ON THE ORDER OF APPROXIMATION OF FUNCTIONS BY THE BIDIMENSIONAL OPERATORS FAVARD-SZÁSZ-MIRAKYAN

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1 An. Şt. Univ. Ovidius Constanţa Vol. 112, 2003, ON THE ORDER OF APPROXIMATION OF FUNCTIONS BY THE BIDIMENSIONAL OPERATORS FAVARD-SZÁSZ-MIRAKYAN Ioana Taşcu and Anca Buie Abstract We will present an approximation result concerning the order o approximation o bivariate unction by means o the bidimensional operator o Favard-Szász-Mirakyan. 1 Introduction Let X be the interval [0, + andr X the space o real unctions deined on X. TheFavard-Szász-Mirakyan operator M m deined on R X is given by M n x =e mx mx k k=0 Consider a bivariate unction deined an I =[0, + [0, +. The parametric extensions or the operators 1.1 are given by x M m x, y =e mx mx k k=0 y M n x, y =e ny ny =0 k. 1.1 m k m,y, 1.2 x,. 1.3 n Since M m is a positive linear operator, it ollows that its parametric extensions x M m, y M n are also positive and linear operators. Key Words: Taylor series, Mirakyan operator, Voronovskaa theorem, parametric extension, modulus o smoothness Mathematical Reviews subect classiication: 41A58, 41A45, 41A63 163

2 164 Ioana Taşcu and Anca Buie Proposition 1.1 The parametric extension x M m, y M n satisy the relation xm m ym n = y M n xm m. Their product is the bidimensional operator M m,n which, or any unction R I, gives the approximant M m,n x, y =e mx e ny mx k ny k m,. 1.4 n k=0 =0 Properties o the bidimensional operator o Mirakyan were studied in [1], [2], [3], [4]. Next, we will present some o them. Theorem 1.1 The bidimensional operator o Favard-Szász-Mirakyan has the ollowing properties: i It is linear and positive. ii M m,n e 0,0 x, y =1; M m,n e 1,0 x, y =x; M m,n e 0,1 x, y =y; M m,n e 2,0 x, y =x 2 + x m ; M m,n e 0,2 x, y =y 2 + y m. iii I a > 0, b > 0 and C [0,a] [0,b], then the sequence {M m,n } m,n N N is uniormly convergent to on [0,a] [0,b]. iv I a>0, b>0 and C [0,a] [0,b], then we have the estimation a b x, y M m,n x, y 4ω m,, n where by ω we denote the irst modulus o smoothness. 2 Main results Lemma 2.1 I δ>0 and a>0, then m me mx mx k 2 k m x =0.

3 ON THE ORDER OF APPROXIMATION OF FUNCTIONS BY THE BIDIMENSIONAL OPERATORS FAVARD-SZÁSZ-MIRAKYAN 165 Proo. By 1.4 we have the next inequality me mx mx k 2 k m x m δ 2 M m,n A simple computation yields M m,n φ 3 x, y = x m 3 +7x2 m 2 +6x3 m + x4, x 4 ; x, y. 2.1 M m,n φ 4 x, y = x m 2 +3x2 m + x3. These equalities and i, ii rom Theorem 1.1. imply: M m,n x 4 ; x, y = 1 3mx 2 m 3 + x. 2.2 By 2.1 and 2.2, we have me mx mx k 2 k m x 3mx2 + x m 2 δ 2. The proo o the lemma is complete. Theorem 2.1 Consider a > 0, b > 0, x, y [0, a] [0, b] and C [0,a] [0,b]. Assume that i the unction has the second partial derivatives; ii the unction has the mixt partial derivatives in x, y. Then min {m, n} [M m,n x, y x, y] x 2 m,n 2 x 2 x, y+y 2 The equality holds when m = n. Proo. Using the corresponding Taylor series, we obtain s, t = x, y+ 1 1! [ s x x, y+t y x, y x y + 1 [ s x 2 2 2! x 2 x, y+s xt y 2 x, y x y + s xt y 2 y x x, y+t 2 ] y2 x, y y2 2 x, y. y2 + s x 2 µ 1 s x+s xt y µ 2 s x, t y + t y 2 µ 3 t y, 2.3 ]

4 166 Ioana Taşcu and Anca Buie where the mappings µ 1,µ 2,µ 3 are bounded and h 0 µ 1 h =0, In 2.3, we choose s = k m, t = n µ 2 h 1,h 2 =0, h 1,h 2 0 µ 3 h =0. h 0 and multiply by mxk ny,weget M m,n x, y x, y = x x, y M m,n x;x, y + y x, y M m,n y;x, y x 2 x, y M m,n x 2 ; x, y x y x, y M m,n x y;x, y y x x, y M m,n x y;x, y y 2 x, y M m,n y 2 ; x, y + R m,n x, y, 2.4 where Rm,n x, y =e mx mx k 2 k k m x µ 1 m x k=0 + e mx e ny mx k ny k km m x n y µ 2 x, n y k=0 =0 + e nx nx 2 n x µ 3 n x. 2.5 =0 Since M m,n x;x, y = 0 M m,n y;x, y = 0 M m,n x y;x, y = 0 M m,n x 2 ; x, y = x m M m,n y 2 ; x, y = y n, the relation 2.4 is equivalent to the ollowing M m,n x, y x, y = x 2 y 2 x, y+ 2m x2 2n y 2 x, y+ R m,n x, y. 2.6

5 ON THE ORDER OF APPROXIMATION OF FUNCTIONS BY THE BIDIMENSIONAL OPERATORS FAVARD-SZÁSZ-MIRAKYAN 167 Now, by multiplying 2.6 by min {m, n}, and crossing to it with respect to m, n we obtain m,n min {m, n} M m,n x, y x, y x 2 2 x 2 x, y+y 2 x, y + 2 y2 m,n min {m, n} R m,n x, y. 2.7 We claim that m,n min {m, n} R m,n x, y =0. Indeed, let ε>0. Since µ 1 h =0,thereexistsδ > 0 such that, or any h 0 h, with h <δ,wehave µ 1 h <ε. From µ 3 k = 0, we obtain that k 0 there exist an δ > 0 such that, or any k with k <δ,wehaveµ 3 k <ε. Considering δ =max{δ,δ }, or every h, k with h <δand k <δ,we have µ 1 h <ε and µ 3 k <ε. Let us use the notations I 1 = { k N; k m x } <δ, I 2 = { k N; k m x } δ, J 1 = { N; n y } <δ, J 2 = { N; n y δ }. Since the maps µ 1,µ 2 and µ 3 are bounded, we can write R m,n x, y e mx mx k 2 x k k m x µ 1 m k I1 +sup µ 1 e mx mx k 2 k m x +sup µ 2 M m,n x y ; x, y k I 2 +e nx J 1 ny 2 n x µ 3 n y +sup µ 3 e nx ny J 2 For k I 1 and J 1,wehave µ 1 k m x <εand µ 3 2 n x n y <ε. 2.8

6 168 Ioana Taşcu and Anca Buie Moreover, M m,n x y ; x, y = 0. Thereore 2.8 becomes R m,n x, y εe mx mx k 2 k m x k I1 +sup µ 1 e mx mx k 2 k m x k I 2 + εe nx J 1 ny 2 n x +sup µ 3 e nx ny J 2 εδ 2 e mx mx k +sup µ 1 e mx mx k k I 1 k I 2 2 n x 2 k m x + εδ 2 e ny I 1 ny +sup µ 3 e ny I 2 ny 2 n y 2εδ 2 +sup µ 1 e mx mx k k m x k I 2 +sup µ 3 e ny ny 2 n y. I 2 Multiplying the previous inequality with min {m, n}, we obtain But min {m, n} R m,n x, y 2εδ 2 min {m, n} +sup µ 1 min{m, n} e mx mx k 2 k m x k I 2 +sup µ 3 min{m, n} e ny ny 2 n y. 2.9 I 2 min {m, n} e mx k I 2 mx k and min {m, n} e ny I 2 ny 2 2 k m x me mx mx k 2 k m x 2 n y ne ny ny 2 n y. n y δ

7 ON THE ORDER OF APPROXIMATION OF FUNCTIONS BY THE BIDIMENSIONAL OPERATORS FAVARD-SZÁSZ-MIRAKYAN 169 Now, by using the Lemma 2.1, we obtain and min {m, n} e mx m,n min {m, n} e ny m,n n y δ mx k ny 2 k m x =0 2 n y =0. So, there exist m 0,n 0 N such that or every m, n N, withm m 0 and n n 0,wehave min {m, n} e mx mx k k m x 1 <ε sup µ 1 and min {m, n} e ny n y δ ny n y 1 <ε sup µ 3. Then, according to 2.13, we can conclude that there exist m 0,n 0 N such that or every m, n N, withm m 0 and n n 0, the next inequality holds min {m, n} R m,n x, y < 2ε δ 2 min {m, n} This is equivalent with Reerences m,n min {m, n} R m,n x, y =0. [1] Agratini, O., Aproximarea prin operatori liniari, Presa Universitară Clueană, Clu-Napoca, [2] Bărbosu, D., Bivariate operators o Mirakyan type, Bulletin or Applied Computer Mathematics, BAM-1794/2000 XCIV, [3] Bărbosu, D., Aproximarea uncţiilor de mai multe variabile prin sume booleene de operatori liniari de tip interpolator, Ed. Risoprint, Clu- Napoca, [4] Ciupa, A., On the remainder in an approximation ormula by Favard-Szasz operators or unctions o two variables, Bul.Şt. Univ. Baia Mare, seria B, Mat-In., vol.x 1994,

8 170 Ioana Taşcu and Anca Buie [5] Stancu, D.D., Coman, Gh., Agratini, O., Trâmbiţaş, R., Analiză numerică şi teoria aproximării, Presa Univ. Clueană, Clu-Napoca, North University o Baia Mare Department o Mathematics and Computer Science Victoriei 78, , Baia Mare, Romania itascu@ubm.ro