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<p>The primary objective of this article is to introduce a sequence of positive linear operators constructed via the Riemann–Liouville fractional integral and Hermite polynomials. Several estimates involving test functions and central moments are established to derive the rate of convergence and the order of approximation. Furthermore, uniform convergence is established using a Korovkin-type theorem, while quantitative estimates are given in terms of the first modulus of smoothness. Approximation results are also investigated through Peetre’s <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K\)</EquationSource> </InlineEquation>-functional, the second modulus of continuity, and Lipschitz-type function spaces. The weighted approximation properties of these operators are further established in terms of the modulus of continuity. The analysis is subsequently extended to a two-dimensional setting, where the corresponding approximation properties are rigorously examined. To support the theoretical developments, numerical examples are presented that illustrate the effectiveness of the proposed operators and confirm the theoretical error estimates. Finally, concluding remarks and possible directions for future research are provided.</p>

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Approximation properties of fractional Szász-Kantorovich operators involving Hermite polynomials

  • 期刊论文

  • 发表日期：

    2026年8月3日

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  • Manoj Kumar,
  • Mohammad Mursaleen

摘要

The primary objective of this article is to introduce a sequence of positive linear operators constructed via the Riemann–Liouville fractional integral and Hermite polynomials. Several estimates involving test functions and central moments are established to derive the rate of convergence and the order of approximation. Furthermore, uniform convergence is established using a Korovkin-type theorem, while quantitative estimates are given in terms of the first modulus of smoothness. Approximation results are also investigated through Peetre’s \(K\) -functional, the second modulus of continuity, and Lipschitz-type function spaces. The weighted approximation properties of these operators are further established in terms of the modulus of continuity. The analysis is subsequently extended to a two-dimensional setting, where the corresponding approximation properties are rigorously examined. To support the theoretical developments, numerical examples are presented that illustrate the effectiveness of the proposed operators and confirm the theoretical error estimates. Finally, concluding remarks and possible directions for future research are provided.