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The iterates of positive linear operators preserving constants. (English) Zbl 1232.41030

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.

MSC:

41A36 Approximation by positive operators
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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
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