1L ; n (f )(z1; z2) + Rn(f ) (z1; z2)
where Rn(f ) (z1; z2) = Wn; (f )(:; z2)(z1) f (z1; z2) z1 n + @f @z1 (z1; z2) z1(1 + z1) 2n @2f @z2 1 (z1; z2) +Wn; (f ) (z1; :) (z2) f (z1; z2) z2 n + @f @z2 (z1; z2) z2(1 + z2) 2n @2f @z2 2 (z1; z2) +z2(1 + z2) 2n W ; n @2f @z2 2 (:; z2) (z1) @2f @z2 2 (z1; z2) +z1(1 + z1) 2n W ; n @2f @z2 1 (z1; :) (z2) @2f @z2 1 (z1; z2) z1(1 + z1) 2n z2(1 + z2) 2n @4f @z2 1@z22 (z1; z2) + z1 n + W ; n @f @z1 (z1; :) (z2) @f @z1 (z1; z2) z2(1 + z2) 2n z1 n + @2 @z2 2 @f @z1 (z1; z2)
By using Theorem 2.4.3 in [8] and Theorem 3 in [6] immediate that kRn(f )kr1;r2 ! 0 as n ! 1. Also, by Theorem (3) we obtain
n2 4 z2L ; n (f )(z1; z2) z1L ; n (f )(z1; z2) r1;r2 M1;r1;r2(f ) 2 + n2 2 (n + )2 6 X k=2 Mk;r1;r2(f ) which implies 2 n n2 4 z2L ; n (f )(z1; z2) z1L ; n (f )(z1; z2) + Rn(f ) r1;r2 ! 0; as n ! 1: Denoting H(z1; z2) = z2(1 + z2) 4 @2f @z2 2 (z1; z2) + n ( z2) 2 (n + ) @f @z2 (z1; z2) +z1(1 + z1) 4 @2f @z2 1 (z1; z2) + n ( z1) 2 (n + ) @f @z1 (z1; z2)
and taking into the inequalities
it follows Wn;n; (f ) f r1;r2 = 2 n z2(1 + z2) 4 @2f @z2 2 (z1; z2) + z1(1 + z1) 4 @2f @z2 1 (z1; z2) +n ( z2) 2 (n + ) @f @z2 (z1; z2) + n ( z1) 2 (n + ) @f @z1 (z1; z2) +2 n n2 4 z2 Ln; (f )(z1; z2) z1L ; n (f )(z1; z2) + Rn(f ) (z1; z2) r1;r2 2 n ( kHkr1;r2 2 n n2 4 z2 Ln; (f )(z1; z2) z1L ; n (f )(z1; z2) + Rn(f ) r1;r2 ) 2 n 1 2kHkr1;r2= 1 nkHkr1;r2
for all n n0, with n0 depending only on f , r1 and r2. We used here that
by hypothesis we have kHkr1;r2 0.For n 2 f1; 2; :::; n0 1g we obviously have W ; n;n(f ) f r1;r2 nN ; r1;r2;f n with nN ; r1;r2;f= n: W ; n;n(f ) f r1;r2 > 0, which …nally implies W ; n;n(f ) f r1;r2 Kr1;;r2;f
n for all n 2 N, where K
; r1;r2;f = minnkHk1r 1;r2; N ; r1;r2;f; 2N ; r1;r2;f; :::; n0 1N ; r1;r2;f o :
Combining Theorem 2 with Theorem 4 we obtain the following exact order. Corollary 1. If f is not a solution of equation (3.1), then the exact order in approximation by the bivariative complex Baskakov-Stancu operator Wn;n; (f ) in n1.
Note that, for = = 0; = = 0, the Theorems 2,3 and 4 become the results in the book [8. pp. 172-179].
Acknowledgement 1. The authors are thankful to the reviewers for making valu-able suggestions, leading to better presentation of the paper
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Current address : Department of Mathematics, Gazi University, 06500 Ankara, Turkey E-mail address : nispir@gazi.edu.tr
Current address : Department of Mathematics, Gazi University, 06500 Ankara, Turkey E-mail address : nmanav@gazi.edu.tr
