On the iterates of positive linear operators. (English) Zbl 1236.41002

Gavrea, Ioan; Ivan, Mircea

J. Approx. Theory 163, No. 9, 1076-1079 (2011).

Let \(U\) be a positive linear operator on \(C[0,1]\) that leaves the linear functions, \(P^1\), invariant. The paper presents a simple short proof of the following:
Theorem: If there is a continuous function \(f\) such that \(Uf-f\) has no zeros in \((1,0)\) then \(U^k g\) converges to the projection onto \(P^1\) that interpolates to \(g\) at \(0\) and \(1\).
The paper presents the theorem for more general domains for the operator than we stated above and applies to subspaces of bounded functions.

Reviewer: Daniel Wulbert (La Jolla)

MSC:

41A05	Interpolation in approximation theory
41A36	Approximation by positive operators

Keywords:

Lagrange interpolation

References:

[1]	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
[2]	Agratini, O., On the iterates of a class of summation-type linear positive operators, Comput. Math. Appl., 55, 1178-1180 (2008) · Zbl 1151.41013
[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]	Badea, C., Bernstein polynomials and operator theory, Results Math., 53, 229-236 (2009) · Zbl 1198.47028
[5]	Cooper, S.; Waldron, S., The eigenstructure of the Bernstein operator, J. Approx. Theory, 105, 133-165 (2000) · Zbl 0963.41006
[6]	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
[7]	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
[8]	Gonska, H.; Piţul, P.; Raşa, I., Over-iterates of Bernstein-Stancu operators, Calcolo, 44, 117-125 (2007) · Zbl 1150.41013
[9]	Gonska, H.; Raşa, I., The limiting semigroup of the Bernstein iterates: degree of convergence, Acta Math. Hungar., 111, 119-130 (2006) · Zbl 1121.41004
[10]	Jessen, B., Bemærkninger om konvekse Funktioner og Uligheder imellem Middelværdier. I., Mat. Tidsskrift B, 17-28 (1931) · JFM 57.0310.01
[11]	Karlin, S.; Ziegler, Z., Iteration of positive approximation operators, J. Approx. Theory, 3, 310-339 (1970) · Zbl 0199.44702
[12]	Kelisky, R. P.; Rivlin, T. J., Iterates of Bernstein polynomials, Pacific J. Math., 21, 511-520 (1967) · Zbl 0177.31302
[13]	Oruç, H.; Tuncer, N., On the convergence and iterates of \(q\)-Bernstein polynomials, J. Approx. Theory, 117, 301-313 (2002) · Zbl 1015.33012
[14]	Ostrovska, S., \(q\)-Bernstein polynomials and their iterates, J. Approx. Theory, 123, 232-255 (2003) · Zbl 1093.41013
[15]	Rasa, I., Asymptotic behaviour of certain semigroups generated by differential operators, Jaen J. Approx., 1, 27-36 (2009) · Zbl 1181.47047
[16]	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
[17]	Rus, I. A., Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl., 292, 259-261 (2004) · Zbl 1056.41004
[18]	Šaškin, J. A., Korovkin systems in spaces of continuous functions, Izv. Akad. Nauk SSSR Ser. Mat., 26, 495-512 (1962)
[19]	Wenz, H. J., On the limits of (linear combinations of) iterates of linear operators, J. Approx. Theory, 89, 219-237 (1997) · Zbl 0871.41014