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<p>This paper introduces two new operators: the Kantorovich form and Bézier variant of sq-Bernstein operators, building upon the sq-Bernstein operators presented in Nouri and Saeidian (Journal of Mathematics 2023:55165, 2023). The essential properties of these operators are investigated, including their uniform convergence and convergence rate, analyzed through the modulus of continuity and Lipschitz class functions. Voronovskaya-type asymptotic formulas are derived for the sq-Bernstein-Kantorovich operators. The approximation capabilities of the Bézier variant are examined using the modulus of continuity, second-order modulus of continuity, and Lipschitz class functions. Illustrative examples demonstrate the behavior and characteristics of each operator. The study establishes a direct approximation theorem using the Ditzian-Totik modulus of smoothness and K-functional. Finally, numerical examples compare the performance of these newly introduced operators with the original sq-Bernstein operators.</p>

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Sq-Bernstein-Kantorovich operators and their Bézier variant

  • 期刊论文

  • 发表日期：

    2025年2月18日

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  • Jamshid Saeidian,
  • Bahareh Nouri

摘要

This paper introduces two new operators: the Kantorovich form and Bézier variant of sq-Bernstein operators, building upon the sq-Bernstein operators presented in Nouri and Saeidian (Journal of Mathematics 2023:55165, 2023). The essential properties of these operators are investigated, including their uniform convergence and convergence rate, analyzed through the modulus of continuity and Lipschitz class functions. Voronovskaya-type asymptotic formulas are derived for the sq-Bernstein-Kantorovich operators. The approximation capabilities of the Bézier variant are examined using the modulus of continuity, second-order modulus of continuity, and Lipschitz class functions. Illustrative examples demonstrate the behavior and characteristics of each operator. The study establishes a direct approximation theorem using the Ditzian-Totik modulus of smoothness and K-functional. Finally, numerical examples compare the performance of these newly introduced operators with the original sq-Bernstein operators.