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<p>This paper mainly introduces <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\lambda ,\mu )$</EquationSource> </InlineEquation>-Bernstein-Kantorovich-Stancu-Bézier operators that are a natural continuation of Stancu-type <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\lambda ,\mu )$</EquationSource> </InlineEquation>-Bernstein-Kantorovich operators constructed by Q.-B. Cai et al. (Enhanced approximation techniques: Stancu-type <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\lambda ,\mu )$</EquationSource> </InlineEquation>-Bernstein-Kantorovich operators, <CitationRef CitationID="CR11">2026</CitationRef>, <a href="https://doi.org/10.21203/rs.3.rs-4689585/v1">https://doi.org/10.21203/rs.3.rs-4689585/v1</a>). For these operators, we first examine the order of approximation in regards to global approximation results using a classical approach, Lipschitz class and the second modulus of continuity. Then, a Voronovskaya-type theorem is provided, which characterizes the asymptotic behavior of these operators. Furthermore, by considering the functions whose first derivatives are of bounded variation, we give the rate of convergence of such operators. Finally, to compare the convergence of such operators both to the <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\lambda ,\mu )$</EquationSource> </InlineEquation>-Bernstein-Kantorovich-Stancu operators and to themselves for different parameters, we provide some graphical and numerical examples.</p>

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A Bézier variant of \((\lambda ,\mu )\)-Bernstein-Kantorovich-Stancu operators

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月23日

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  • Xiu-Liang Qiu,
  • Murat Bodur,
  • Qing-Bo Cai

摘要

This paper mainly introduces ( λ , μ ) $(\lambda ,\mu )$ -Bernstein-Kantorovich-Stancu-Bézier operators that are a natural continuation of Stancu-type ( λ , μ ) $(\lambda ,\mu )$ -Bernstein-Kantorovich operators constructed by Q.-B. Cai et al. (Enhanced approximation techniques: Stancu-type ( λ , μ ) $(\lambda ,\mu )$ -Bernstein-Kantorovich operators, 2026, https://doi.org/10.21203/rs.3.rs-4689585/v1). For these operators, we first examine the order of approximation in regards to global approximation results using a classical approach, Lipschitz class and the second modulus of continuity. Then, a Voronovskaya-type theorem is provided, which characterizes the asymptotic behavior of these operators. Furthermore, by considering the functions whose first derivatives are of bounded variation, we give the rate of convergence of such operators. Finally, to compare the convergence of such operators both to the ( λ , μ ) $(\lambda ,\mu )$ -Bernstein-Kantorovich-Stancu operators and to themselves for different parameters, we provide some graphical and numerical examples.