Geodesic
On modified Meyer-König and Zeller operators of functions of two variables
Archivum mathematicum, Tome 42 (2006) no. 3, pp. 273-284
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This paper is motivated by Kirov results on generalized Bernstein polynomials given in (Kirov, G. H., A generalization of the Bernstein polynomials, Math. Balk. New Ser. bf 6 (1992), 147–153.). We introduce certain modified Meyer-König and Zeller operators in the space of differentiable functions of two variables and we study approximation properties for them. Some approximation properties of the Meyer-König and Zeller operators of differentiable functions of one variable are given in (Rempulska, L., Tomczak, K., On certain modified Meyer-König and Zeller operators, Grant PB-43-71/2004.) and (Rempulska, L., Skorupka, M., On strong approximation by modified Meyer-König and Zeller operators, Tamkang J. Math. (in print).).
This paper is motivated by Kirov results on generalized Bernstein polynomials given in (Kirov, G. H., A generalization of the Bernstein polynomials, Math. Balk. New Ser. bf 6 (1992), 147–153.). We introduce certain modified Meyer-König and Zeller operators in the space of differentiable functions of two variables and we study approximation properties for them. Some approximation properties of the Meyer-König and Zeller operators of differentiable functions of one variable are given in (Rempulska, L., Tomczak, K., On certain modified Meyer-König and Zeller operators, Grant PB-43-71/2004.) and (Rempulska, L., Skorupka, M., On strong approximation by modified Meyer-König and Zeller operators, Tamkang J. Math. (in print).).
Classification : 41A35, 41A36
Keywords: Meyer-König and Zeller operator; function of two variables; approximation theorem 
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Rempulska, Lucyna; Skorupka, Mariola. On modified Meyer-König and Zeller operators of functions of two variables. Archivum mathematicum, Tome 42 (2006) no. 3, pp. 273-284. http://geodesic.mathdoc.fr/item/ARM_2006_42_3_a9/

[1] Abel U.: The moments for the Meyer-König and Zeller operators. J. Approx. Theory 82 (1995), 352–361. | MR | Zbl

[2] Alkemade J. A. H.: The second moment for the Meyer-König and Zeller operators. J. Approx. Theory 40 (1984), 261–273. | MR | Zbl

[3] Abel U., Della Vecchia B.: Enhanced asymptotic approximation by linear operators. Facta Univ., Ser. Math. Inf. 19 (2004), 37–51.

[4] Becker M., Nessel R. J.: A global approximation theorem for Meyer-König and Zeller operator. Math. Z. 160 (1978), 195–206. | MR

[5] Chen W.: On the integral type Meyer-König and Zeller operators. Approx. Theory Appl. 2(3) (1986), 7–18. | MR | Zbl

[6] De Vore R. A.: The Approximation of Continuous Functions by Positive Linear operators. New York, 1972.

[7] Fichtenholz G. M.: Calculus. Vol. 1, Warsaw, 1964.

[8] Guo S.: On the rate of convergence of integrated Meyer-König and Zeller operators for functions of bounded variation. J. Approx. Theory 56 (1989), 245–255. | MR

[9] Gupta V.: A note on Meyer-König and Zeller operators for functions of bounded variation. Approx. Theory Appl. 18(3) (2002), 99–102. | MR | Zbl

[10] Hölzle G. E.: On the degree of approximation of continuous functions by a class of sequences of linear positive operators. Indag. Math. 42 (1980), 171–181. | MR | Zbl

[11] Kirov G. H.: A generalization of the Bernstein polynomials. Math. Balk. New Ser. bf 6 (1992), 147–153. | MR | Zbl

[12] Kirov G. H., Popova L.: A generalization of the linear positive operators. Math. Balk. New Ser. 7 (1993), 149–162. | MR | Zbl

[13] Lupas A.: Approximation properties of the $M_{n}$-operators. Aequationes Math. 5 (1970), 19–37. | MR

[14] Meyer-König W., Zeller K.: Bernsteinche Potenzreihen. Studia Math. 19 (1960), 89–94. | MR

[15] Rempulska L., Tomczak K.: On certain modified Meyer-König and Zeller operators. Grant PB-43-71/2004. | Zbl

[16] Rempulska L., Skorupka M.: On strong approximation by modified Meyer-König and Zeller operators. Tamkang J. Math. (in print). | MR | Zbl

[17] Timan A. F.: Theory of Approximation of Functions of a Real Variable. Moscow, 1960 (Russian). | MR