Gavrea, Ioan; Ivan, Mircea The iterates of positive linear operators preserving constants. (English) Zbl 1232.41030 Appl. Math. Lett. 24, No. 12, 2068-2071 (2011). Summary: We introduce a simple and efficient technique for studying the asymptotic behavior of the iterates of a large class of positive linear operators preserving constant functions. Cited in 7 Documents MSC: 41A36 Approximation by positive operators Keywords:iterates; linear positive operator; Cesàro mean operator PDF BibTeX XML Cite \textit{I. Gavrea} and \textit{M. Ivan}, Appl. Math. Lett. 24, No. 12, 2068--2071 (2011; Zbl 1232.41030) Full Text: DOI References: [1] Kelisky, R. P.; Rivlin, T. J., Iterates of Bernstein polynomials, Pacific J. Math., 21, 511-520 (1967) · Zbl 0177.31302 [2] Karlin, S.; Ziegler, Z., Iteration of positive approximation operators, J. Approximation Theory, 3, 310-339 (1970) · Zbl 0199.44702 [3] Altomare, F.; Campiti, M., Korovkin-type approximation theory and its applications, (de Gruyter Studies in Mathematics, vol. 17 (1994), Walter de Gruyter & Co.: Walter de Gruyter & Co. Berlin) · Zbl 0924.41001 [4] Gonska, H. H.; Zhou, X. L., Approximation theorems for the iterated Boolean sums of Bernstein operators, J. Comput. Appl. Math., 53, 21-31 (1994) · Zbl 0816.41020 [5] Oruç, H.; Tuncer, N., On the convergence and iterates of \(q\)-Bernstein polynomials, J. Approx. Theory, 117, 301-313 (2002) · Zbl 1015.33012 [6] Ostrovska, S., \(q\)-Bernstein polynomials and their iterates, J. Approx. Theory, 123, 232-255 (2003) · Zbl 1093.41013 [7] King, J. P., Positive linear operators which preserve \(x^2\), Acta Math. Hungar., 99, 203-208 (2003) · Zbl 1027.41028 [8] Rus, I. A., Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl., 292, 259-261 (2004) · Zbl 1056.41004 [9] Itai, U., On the eigenstructure of the Bernstein kernel, Electron. Trans. Numer. Anal., 25, 431-438 (2006) · Zbl 1160.42317 [10] Gonska, H.; Kacsó, D.; Piţul, P., The degree of convergence of over-iterated positive linear operators, J. Appl. Funct. Anal., 1, 403-423 (2006) · Zbl 1099.41011 [11] Gonska, H.; Raşa, I., The limiting semigroup of the Bernstein iterates: degree of convergence, Acta Math. Hungar., 111, 119-130 (2006) · Zbl 1121.41004 [12] Gonska, H.; Piţul, P.; Raşa, I., Over-iterates of Bernstein-Stancu operators, Calcolo, 44, 117-125 (2007) · Zbl 1150.41013 [13] Wenz, H. J., On the limits of (linear combinations of) iterates of linear operators, J. Approx. Theory, 89, 219-237 (1997) · Zbl 0871.41014 [14] Agratini, O., On the iterates of a class of summation-type linear positive operators, Comput. Math. Appl., 55, 1178-1180 (2008) · Zbl 1151.41013 [15] Galaz Fontes, F.; Solís, F. J., Iterating the Cesàro operators, Proc. Am. Math. Soc., 136, 2147-2153 (2008) · Zbl 1146.47019 [16] Abel, U.; Ivan, M., Over-iterates of Bernstein’s operators: a short and elementary proof, Amer. Math. Monthly, 116, 535-538 (2009) · Zbl 1229.41001 [17] Gavrea, I.; Ivan, M., On the iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl., 372, 366-368 (2010) · Zbl 1196.41014 [18] Rasa, I., Asymptotic behaviour of certain semigroups generated by differential operators, Jaen J. Approx., 1, 27-36 (2009) · Zbl 1181.47047 [19] Rasa, I., \(C_0\) semigroups and iterates of positive linear operators: asymptotic behaviour, Rend. Circ. Mat. Palermo (2) Suppl., 82, 123-142 (2010) · Zbl 1470.41023 [20] Stancu, D. D., Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl., 13, 1173-1194 (1968) · Zbl 0167.05001 [21] Cheney, E. W.; Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma (2), 5, 77-84 (1964) · Zbl 0146.08202 [22] Aldaz, J. M.; Kounchev, O.; Render, H., Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces, Numer. Math., 114, 1-25 (2009) · Zbl 1184.41011 [23] Jichang, K., The norm inequalities for the weighted Cesaro mean operators, Comput. Math. Appl., 56, 2588-2595 (2008) · Zbl 1165.42305 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.