Abstract
The starting point of this paper is the construction of a general family \(\left( L_{n}\right) _{n\geq 1}\) of positive linear operators of discrete type. Considering \(L_{n\left( k\geq 1\right) }^{k}\) the sequence of iterates of one of such operators,\(L_{n}\) our goal is to find an expression of the upper edge of the error \(\left \Vert L_{n}^{k}f-f^{\ast }\right \Vert ,f\in\left[ 0,1\right]\),where \(f^{\ast }\), is the fixed point of \(L_{n}\).The estimate makes use of the error formula for the sequence of successive approximations in Banach’s fixed point theorem and the error of approximation of the operator \(L_{n}\).Examples of special operators are inserted. Some extensions to multidimensional approximation operators are also given.
Authors
Octavian Agratini
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania
Radu Precup
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, Cluj-Napoca, 400084, Romania
Keywords
Positive linear operator; Bernstein operator; Stancu operator; Cheney–Sharma operator; Banach fixed point theorem; Perov fixed point theorem
Paper coordinates
O. Agratin, R. Precup, Estimates related to the iterates of positive linear operators and their multidimensional analogues, Positivity, 28 (2024) art. no. 27,
About this paper
Print ISSN
Online ISSN
google scholar link
[1] Agratini, O., Precup, R., Iterates of multidimensional operators via Perov theorem, Carpathian J. Math., 38(2022), No. 3, 539-546
[2] Agratini, O., Rus, I.A., Iterates of a class of discrete operators via contraction principle, Comment. Math. Univ. Carolinae, 44(2003), No. 3, 555-563
[3] Banach, S.: Sur les opérations dans les ensemble abstraits et leur application aux équations intégrales. Fund. Math. 3, 133–181 (1922) Article MathSciNet Google Scholar
[4] Cheney, E.W., Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma, 5(1964), No. 2, 77-84
[5] Kelinsky, R.P., Rivlin, T.J., Iterates of Bernstein polynomials, Pacific J. Math., 21(1967), No. 3, 511-520
[6] Păltănea, R., Optimal constant in approximation by Bernstein operators, J. Comput. Analysis Appl., 5(2003), No. 2, 195-235
[7] Precup, R.: Methods in Nonlinear Integral Equations. Springer, Dordrecht (2002) Book Google Scholar
[8] Rus, I.A.: Iterates of Bernstein operators via contraction principle. J. Math. Anal. Appl. 292, 259–261 (2004) Article MathSciNet Google Scholar
[9] Shisha, O., Mond, B.: The degree of convergence of linear positive operators. Proc. Nat. Acad. Sci. USA 60, 1196–1200 (1968) Article MathSciNet Google Scholar
[10] Sikkema, P.C., Der,: Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen. Numer. Math. 3, 107–116 (1961)
[11] Stancu, D.D.: Approximation of functions by a new class of linear polynomial operators. Rev. Roum. Math. Pures Appl. 13(8), 1173–1194 (1968) MathSciNet Google Scholar
[12] Stancu, D.D., Cismaşiu, C., On an approximating linear positive operator of Cheney-Sharma, Rev. Anal. Numér. Théor. Approx., 26(1997), Nos. 1-2, 221-227