Document Zbl 1338.41013

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Math. Comput.</a> <a href="/?q=in%3A356357" title="Articles in this Issue">248, 342-353 (2014)</a>. </div> <div class="abstract">Summary: This paper deals with two families of Bernstein-Durrmeyer type operators that arise as integral modifications respectively from the classical Bernstein operators and from Bernstein-type operators based on the Polya distribution. An important role is played by the so-called genuine elements of each class, the only ones that reproduce linear functions. Asymptotic formulae and direct convergence results are stated. Finally some modifications of the sequences of the families are introduced, in such a way that the resulting operators reproduce linear functions and are comparable with the genuine sequences.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6595634">Cited in <strong>14</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A41A35" title="MSC2020">41A35</a> </td> <td class="space"> Approximation by operators (in particular, by integral operators) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ABernstein+polynomials">Bernstein polynomials</a>; <a href="/?q=ut%3ABernstein-Durrmeyer+type+operators">Bernstein-Durrmeyer type operators</a>; <a href="/?q=ut%3Aasymptotic+formula">asymptotic formula</a>; <a href="/?q=ut%3Amodulus+of+continuity">modulus of continuity</a>; <a href="/?q=ut%3Abounded+variation">bounded variation</a>; <a href="/?q=ut%3Apolya+distribution">polya distribution</a></div>  <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">×</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;">  </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">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1338.41013" data-ciurl="/ci/06595634" data-biburl="/bibtex/06595634.bib" data-amsurl="/amsrefs/06595634.bib" data-xmlurl="/xml/06595634.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06595634.pdf" title="Zbl 1338.41013 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/10.1016/j.amc.2014.09.094" aria-label="DOI for “Two families of Bernstein-Durrmeyer type operators”" title="10.1016/j.amc.2014.09.094">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">Abel, U.; Gupta, V.; Mohapatra, R. N., Local approximation by a variant of Bernstein-Durrmeyer operators, Nonlinear Anal., 68, 3372-3381 (2008) · <a href="/1175.41020" class="nowrap">Zbl 1175.41020</a></td> </tr><tr> <td>[2]</td> <td class="space">Cárdenas-Morales, D.; Garrancho, P.; Raşa, I., Asymptotic formulae via a Korovkin-type result, Abstr. Appl. Anal., 2012, 12 (2012), Article ID 217464 · <a href="/1241.41012" class="nowrap">Zbl 1241.41012</a></td> </tr><tr> <td>[3]</td> <td class="space">Cárdenas-Morales, D.; Garrancho, P.; Raşa, I., Approximation properties of Bernstein-Durrmeyer type operators, Appl. Math. Comput., 232, 1-8 (2014) · <a href="/1410.41030" class="nowrap">Zbl 1410.41030</a></td> </tr><tr> <td>[5]</td> <td class="space">Derrienic, M. M., Sur l’approximation des fonctions integrables sur $[0, 1]$ par des polynomes de Bernstein modifies, J. Approx. Theory, 31, 325-343 (1981) · <a href="/0475.41025" class="nowrap">Zbl 0475.41025</a></td> </tr><tr> <td>[6]</td> <td class="space">DeVore, R. A.; Lorentz, G. G., Constructive Approximation. Constructive Approximation, Grundlehren der Mathematischen Wissenschaften, Band 303 (1993), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York and London · <a href="/0797.41016" class="nowrap">Zbl 0797.41016</a></td> </tr><tr> <td>[8]</td> <td class="space">Gonska, H.; Kacsó, D.; Raşa, I., On genuine Bernstein-Durrmeyer operators, Results Math., 50, 213-225 (2007) · <a href="/1145.41004" class="nowrap">Zbl 1145.41004</a></td> </tr><tr> <td>[10]</td> <td class="space">Guo, S., On the rate of convergence of the Durrmeyer operator for functions of bounded variation, J. Approx. Theory, 51, 183-192 (1987) · <a href="/0666.41017" class="nowrap">Zbl 0666.41017</a></td> </tr><tr> <td>[11]</td> <td class="space">Gupta, V., A note on the rate of convergence of Durrmeyer type operators for functions of bounded variation, Soochow J. 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U.S.A., 60, 1196-1200 (1968) · <a href="/0164.07102" class="nowrap">Zbl 0164.07102</a></td> </tr><tr> <td>[25]</td> <td class="space">Stancu, D. D., Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pure. Appl., 13, 1173-1194 (1968) · <a href="/0167.05001" class="nowrap">Zbl 0167.05001</a></td> </tr><tr> <td>[26]</td> <td class="space">Waldron, S., A generalised beta integral and the limit of the Bernstein-Durrmeyer operator with Jacobi weights, J. Approx. Theory, 122, 141-150 (2003) · <a href="/1024.41014" class="nowrap">Zbl 1024.41014</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 Open. 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