Approximation by modified Szász-Durrmeyer operators

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• 作者：Tuncer Acar ; Gulsum Ulusoy
• 关键词：Szász ; Durrmeyer operators ; Weighted modulus of continuity ; Quantitative Voronovskaya theorem
• 刊名：Periodica Mathematica Hungarica
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：72
• 期：1
• 页码：64-75
• 全文大小：450 KB
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  4.T. Acar, A. Aral, I. Raşa, The new forms of Voronovskaya’s theorem in weighted spaces. Positivity (2015). doi:10.​1007/​s11117-015-0338-4
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• 作者单位：Tuncer Acar (1)
  Gulsum Ulusoy (1)
  
  1. Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450, Yahsihan, Kirikkale, Turkey
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Sciences
  Mathematics
  
• 出版者：Akad茅miai Kiad贸, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic
• ISSN：1588-2829

文摘

The main goal of this paper is to introduce Durrmeyer modifications for the generalized Szász–Mirakyan operators defined in (Aral et al., in Results Math 65:441–452, 2014). The construction of the new operators is based on a function \(\rho \) which is continuously differentiable \(\infty \) times on \( \left[ 0,\infty \right) ,\) such that \(\rho \left( 0\right) =0\) and \( \inf _{x\in \left[ 0,\infty \right) }\rho ^{\prime }\left( x\right) \ge 1.\) Involving the weighted modulus of continuity constructed using the function \( \rho \), approximation properties of the operators are explored: uniform convergence over unbounded intervals is established and a quantitative Voronovskaya theorem is given. Moreover, we obtain direct approximation properties of the operators in terms of the moduli of smoothness. Our results show that the new operators are sensitive to the rate of convergence to f,  depending on the selection of \(\rho .\) For the particular case \(\rho \left( x\right) =x\), the previous results for classical Szász-Durrmeyer operators are captured.