{"title":"Better degree of approximation by modified Bernstein-Durrmeyer type operators","authors":"P. Agrawal, S. Güngör, Abhishek Kumar","doi":"10.3934/mfc.2021024","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function <inline-formula><tex-math id=\"M1\">\\begin{document}$ \\tau(x), $\\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\tau $\\end{document}</tex-math></inline-formula> is infinitely differentiable function on <inline-formula><tex-math id=\"M3\">\\begin{document}$ [0, 1], \\; \\tau(0) = 0, \\tau(1) = 1 $\\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\tau^{\\prime }(x)>0, \\;\\forall\\;\\; x\\in[0, 1]. $\\end{document}</tex-math></inline-formula> We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function <inline-formula><tex-math id=\"M5\">\\begin{document}$ \\tau(x) $\\end{document}</tex-math></inline-formula> leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [<xref ref-type=\"bibr\" rid=\"b11\">11</xref>].</p>","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical foundations of computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/mfc.2021024","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 4
Abstract
In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function \begin{document}$ \tau(x), $\end{document} where \begin{document}$ \tau $\end{document} is infinitely differentiable function on \begin{document}$ [0, 1], \; \tau(0) = 0, \tau(1) = 1 $\end{document} and \begin{document}$ \tau^{\prime }(x)>0, \;\forall\;\; x\in[0, 1]. $\end{document} We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function \begin{document}$ \tau(x) $\end{document} leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [11].
In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function \begin{document}$ \tau(x), $\end{document} where \begin{document}$ \tau $\end{document} is infinitely differentiable function on \begin{document}$ [0, 1], \; \tau(0) = 0, \tau(1) = 1 $\end{document} and \begin{document}$ \tau^{\prime }(x)>0, \;\forall\;\; x\in[0, 1]. $\end{document} We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function \begin{document}$ \tau(x) $\end{document} leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [11].