APPROXIMATION OF BIVARIATE FUNCTIONS BY OPERATORS OF STANCU HURWITZ TYPE. 1. Introduction

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1 FACTA UNIVERSITATIS NIŠ Ser. Math. Inor. 5, APPROXIMATION OF BIVARIATE FUNCTIONS BY OPERATORS OF STANCU HURWITZ TYPE Ioana Taşcu Abstract. The ai o this paper is to introduce and study a linear positive approxiation operator o Stancu-Hurwitz type [1], depending on several nonnegative paraeters, useul in the approxiation o unctions o two variables. The corresponding approxiation orula 3.1 has the degree o exactness 1, 1. For the reainder o this orula we give several representations, by starting ro a ethod o T. Popoviciu [7] or representation o the reainder ter in linear approxiation orulas, by using the divided dierences. 1. Introduction In a paper published in by D. D. Stancu [11] there has been constructed an approxiation linear positive operator, denoted by S β 1,...,β, which was deined, or any unction C[, 1], by the ollowing orula 1.1 where 1. S β 1,...,β x = k= 1 + β β 1 w β 1,...,β,k x w β 1,...,β k,k x, = xx + β β ik k1 1 x1 x + β j1 + + β jk k1. On the other hand, β 1,..., β are nonnegative paraeters. Received June 14, 5. Matheatics Subject Classiication. 41A1, 41A36 33

2 34 I. Taşcu These basis polynoials were constructed in [11] by starting ro an identity o Hurwitz [3], generalizing the classical identity o Abel-Jensen, naely u + vu + v + β β 1 = uu + β i1 + + β ik k1 vv + β j1 + + β jk k1, which in the special case β 1 = β = = β = β reduces to the identity o Abel-Jensen [4]: u + vu + v + β 1 = uu + kβ k1 vv + kβ k1. k k=. The Bivariate Polynoial Operator o Stancu-Hurwitz Type In this paper we consider the space o real-valued bivariate unctions CD, continuous on the unit square D = [, 1] [, 1], and we associate the Stancu-Hurwitz type bivariate polynoials.1 S β 1,...,β ;γ 1,...,γ n,n = x, y n k= ν= w β 1,...,β,k xv γ 1,...,γ n n,ν y k, ν, n where w β 1,...,β,k x is deined by orula 1. and v γ 1,...,γ n n,ν y is given by a siilar orula 1 + γ γ n v γ 1,...,γ n n,ν y = y y + γ γ sν ν1 1 y1 y + γ t1 + + γ nγ1 t nν, γ 1, γ,..., γ n being nonnegative paraeters. In the special cases β 1 = = β = β and γ 1 = = γ n = γ, we obtain the Cheney-Shara-Stancu type bivariate linear positive operator deined by the ollowing orula where S,n β;γ x, y = 1 + β 1 w β,k x = k n k= ν= w β k,k xvγ n,νy, ν, n xx + kβ k1 1 x1 x + kβ k1

3 Approxiation o Bivariate Functions by Operators and 1 + nγ n1 v γ n,νy = n yy + νγ ν1 1 y + n νγ nν1. γ The operator S,n β;γ represents an extension to two variables o the second operator o Cheney-Shara [1]. 3. Approxiation o Bivariate Functions by Means o a Polynoial Operator o Stancu-Hurwitz Type It is easy to see that the polynoial deined at.1 is interpolatory in the corners o the square D, that is it reproduces de values o the unction CD in the our points:,, 1,, 1, 1,, 1. Consequently, the approxiation orula 3.1 x, y = x, y + S β 1,...,β ;γ 1,...,γ n,n has the degree o exactness 1, 1. R β 1,...,β ;γ 1,...,γ n,n x, y Now i we use a theore o Peano-Milne-Stancu type, given in D. D. Stancu [9], we can give an integral representation or the reainder ter o the approxiation orula 3.1. Theore 3.1. I the unction has continuous second-order partial derivatives on the square D, then the reainder o the approxiation orula 3.1 can be represented under the ollowing integral or 3. x, y R β 1,...,β ;γ 1,...,γ n,n = G β 1,...,β t, x, t, ydt + where the Peano kernels are G β 1,...,β H γ 1,...,γ n n z, y, x, zdz t, xh γ 1,...,γ n n z, y, t, zdtdz G β 1,...,β t, x = R β 1,...,β ϕ x t, with ϕ x t = x t + x t = x t +,

4 36 I. Taşcu and with H γ 1,...,γ n n z, y = R γ 1,...,γ n n ψ y z, ψ y z = y z + y z = y z +. We have used above the notation It ollows that we can write explicitly r,s u, v = r+s u, v u r v s. G β 1,...,β t, x = x t + H γ 1,...,γ n n z, y = y z + w β 1,...,β k,k x t, + n v γ 1,...,γ n ν n,ν y n z. + Using these explicit expressions or the partial Peano kernels, we can see that they represent polygonal lines situated beneath the t-axis, respectively the z-axis, which joins the points, and, 1, respectively the points, and 1,. [ r 1 I we assue that x, r ], we can give or the irst Peano kernel the ollowing expression: k= ν= G β 1,...,β t, x = i1 k= r1 k= k=r k=i w β 1,...,β,k x t k w β 1,...,β,k x t k w β 1,...,β k,k x t w β 1,...,β k,k x t [ i 1 i t, i ] [ ] r 1 i t, x, [ i t x, r ], [ i 1 i t, i ] 1 i r 1, r i. The dual Peano kernel H γ 1,...,γ n n z, y has a siilar expression.

5 Approxiation o Bivariate Functions by Operators Now i we take into account that on the square D, we have G β 1,...,β t, x and H γ 1,...,γ n n z, y, then we can apply the irst law o the ean to the integrals and we can ind that R β 1,...,β ;γ 1,...,γ n,n x, y =, ξ, y +, x, η, ξ, η [ 1 H γ 1,...,γ n n z, ydz G β 1,...,β n t, xdt ][ 1 G β 1,...,β t, xdt where ξ and η are certain points ro the interval, 1. and It is easy to see that we have G β 1,...,β t, xdt = 1 R β 1,...,β e, x H γ 1,...,γ n n z, ydz = 1 R γ 1,...,γ n n e, y, where we have considered the univariate reainders ] H γ 1,...,γ n n z, ydz, R β 1,...,β = I S β 1,...,β, R γ 1,...,γ n n = I S γ 1,...,γ n n. Now we can state the ollowing result: Theore 3.. I C, D, then the reainder o the approxiation orula 3.1 can be represented under the ollowing Cauchy or: 3.3 R β 1,...,β ;γ 1,...,γ n,n x, y = 1 R β 1,...,β e, x, ξ, y R γ 1,...,γ n n e, y, x, η R β 1,...,β e, x R γ 1,...,γ n n e, y, ξ, η.

6 38 I. Taşcu Because S β 1,...,β x and S γ 1,...,γ n n y are interpolatory at both sides o the interval [, 1], we can conclude that R β 1,...,β e, x contains the actor xx 1, while Since R γ 1,...,γ n n e, y has the actor yy 1. R β 1,...,β ;γ 1,...,γ n,n e, x, y = and the reainder is dierent ro zero or any convex unction o the irst order, we can apply a criterion o T. Popoviciu [7] and we ind that the reainder o the approxiation orula 3.1 is o siple or. Consequently, we can state: Theore 3.3. I the second-order divided dierences o the unction CD are bounded on the square D, then we can give an expression o the reainder o the orula 3.1 in ters o divided dierences under the ollowing or 3.4 R β 1,...,β ;γ 1,...,γ n,n x, y = R β 1,...,β e, x [x,1, x,, x,3 ; t, y] +R γ 1,...,γ n n e, y [y n,1, y n,, y n,3 ; x, z] [ ] e, xr γ 1,...,γ n x,1, x n e, y,, x,3 ; t, z, y n,1, y n,, y n,3 R β 1,...,β where x,1, x,, x,3, respectively y n,1, y n,, y n,3 are certain points in the interval [, 1]. Now i we consider that C, D, then we can apply the ean value theores to the divided dierences and we arrive at the expression 3.3 or the reainder o approxiation orula 3.1. Finally, we ention that orulas 3., 3.3 and 3.4 can be extended to unctions ore than two variables without any diiculty. R E F E R E N C E S 1. E.W. Cheney and A. Shara: On a generalization o Bernstein polynoials. Rev. Mat. Univ. Para , I. Horova and M. Budikova: A note on D. D. Stancu operators. Ricerche di Mate ,

7 Approxiation o Bivariate Functions by Operators A. Hurwitz: Über Abel s Vereingeeinerung der Binoischen Forel. Acta Math. 6 19, L.W. Jensen: Sur une identité d Abel et sur d autres orules analogues. Acta Math. 6 19, W.E. Milne: The reainder in linear ethods o approxiation. J. Res. Mat. Bur. Standards , G. Peano: Resto nelle orule di quadratura expresso con un integrale deinito. Atti Acad. Naz. Lincei Rend. 1913, T. Popoviciu: Sur le reste dans certaines orules linéaires d approxiation de l analyse. Matheatica, Cluj , D.D. Stancu: Evaluation o the reainder ter in approxiation orulas by Bernstein polynoials. Math. Cop , D.D. Stancu: The reainder o certain linear approxiation orulas in two variables. J. SIAM Nuer. Anal. B, , D.D. Stancu: Methods or construction o linear positive operators o approxiation. In: Nuerical Analysis and Approxiation Theory, Proc. o the International Syposiu R. T. Trîbiţaş, ed., Cluj University Press,, pp D.D. Stancu: Use o an identity o A. Hurwitz or construction o a linear positive operator o approxiation. Rev. Anal. Nuér. Théor. Approx. 31, D.D. Stancu and C. Cişaşiu: On an approxiating linear positive operator o Cheney-Shara. Rev. Anal. Nuér. Théor. Approx , 1 7. North University o Baia Mare Departent o Matheatics and Coputer Science Victoriei 76, 431 Baia Mare, Roania itascu@yahoo.co