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Further results concerning some general Durrmeyer type operators

  • Tarul Garg
  • Ana Maria AcuEmail author
  • Purshottam Narain Agrawal
Original Paper
  • 22 Downloads

Abstract

The present paper deals with the approximation properties of the Miheşan-Durrmeyer type operators due to Kajla (Adv Oper Theory 2:162–178, 2017) which include well-known operators like Phillips operators (Ann Math 59:325–356, 1954). and Pǎltǎnea operators (Carpathian J Math 24:378–385, 2008) as particular cases. We establish the degree of approximation of the operators in a weighted space and then extend the study to the approximation in a weighted space and the order of convergence of the operators by means of the Ditzian–Totik modulus of smoothness, quantitative and Grüss Voronovskaya type approximation. We construct a variant of our operator that preserves the test functions \(e_0\) and \(e_2\) and show that the new operators present a better rate of convergence than the original ones. Next we define an operator which preserves the functions \(e_0\) and \(e^{-x}\) and show by numerical examples and illustrations that at least one of the two newly constructed operators presents a better rate of convergence than the original operator.

Keywords

Ditzian–Totik modulus of smoothness Rate of convergence Weighted space Voronovskaya type theorem 

Mathematics Subject Classification

41A10 41A25 41A36 41A60 

Notes

Acknowledgements

The authors are extremely grateful to the learned reviewers for a very critical reading of the paper and making valuable comments and suggestions leading to a better presentation of the paper. The first author is thankful to “The Ministry of Human Resource and Development”, India for the financial support to carry out her research work and the work of the second author was financed from Lucian Blaga University of Sibiu research grant LBUS-IRG-2018-04.

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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia
  2. 2.Department of Mathematics and InformaticsLucian Blaga University of SibiuSibiuRomania

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