Approximation properties of a moment-based modification of Bernstein operators

Şule Yüksel Güngör; Şule Yüksel Güngör

doi:10.3934/math.2025970

AIMS Mathematics

2025, Volume 10, Issue 9: 21820-21834. doi: 10.3934/math.2025970

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Research article

Approximation properties of a moment-based modification of Bernstein operators

Şule Yüksel Güngör ^,

Department of Mathematics, Gazi University, Ankara 06500, Turkey

Received: 04 August 2025 Revised: 31 August 2025 Accepted: 10 September 2025 Published: 18 September 2025
MSC : 41A25, 41A36

In this paper, a moment-based modification of the classical Bernstein operators is introduced. The proposed operators incorporate both a domain transformation and an adjustment by the second central moment to improve the approximation properties. We investigate their convergence behavior using tools such as the Lipschitz class and Peetre's $ \kappa- $functional. Quantitative estimates are established based on the classical and second-order modulus of continuity. Furthermore, a Voronovskaja-type theorem is provided to analyze the asymptotic behavior. Theoretical results are supported by numerical examples and graphical illustrations that demonstrate the effectiveness of the operators.
- Bernstein operators,
- rate of convergence,
- modulus of continuity,
- Voronovskaja-type theorem
Citation: Şule Yüksel Güngör. Approximation properties of a moment-based modification of Bernstein operators[J]. AIMS Mathematics, 2025, 10(9): 21820-21834. doi: 10.3934/math.2025970

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Abstract

In this paper, a moment-based modification of the classical Bernstein operators is introduced. The proposed operators incorporate both a domain transformation and an adjustment by the second central moment to improve the approximation properties. We investigate their convergence behavior using tools such as the Lipschitz class and Peetre's $ \kappa- $functional. Quantitative estimates are established based on the classical and second-order modulus of continuity. Furthermore, a Voronovskaja-type theorem is provided to analyze the asymptotic behavior. Theoretical results are supported by numerical examples and graphical illustrations that demonstrate the effectiveness of the operators.

References

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