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<p>Several authors have defined and studied various generalizations of the well-known Bernstein type operators. For functions that may not be continuous (according to Morigi and Neamtu), the traditional operators were replaced with their integral extensions defined in the Kantorovich and/ or Durrmeyer sense. In 2019, Aral et al. reconstructed the Bernstein-Kantorovich exponential-type polynomials and obtained some approximation results. In this work, we develop the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(q\)</EquationSource> </InlineEquation>-analogue of the Bernstein-Kantorovich exponential-type operators. These newly defined operators provide an approximation process within exponentially weighted spaces, like <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(H_{p,\mu}[0,1]\)</EquationSource> </InlineEquation>. By employing the modulus of continuity alongside K-functionals, we obtain various precise quantitative estimates and further deduce a quantitative version of the Voronovskaja-type theorem. Finally, for graphical representation, we use MATLAB (R2025a).</p>

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Exponential-type Bernstein-Kantorovich polynomials in \(q\)-analogue with graphical analysis

  • 期刊论文

  • 发表日期：

    2026年5月19日

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  • Mohd. Ahasan,
  • Mohammad Mursaleen,
  • Yogesh Nagar

摘要

Several authors have defined and studied various generalizations of the well-known Bernstein type operators. For functions that may not be continuous (according to Morigi and Neamtu), the traditional operators were replaced with their integral extensions defined in the Kantorovich and/ or Durrmeyer sense. In 2019, Aral et al. reconstructed the Bernstein-Kantorovich exponential-type polynomials and obtained some approximation results. In this work, we develop the \(q\) -analogue of the Bernstein-Kantorovich exponential-type operators. These newly defined operators provide an approximation process within exponentially weighted spaces, like \(H_{p,\mu}[0,1]\) . By employing the modulus of continuity alongside K-functionals, we obtain various precise quantitative estimates and further deduce a quantitative version of the Voronovskaja-type theorem. Finally, for graphical representation, we use MATLAB (R2025a).