Document Zbl 1499.41070

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H. Ismail Mourad</span> and <span class="zbmathjax-textit">C. P. May</span>, J. Math. Anal. Appl. 63, 446–462 (1978; <a href="/0375.41011">Zbl 0375.41011</a>)] based on a non-negative real parameter $\lambda $. We provide some graphs and error estimation table for a numerical example depicting the convergence of our proposed operators. We further define the Bézier variant of these operators and give a direct approximation theorem using Ditizan-Totik modulus of smoothness and a Voronovoskaya type asymptotic theorem. We also study the error in approximation of functions having derivatives of bounded variation. Lastly, we introduce the bivariate generalization of Ismail May operators and estimate its rate of convergence for functions of Lipschitz class.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7541470">Cited in <strong>4</strong> Documents</a></div></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%3A41A25" title="MSC2020">41A25</a> </td> <td class="space"> Rate of convergence, degree of approximation </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Arate+of+convergence">rate of convergence</a>; <a href="/?q=ut%3AVoronovoskaya+theorem">Voronovoskaya theorem</a>; <a href="/?q=ut%3APeetre%27s+%5C%28K%5C%29-functional">Peetre’s $K$-functional</a>; <a href="/?q=ut%3Abounded+variation">bounded variation</a>; <a href="/?q=ut%3Apartial+and+mixed+modulus+of+smoothness">partial and mixed modulus of smoothness</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0375.41011">Zbl 0375.41011</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 1499.41070" data-ciurl="/ci/07541470" data-biburl="/bibtex/07541470.bib" data-amsurl="/amsrefs/07541470.bib" data-xmlurl="/xml/07541470.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07541470.pdf" title="Zbl 1499.41070 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/10.2298/FIL2012043L" aria-label="DOI for “General family of exponential operators”" title="10.2298/FIL2012043L">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">O. Agratini, On a class of linear positive bivariate operators of King type, Studia Universitatis Babes¸-Bolyai Mathematica 51(4) (2006) 13-22. · <a href="/1164.41337" class="nowrap">Zbl 1164.41337</a></td> </tr><tr> <td>[2]</td> <td class="space">P. N. Agrawal, N. Ispir, Degree of approximation for bivariate Chlodowsky-Sz´asz-Charlier type operators, Results in Mathematics 69 (2016) 369-385. · <a href="/1339.41029" class="nowrap">Zbl 1339.41029</a></td> </tr><tr> <td>[3]</td> <td class="space">P. N. Agrawal, N. Ispir, A. Kajla, Approximation properties of B´ezier-summation-integral type operators based on Polya-Bernstein functions, Applied Mathematics and Computation 259 (2015) 533-539. · <a href="/1390.41018" class="nowrap">Zbl 1390.41018</a></td> </tr><tr> <td>[4]</td> <td class="space">N. Deo, N. Bhardwaj, Some approximation theorems for multivariate Bernstein operators, Southeast Asian Bulletin of Mathematics 34 (2010) 1023-1034. · <a href="/1240.41062" class="nowrap">Zbl 1240.41062</a></td> </tr><tr> <td>[5]</td> <td class="space">N. Deo, M. Dhamija, D. Miclˇaus¸, New modified Baskakov operators based on the inverse P ´olya-Eggenberger distribution, Filomat 33(11) (2019) 3537-3550. · <a href="/1499.41055" class="nowrap">Zbl 1499.41055</a></td> </tr><tr> <td>[6]</td> <td class="space">S. Deshwal, A. M. Acu, P. N. Agrawal, Pointwise approximation by B´ezier Variant of an operator based on Laguerre polynomials, Journal of Mathematical Inequalities 12(3) (2018) 693-707 · <a href="/1403.41007" class="nowrap">Zbl 1403.41007</a></td> </tr><tr> <td>[7]</td> <td class="space">R. A. DeVore, G. G. Lorentz, Constructive Approximation, Springer, Berlin, 1993. · <a href="/0797.41016" class="nowrap">Zbl 0797.41016</a></td> </tr><tr> <td>[8]</td> <td class="space">M. Dhamija, N. Deo, Approximation by generalized positive linear Kantorovich operators, Filomat 31(14) (2017), 4353-4368. · <a href="/1365.41012" class="nowrap">Zbl 1365.41012</a></td> </tr><tr> <td>[9]</td> <td class="space">M. Dhamija, Durrmeyer modification of Lupas type Baskakov operators based on IPED, Mathematical Analysis I: Approximation Theory, Springer Proceedings in Mathematics & Statistics (eds. Deo et al.) 306 (2020) 111-120.</td> </tr><tr> <td>[10]</td> <td class="space">Z. Ditzian, V. Totik, Moduli of Smoothness, Springer, New York, 1987. · <a href="/0666.41001" class="nowrap">Zbl 0666.41001</a></td> </tr><tr> <td>[11]</td> <td class="space">Z. Finta, Remark on Voronovskaja theorem for q-Bernstein operators, Studia Universitatis Babes¸-Bolyai Mathematica 56 (2011) 335-339. · <a href="/1249.41013" class="nowrap">Zbl 1249.41013</a></td> </tr><tr> <td>[12]</td> <td class="space">V. Gupta, N. Deo, On the rate of convergence for bivariate Beta operators, General Mathematics 13(3) (2005) 107-114. · <a href="/1109.41012" class="nowrap">Zbl 1109.41012</a></td> </tr><tr> <td>[13]</td> <td class="space">V. Gupta, N. Ispir, On the B´ezier variant of generalized Kantorovich type Bal´azs operators, Applied Mathematics Letters 18 (2005) 1052-1061. · <a href="/1088.41012" class="nowrap">Zbl 1088.41012</a></td> </tr><tr> <td>[14]</td> <td class="space">V. Gupta, H. M. Srivastava, A general family of the Srivastava-Gupta operators preserving linear functions, European Journal of Pure and Applied Mathematics 11(3) (2018) 575-579. · <a href="/1413.41012" class="nowrap">Zbl 1413.41012</a></td> </tr><tr> <td>[15]</td> <td class="space">V. Gupta, X. M. Zeng, Approximation by B´ezier variant of the Sz´asz−Kantorovich operator in case 0<a<1, Georgian Mathematical Journal 17 (2010) 253-260. · <a href="/1195.41026" class="nowrap">Zbl 1195.41026</a></td> </tr><tr> <td>[16]</td> <td class="space">E. H. Ismail Mourad, C. P. May, On a family of approximation operators, Journal of Mathematical Analysis and Applications 63 (1978) 446-462. · <a href="/0375.41011" class="nowrap">Zbl 0375.41011</a></td> </tr><tr> <td>[17]</td> <td class="space">N. Ispir, I. B ¨uy ¨ukyazici, Quantitative estimates for a certain bivariate Chlodowsky-Sz´asz-Kantorovich type operators, Mathematical Communications 21 (2016) 31-44. · <a href="/1342.41028" class="nowrap">Zbl 1342.41028</a></td> </tr><tr> <td>[18]</td> <td class="space">N. Ispir, I. Yuksel, On the B´ezier variant of Srivastava-Gupta operators, Applied Mathematics E-Notes 5 (2005) 129-137. · <a href="/1084.41018" class="nowrap">Zbl 1084.41018</a></td> </tr><tr> <td>[19]</td> <td class="space">H. Karsli, M. A. ¨Ozarslan, Direct local and global approximation results for the operators of gamma type, Hacettepe Journal of Mathematics and Statistics 39(2) (2010) 241-253. · <a href="/1203.41009" class="nowrap">Zbl 1203.41009</a></td> </tr><tr> <td>[20]</td> <td class="space">B. Lenze, On Lipschitz-type maximal functions and their smoothness spaces, Nederl Akad Wetensch Indag Math 50(1) (1988) 53-63. · <a href="/0652.42004" class="nowrap">Zbl 0652.42004</a></td> </tr><tr> <td>[21]</td> <td class="space">C. P. May, Saturation and inverse theorems for combinations of a class of exponential type operators, Canadian Journal of Mathematics 28 (1976) 1224-1250. · <a href="/0342.41018" class="nowrap">Zbl 0342.41018</a></td> </tr><tr> <td>[22]</td> <td class="space">N. S. Mishra, N. Deo, Kantorovich variant of Ismail-May operators, Iranian Journal of Science and Technology. Transaction A. Science (2020) In-press.</td> </tr><tr> <td>[23]</td> <td class="space">E. ¨Ozergin, M. A. ¨Ozarslan, H. M. Srivastava, Some families of generating functions for a class of bivariate polynomials, Mathematical and Computer Modelling (50) (2009) 1113-1120. · <a href="/1185.33014" class="nowrap">Zbl 1185.33014</a></td> </tr><tr> <td>[24]</td> <td class="space">G. P ´olya, G. Szeg ¨o, Problems and Theorems in Analysis, Vol.I (English translation), Springer-Verlag, New York, 1972. · <a href="/0236.00003" class="nowrap">Zbl 0236.00003</a></td> </tr><tr> <td>[25]</td> <td class="space">R. Pratap, N. Deo, Rate of convergence of Gupta-Srivastava operators based on certain parameters, Journal of Classical Analysis 14(2) (2019) 137-153. · <a href="/1449.41025" class="nowrap">Zbl 1449.41025</a></td> </tr><tr> <td>[26]</td> <td class="space">P. Sharma, V. N. Mishra, Bivariate generalization ofq-Bernstein-Kantorovich type operator, Cogent Mathematics (3) (2016) Article 1160587. · <a href="/1426.41018" class="nowrap">Zbl 1426.41018</a></td> </tr><tr> <td>[27]</td> <td class="space">H. M. Srivastava, Z. Finta, V. Gupta, Direct results for a certain family of summation-integral type operators, Applied Mathematics and Computation 190 (2007) 449-457. · <a href="/1124.41020" class="nowrap">Zbl 1124.41020</a></td> </tr><tr> <td>[28]</td> <td class="space">H. M. Srivastava, V. Gupta., A certain family of summation-integral type operators, Mathematical and Computer Modelling 37 (2003) 1307-1315. · <a href="/1058.41015" class="nowrap">Zbl 1058.41015</a></td> </tr><tr> <td>[29]</td> <td class="space">H. M. Srivastava, V. Gupta, Rate of convergence for B´ezier variant of the Bleimann-Butzer-Hahn operators, Applied Mathematics Letters 18(8) (2005) 849-857. · <a href="/1084.41010" class="nowrap">Zbl 1084.41010</a></td> </tr><tr> <td>[30]</td> <td class="space">H. M. Srivastava, G. Ic¸ ¨oz, B. C¸ ekim, Approximation properties of an extended family of the Sz´asz-Mirakjan Beta-type operators, Axioms 8(4) (2019) Article 111.</td> </tr><tr> <td>[31]</td> <td class="space">H. M. Srivastava, M. Mursaleen, A. M. Alotaibi, M. Nasiruzzaman, A. A. H. Al-Abied, Some approximation results involving the q-Sz´asz-Mirakjan-Kantorovich type operators via Dunkl’s generalization, Mathematical Methods in the Applied Sciences 40(15) (2017) 5437-5452. · <a href="/1384.41015" class="nowrap">Zbl 1384.41015</a></td> </tr><tr> <td>[32]</td> <td class="space">H. M. Srivastava, F. ¨Ozger, S. A. Mohiuddine, Construction of Stancu-type Bernstein operators based on B´ezier bases with shape parameterλ, Symmetry 11(3) (2019) Article 316. · <a href="/1423.41022" class="nowrap">Zbl 1423.41022</a></td> </tr><tr> <td>[33]</td> <td class="space">H. M. Srivastava, X. M. Zeng, Approximation by means of the Sz´asz-B´ezier integral operators, International Journal of Pure and Applied Mathematics 14 (2004) 283-294. · <a href="/1056.41012" class="nowrap">Zbl 1056.41012</a></td> </tr><tr> <td>[34]</td> <td class="space">X. M. Zeng, A. Piriou, On the rate of convergence of two Bernstein-B´ezier type operators for bounded variation functions, Journal of Approximation Theory 95(3) (1998) 369-387 · <a href="/0918.41016" class="nowrap">Zbl 0918.41016</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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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