Abstract
The starting point of this paper is the construction of a general family \( (L_{n})_{n\ge 1}\) of positive linear operators of discrete type. Considering \((L_{n}^{k})_{k\ge 1}\) 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 \(\Vert L_{n}^{k}f-f^{*}\Vert \), \(f\in C[0,1]\), where \(f^{*} \) 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.
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Octavian Agratini constructed the class of positive operators and investigated their approximation properties. Also, three special cases have been highlighted. Radu Precup estimated the approximation error using Banach’s and Perov’s theorems in one and multi-dimensional cases. Both authors investigated contraction property of the class of operators and were involved in the technical editing of the manuscript. Also, all authors reviewed the manuscript.
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Agratini, O., Precup, R. Estimates related to the iterates of positive linear operators and their multidimensional analogues. Positivity 28, 27 (2024). https://doi.org/10.1007/s11117-024-01045-4
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DOI: https://doi.org/10.1007/s11117-024-01045-4
Keywords
- Positive linear operator
- Bernstein operator
- Stancu operator
- Cheney–Sharma operator
- Banach fixed point theorem
- Perov fixed point theorem