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<p>In this work, we analyze the simultaneous approximation properties of the established King-type modifications of Bernstein operators, with a focus on both integer and fractional-order derivatives. We derive the necessary conditions for the existence of simultaneous approximation and rigorously analyze their implications. We extend classical convergence results to the fractional setting and thereby clarify the convergence behavior of these operators. Theoretical results are also supported by numerical examples and graphical visualizations which illustrate the effectiveness of the approach. This work contributes to the refinement of approximation theory and its potential applications in computational mathematics and fractional calculus.</p>

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Simultaneous Approximation Via Modified Bernstein Operators: From Integer to Fractional Order Derivatives

  • 期刊论文

  • 发表日期：

    2026年3月5日

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  • Oktay Duman,
  • Mehmet Ali Özarslan

摘要

In this work, we analyze the simultaneous approximation properties of the established King-type modifications of Bernstein operators, with a focus on both integer and fractional-order derivatives. We derive the necessary conditions for the existence of simultaneous approximation and rigorously analyze their implications. We extend classical convergence results to the fractional setting and thereby clarify the convergence behavior of these operators. Theoretical results are also supported by numerical examples and graphical visualizations which illustrate the effectiveness of the approach. This work contributes to the refinement of approximation theory and its potential applications in computational mathematics and fractional calculus.