Sharp inequalities for the rates of convergence of the iterates of some operators which preserve the constants

Marius-Mihai Birou

Abstract


In this paper we give estimates for the rates of convergence
for the iterates of some positive linear operators which preserve only the
constants. We obtain sharp inequalities when we use both continuous
functions and differentiable functions. We present some optimal results
for the Cesaro, Stancu and Schurer operators.


Keywords


positive linear operators, iterates, convergence, sharp in- equalities

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References


Abel U., Ivan M., New representation of the remainder in the Bernstein approximation, J. Math. Anal. Appl., 381 (2011), 952-956.

Altomare F., Campiti M., Korovkin-Type Approximation Theory and Its Applications, vol. 17 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1994.

Galaz Fontes F., Solis F.J., Iterating the Cesaro operators, Proceedings of the AmericanMathematical Society, 136(2008), no. 6, 2147-2153.

Gavrea I., Ivan M., The iterates of positive linear operators preserving constants, Appl. Math. Lett., 24 (2011), 2068-2071.

Gavrea I., Ivan M., Asymptotic Behaviour of the Iterates of Positive Linear Operators, Hindawi Publishing Corporation, Abstr. Appl. Anal., Volume 2011, Article ID 670509, 11 pages.

Gonska H., Pitul P., Rasa I., Over-iterates of Bernstein-Stancu operators, Calcolo, 44(2007), no. 2, 117-125.

Kelisky R.P., Rivlin T.J., Iterates of Bernstein polynomials, Paci¯c Journal of Mathematics, 21(1967), 511-520.

Karlin S., Ziegler Z., Iteration of positive approximation operators, J. Approx. Theory, 3(1970), 310-339.

Mahmudov N.I.: Asymptotic properties of iterates of certain positive linear operators. Math. Comp. Model., 57(2013), 1480-1488.

P¸al¸tanea R., Optimal estimates with modulli of continuity, Results Math., 32(1997), 318-331.

Rus I.A., Iterates of Stancu operators, via contraction principle, Studia Univ. Babe»s-Bolyai Math., 47(2002), no. 4, 101-104.

Rus I.A., Iterates of Stancu operators (via ¯xed point principles) revisited, Fixed Point Theory, 11(2010), no. 2, 369-374.

Stancu D.D., Asupra unei generaliz¸ari a polinoamelor lui Bernstein, Studia Univ. Babe»s-Bolyai Math., 14(1969), no. 2, 31-45.




DOI: http://dx.doi.org/10.24193/subbmath.2021.2.05

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