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In this chapter, a novel extension of Bernstein type operators are introduced based on multiple shape parameters \(\lambda _i\) , \(i=1,2,\ldots , n\) . Convergence properties of the operators are investigated and local and global approximation theorems are established via Ditzian-Totik modulus of smoothness of first and second order. The rate of convergence is estimated with the help of Lipschitz-type function involving two parameters. Furthermore, statistical approximation properties are given using weighted mean matrix method, and Voronovskaja type asymptotic approximation results are presented. Finally, some illustrative graphs and numerical examples are provided to show the efficiency, applicability and validity of proposed operators.

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Generalized Bernstein Operators Associated with Multiple Shape Parameters

  • 章节

  • 发表日期：

    2025年

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  • Uğur Kadak,
  • Faruk Özger

摘要

In this chapter, a novel extension of Bernstein type operators are introduced based on multiple shape parameters \(\lambda _i\) , \(i=1,2,\ldots , n\) . Convergence properties of the operators are investigated and local and global approximation theorems are established via Ditzian-Totik modulus of smoothness of first and second order. The rate of convergence is estimated with the help of Lipschitz-type function involving two parameters. Furthermore, statistical approximation properties are given using weighted mean matrix method, and Voronovskaja type asymptotic approximation results are presented. Finally, some illustrative graphs and numerical examples are provided to show the efficiency, applicability and validity of proposed operators.