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(2024)</a>. </div> <div class="abstract"><div class="pre">The starting point of this paper is the construction of a general family of positive linear operators of discrete type. Considering the sequence of iterates of one of such operators, \(L_n\), the goal to find an expression of the upper edge of the error is achieved. The estimate makes use of the error formula for the sequence of successive approximations in Banach&rsquo;s fixed point theorem and the error of approximation of the operator \(L_n\). Some extensions to multidimensional approximation operators are also given.</div><div class="reviewer"> Reviewer:&nbsp;<a href="/authors/?q=rv%3A10874">Vijay Gupta (New Delhi)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A41A36" title="MSC2020">41A36</a> </td> <td class="space"> Approximation by positive operators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A41A30" title="MSC2020">41A30</a> </td> <td class="space"> Approximation by other special function classes </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Apositive+linear+operator">positive linear operator</a>; <a href="/?q=ut%3ABernstein+operator">Bernstein operator</a>; <a href="/?q=ut%3AStancu+operator">Stancu operator</a>; <a href="/?q=ut%3ACheney-Sharma+operator">Cheney-Sharma operator</a>; <a href="/?q=ut%3ABanach+fixed+point+theorem">Banach fixed point theorem</a>; <a href="/?q=ut%3APerov+fixed+point+theorem">Perov fixed point theorem</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">&times;</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1535.41023" data-ciurl="/ci/07829526" data-biburl="/bibtex/07829526.bib" data-amsurl="/amsrefs/07829526.bib" data-xmlurl="/xml/07829526.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07829526.pdf" title="Zbl 1535.41023 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s11117-024-01045-4" aria-label="DOI for “Estimates related to the iterates of positive linear operators and their multidimensional analogues”" title="10.1007/s11117-024-01045-4">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Agratini, O., Precup, R., Iterates of multidimensional operators via Perov theorem, Carpathian J. Math., 38(2022), No. 3, 539-546 &middot; <a href="/1538.41041" class="nowrap">Zbl 1538.41041</a></td> </tr><tr> <td>[2]</td> <td class="space">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 &middot; <a href="/1096.41015" class="nowrap">Zbl 1096.41015</a></td> </tr><tr> <td>[3]</td> <td class="space">Banach, S., Sur les opérations dans les ensemble abstraits et leur application aux équations intégrales, Fund. Math., 3, 133-181, 1922 &middot; <a href="/48.0201.01" class="nowrap">JFM 48.0201.01</a>&nbsp;&middot; <a href="https://doi.org/10.4064/fm-3-1-133-181" class="nowrap">doi:10.4064/fm-3-1-133-181</a></td> </tr><tr> <td>[4]</td> <td class="space">Cheney, E.W., Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma, 5(1964), No. 2, 77-84 &middot; <a href="/0146.08202" class="nowrap">Zbl 0146.08202</a></td> </tr><tr> <td>[5]</td> <td class="space">Kelinsky, R.P., Rivlin, T.J., Iterates of Bernstein polynomials, Pacific J. Math., 21(1967), No. 3, 511-520 &middot; <a href="/0177.31302" class="nowrap">Zbl 0177.31302</a></td> </tr><tr> <td>[6]</td> <td class="space">Păltănea, R., Optimal constant in approximation by Bernstein operators, J. Comput. Analysis Appl., 5(2003), No. 2, 195-235 &middot; <a href="/1033.41008" class="nowrap">Zbl 1033.41008</a></td> </tr><tr> <td>[7]</td> <td class="space">Precup, R., Methods in Nonlinear Integral Equations, 2002, Dordrecht: Springer, Dordrecht &middot; <a href="/1060.65136" class="nowrap">Zbl 1060.65136</a>&nbsp;&middot; <a href="https://doi.org/10.1007/978-94-015-9986-3" class="nowrap">doi:10.1007/978-94-015-9986-3</a></td> </tr><tr> <td>[8]</td> <td class="space">Rus, IA, Iterates of Bernstein operators via contraction principle, J. Math. Anal. Appl., 292, 259-261, 2004 &middot; <a href="/1056.41004" class="nowrap">Zbl 1056.41004</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.jmaa.2003.11.056" class="nowrap">doi:10.1016/j.jmaa.2003.11.056</a></td> </tr><tr> <td>[9]</td> <td class="space">Shisha, O.; Mond, B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. USA, 60, 1196-1200, 1968 &middot; <a href="/0164.07102" class="nowrap">Zbl 0164.07102</a>&nbsp;&middot; <a href="https://doi.org/10.1073/pnas.60.4.1196" class="nowrap">doi:10.1073/pnas.60.4.1196</a></td> </tr><tr> <td>[10]</td> <td class="space">Sikkema, P.C., Der,: Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen. Numer. Math. 3, 107-116 (1961) &middot; <a href="/0099.04701" class="nowrap">Zbl 0099.04701</a></td> </tr><tr> <td>[11]</td> <td class="space">Stancu, DD, Approximation of functions by a new class of linear polynomial operators, Rev. Roum. Math. Pures Appl., 13, 8, 1173-1194, 1968 &middot; <a href="/0167.05001" class="nowrap">Zbl 0167.05001</a></td> </tr><tr> <td>[12]</td> <td class="space">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 &middot; <a href="/1047.41017" class="nowrap">Zbl 1047.41017</a></td> </tr></table> <div class="reference_disclaimer"> 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. In some cases that data have been complemented/enhanced by data from zbMATH&nbsp;Open. 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