{"title":"由$ \\ α - $Baskakov算子的Durrmeyer变体得到更好的近似","authors":"P. Agrawal, J. Singh","doi":"10.3934/mfc.2021040","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>The aim of this paper is to study some approximation properties of the Durrmeyer variant of <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\alpha $\\end{document}</tex-math></inline-formula>-Baskakov operators <inline-formula><tex-math id=\"M3\">\\begin{document}$ M_{n,\\alpha} $\\end{document}</tex-math></inline-formula> proposed by Aral and Erbay [<xref ref-type=\"bibr\" rid=\"b3\">3</xref>]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr<inline-formula><tex-math id=\"M4\">\\begin{document}$ \\ddot{u} $\\end{document}</tex-math></inline-formula>ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions <inline-formula><tex-math id=\"M5\">\\begin{document}$ e_0 $\\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M6\">\\begin{document}$ e_2 $\\end{document}</tex-math></inline-formula> and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators <inline-formula><tex-math id=\"M7\">\\begin{document}$ M_{n,\\alpha} $\\end{document}</tex-math></inline-formula> and show the comparison of its rate of approximation vis-a-vis the modified operators.</p>","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Better approximation by a Durrmeyer variant of $ \\\\alpha- $Baskakov operators\",\"authors\":\"P. Agrawal, J. Singh\",\"doi\":\"10.3934/mfc.2021040\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p style='text-indent:20px;'>The aim of this paper is to study some approximation properties of the Durrmeyer variant of <inline-formula><tex-math id=\\\"M2\\\">\\\\begin{document}$ \\\\alpha $\\\\end{document}</tex-math></inline-formula>-Baskakov operators <inline-formula><tex-math id=\\\"M3\\\">\\\\begin{document}$ M_{n,\\\\alpha} $\\\\end{document}</tex-math></inline-formula> proposed by Aral and Erbay [<xref ref-type=\\\"bibr\\\" rid=\\\"b3\\\">3</xref>]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr<inline-formula><tex-math id=\\\"M4\\\">\\\\begin{document}$ \\\\ddot{u} $\\\\end{document}</tex-math></inline-formula>ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions <inline-formula><tex-math id=\\\"M5\\\">\\\\begin{document}$ e_0 $\\\\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\\\"M6\\\">\\\\begin{document}$ e_2 $\\\\end{document}</tex-math></inline-formula> and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators <inline-formula><tex-math id=\\\"M7\\\">\\\\begin{document}$ M_{n,\\\\alpha} $\\\\end{document}</tex-math></inline-formula> and show the comparison of its rate of approximation vis-a-vis the modified operators.</p>\",\"PeriodicalId\":93334,\"journal\":{\"name\":\"Mathematical foundations of computing\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical foundations of computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3934/mfc.2021040\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical foundations of computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/mfc.2021040","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 4
摘要
The aim of this paper is to study some approximation properties of the Durrmeyer variant of \begin{document}$ \alpha $\end{document}-Baskakov operators \begin{document}$ M_{n,\alpha} $\end{document} proposed by Aral and Erbay [3]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr\begin{document}$ \ddot{u} $\end{document}ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions \begin{document}$ e_0 $\end{document} and \begin{document}$ e_2 $\end{document} and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators \begin{document}$ M_{n,\alpha} $\end{document} and show the comparison of its rate of approximation vis-a-vis the modified operators.
Better approximation by a Durrmeyer variant of $ \alpha- $Baskakov operators
The aim of this paper is to study some approximation properties of the Durrmeyer variant of \begin{document}$ \alpha $\end{document}-Baskakov operators \begin{document}$ M_{n,\alpha} $\end{document} proposed by Aral and Erbay [3]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr\begin{document}$ \ddot{u} $\end{document}ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions \begin{document}$ e_0 $\end{document} and \begin{document}$ e_2 $\end{document} and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators \begin{document}$ M_{n,\alpha} $\end{document} and show the comparison of its rate of approximation vis-a-vis the modified operators.