We propose certain Durrmeyer-type operators for Apostol-Genocchi polynomials in this research. We explore these operators' approximation attributes and measure the rate of convergence. In addition, we present a direct approximation theorem based on first and second-order modulus of continuity, local approximation findings for Lipschitz class functions and a direct theorem based on the typical modulus of continuity. Finally, we showed a graph illustrating the convergence of the suggested operators and an error table.
{"title":"Integral modification of Beta-Apostol-Genocchi operators","authors":"N. Bhardwaj, N. Deo","doi":"10.3934/mfc.2022039","DOIUrl":"https://doi.org/10.3934/mfc.2022039","url":null,"abstract":"We propose certain Durrmeyer-type operators for Apostol-Genocchi polynomials in this research. We explore these operators' approximation attributes and measure the rate of convergence. In addition, we present a direct approximation theorem based on first and second-order modulus of continuity, local approximation findings for Lipschitz class functions and a direct theorem based on the typical modulus of continuity. Finally, we showed a graph illustrating the convergence of the suggested operators and an error table.","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"88 1","pages":"474-483"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83533119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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} $end{document} proposed by Aral and Erbay [3]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Grbegin{document}$ 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} $end{document} and show the comparison of its rate of approximation vis-a-vis the modified operators.
The aim of this paper is to study some approximation properties of the Durrmeyer variant of begin{document}$ alpha $end{document}-Baskakov operators begin{document}$ M_{n,alpha} $end{document} proposed by Aral and Erbay [3]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Grbegin{document}$ 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.
{"title":"Better approximation by a Durrmeyer variant of $ alpha- $Baskakov operators","authors":"P. Agrawal, J. Singh","doi":"10.3934/mfc.2021040","DOIUrl":"https://doi.org/10.3934/mfc.2021040","url":null,"abstract":"<p style='text-indent:20px;'>The aim of this paper is to study some approximation properties of the Durrmeyer variant of <inline-formula><tex-math id=\"M2\">begin{document}$ 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":"10 1","pages":"108-122"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80868412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Here we study the univariate fuzzy fractional quantitative approximation of fuzzy real valued functions on a compact interval by quasi-interpolation arctangent-algebraic-Gudermannian-generalized symmetrical activation function relied fuzzy neural network operators. These approximations are derived by establishing fuzzy Jackson type inequalities involving the fuzzy moduli of continuity of the right and left Caputo fuzzy fractional derivatives of the involved function. The approximations are fuzzy pointwise and fuzzy uniform. The related feed-forward fuzzy neural networks are with one hidden layer. We study also the fuzzy integer derivative and just fuzzy continuous cases. Our fuzzy fractional approximation result using higher order fuzzy differentiation converges better than in the fuzzy just continuous case.
{"title":"Fuzzy fractional more sigmoid function activated neural network approximations revisited","authors":"G. Anastassiou","doi":"10.3934/mfc.2022031","DOIUrl":"https://doi.org/10.3934/mfc.2022031","url":null,"abstract":"Here we study the univariate fuzzy fractional quantitative approximation of fuzzy real valued functions on a compact interval by quasi-interpolation arctangent-algebraic-Gudermannian-generalized symmetrical activation function relied fuzzy neural network operators. These approximations are derived by establishing fuzzy Jackson type inequalities involving the fuzzy moduli of continuity of the right and left Caputo fuzzy fractional derivatives of the involved function. The approximations are fuzzy pointwise and fuzzy uniform. The related feed-forward fuzzy neural networks are with one hidden layer. We study also the fuzzy integer derivative and just fuzzy continuous cases. Our fuzzy fractional approximation result using higher order fuzzy differentiation converges better than in the fuzzy just continuous case.","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"20 1","pages":"320-353"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78100258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Symbolic computation of recurrence coefficients for polynomials orthogonal with respect to the Szegő-Bernstein weights","authors":"G. Milovanović","doi":"10.3934/mfc.2022049","DOIUrl":"https://doi.org/10.3934/mfc.2022049","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"30 1","pages":"460-473"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88871752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Complex network pinning control based On DR algorithm","authors":"Haiyi Sun, Limeng Zhang, Lei Ji","doi":"10.3934/mfc.2023013","DOIUrl":"https://doi.org/10.3934/mfc.2023013","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70220403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present article, we construct the Szász-Jakimovski-Leviatan operators in parametric form by including the sequences of continuous functions and then investigate the approximation properties. We have successfully estimated the convergence by use of modulus of continuity in the spaces of Lipschitz functions, Peetres begin{document}$ K $end{document}-functional and weighted functions.
In the present article, we construct the Szász-Jakimovski-Leviatan operators in parametric form by including the sequences of continuous functions and then investigate the approximation properties. We have successfully estimated the convergence by use of modulus of continuity in the spaces of Lipschitz functions, Peetres begin{document}$ K $end{document}-functional and weighted functions.
{"title":"Convergence on sequences of Szász-Jakimovski-Leviatan type operators and related results","authors":"M. Nasiruzzaman","doi":"10.3934/mfc.2022019","DOIUrl":"https://doi.org/10.3934/mfc.2022019","url":null,"abstract":"<p style='text-indent:20px;'>In the present article, we construct the Szász-Jakimovski-Leviatan operators in parametric form by including the sequences of continuous functions and then investigate the approximation properties. We have successfully estimated the convergence by use of modulus of continuity in the spaces of Lipschitz functions, Peetres <inline-formula><tex-math id=\"M1\">begin{document}$ K $end{document}</tex-math></inline-formula>-functional and weighted functions.</p>","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"18 1","pages":"218-230"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75494743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function begin{document}$ 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;; xin[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;; xin[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) $end{document} leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [11].
{"title":"Better degree of approximation by modified Bernstein-Durrmeyer type operators","authors":"P. Agrawal, S. Güngör, Abhishek Kumar","doi":"10.3934/mfc.2021024","DOIUrl":"https://doi.org/10.3934/mfc.2021024","url":null,"abstract":"<p style='text-indent:20px;'>In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function <inline-formula><tex-math id=\"M1\">begin{document}$ 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;; xin[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":"17 42","pages":"75"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72489625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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