On infinite products of positive linear operators reproducing linear functions. (English) Zbl 1271.41006

Gonska, Heiner; Raşa, Ioan

Positivity 17, No. 1, 67-79 (2013).

Summary: Infinite products of positive linear operators reproducing linear functions are considered from a quantitative point of view. Refining and generalizing convergence theorems of Gwóźdź-Łukawska, Jachymski, Gavrea, Ivan and the present authors, it is shown that infinite products of certain positive linear operators, all taken from a finite set of mappings reproducing linear functions, weakly converge to the first Bernstein operator. A discussion of products of Meyer-König and Zeller operators is included.

MSC:

41A17	Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A25	Rate of convergence, degree of approximation
41A36	Approximation by positive operators

Keywords:

infinite operator products; positive linear operators; degree of approximation; second-order modulus of smoothness; Bernstein operators; Meyer-König and Zeller operators

References:

[1]	Becker M., Nessel R.J.: A global approximation theorem for Meyer-König and Zeller operators. Math. Z. 160, 195–206 (1978) · Zbl 0376.41007
[2]	Gavrea I., Ivan M.: On the iterates of positive linear operators preserving the affine functions. J. Math. Anal. Appl. 372(2), 366–368 (2010) · Zbl 1196.41014
[3]	Gavrea, I., Ivan, M.: Asymptotic behaviour of the iterates of positive linear operators. Abstr. Appl. Anal. 2011, (2011). doi: 10.1155/2011/670509 · Zbl 1223.41015
[4]	Gavrea I., Ivan M.: On the iterates of positive linear operators. J. Approx. Theory 163, 1076–1079 (2011) · Zbl 1236.41002
[5]	Gonska, H.: On Mamedov estimates for the approximation of finitely defined operators. Approximation theory, III. In: Proceedings of Conference. University of Texas, Austin, pp. 443–448. Academic Press, New York (1980) · Zbl 0484.41025
[6]	Gonska, H.: On the composition and decomposition of positive linear operators. In: Kovtunets, V.V., et al. (eds.) Approximation Theory and its Applications. Proceedings of the International Conference Dedicated to the Memory of Vladislav Kirillovich Dzyadyk held in Kiev, Ukraine, May 27–31, 1999. Kyïv: Instytut Matematyky NAN Ukraïny. Pr. Inst. Mat. Nats. Akad. Nauk Ukr., Mat. Zastos., vol. 31, pp. 161–180 (2000)
[7]	Gonska H., Kacsó D., Piţul P.: The degree of convergence of over-iterated positive linear operators. J. Appl. Funct. Anal. 1(4), 403–423 (2006) · Zbl 1099.41011
[8]	Shunsheng G., Qiulan Q.: The moments for the Meyer-König and Zeller operators. Appl. Math. Lett. 20(7), 719–722 (2007) · Zbl 1132.41324
[9]	Gwóźdź-Łukawska G., Jachymski J.: IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem. J. Math. Anal. Appl. 356(2), 453–463 (2009) · Zbl 1171.28002
[10]	Karlin S., Ziegler Z.: Iteration of positive approximation operators. J. Approx. Theory 3, 310–339 (1970) · Zbl 0199.44702
[11]	Kelisky R.P., Rivlin T.J.: Iterates of Bernstein polynomials. Pacific J. Math. 21, 511–520 (1967) · Zbl 0177.31302
[12]	Raşa I.: Asymptotic behaviour of iterates of positive linear operators. Jaen J. Approx. 1(2), 195–204 (2009)
[13]	Raşa, I.: C 0-semigroups and iterates of positive linear operators: asymptotic behaviour. Rend. Circ. Mat. Palermo 2(Suppl 82), 123–142 (2010)