Blending-type modifications of $ \alpha $-Bernstein polynomials and their approximation behavior

Harun ÇİÇEK; Harun ÇİÇEK

doi:10.3934/math.20261349

AIMS Mathematics

2026, Volume 11, Issue 10: 34615-34634. doi: 10.3934/math.20261349

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Research article

Blending-type modifications of $ \alpha $-Bernstein polynomials and their approximation behavior

Harun ÇİÇEK ^,

Department of Mathematics, Science and Arts Faculty, Rahva Campus, Bitlis Eren University, Bitlis, Türkiye

Received: 13 April 2026 Revised: 22 September 2026 Accepted: 05 October 2026 Published: 23 October 2026
MSC : 41A10, 41A25, 41A36

This paper introduced a modified version of the classical Bernstein operators that involved a nonnegative mixing parameter denoted by $ \alpha $. We also demonstrated that these operators satisfy important mathematical properties, such as linearity and positivity. By applying the Bohman-Korovkin theorem, we established conditions under which the operators uniformly approximated a given continuous function. Based on this, we derived formulas that described the rate of convergence. Additionally, we proved a Voronovskaya type theorem, which provided an asymptotic description of the approximation error. We also provided estimates related to the smoothness of the function. By using the classical modulus of continuity, we explained how the error behaved when the function had local variations, and expressed this in terms of first-order properties like monotonicity and convexity. Finally, we showed that our operators preserved these shape-related features, which gave theoretical strength to the results and highlighted the practical value of the operator.
- approximation operators,
- blending-type operators,
- Voronovskaja theorem,
- modulus of continuity,
- shape-preserving properties
Citation: Harun ÇİÇEK. Blending-type modifications of $ \alpha $-Bernstein polynomials and their approximation behavior[J]. AIMS Mathematics, 2026, 11(10): 34615-34634. doi: 10.3934/math.20261349

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Abstract

This paper introduced a modified version of the classical Bernstein operators that involved a nonnegative mixing parameter denoted by $ \alpha $. We also demonstrated that these operators satisfy important mathematical properties, such as linearity and positivity. By applying the Bohman-Korovkin theorem, we established conditions under which the operators uniformly approximated a given continuous function. Based on this, we derived formulas that described the rate of convergence. Additionally, we proved a Voronovskaya type theorem, which provided an asymptotic description of the approximation error. We also provided estimates related to the smoothness of the function. By using the classical modulus of continuity, we explained how the error behaved when the function had local variations, and expressed this in terms of first-order properties like monotonicity and convexity. Finally, we showed that our operators preserved these shape-related features, which gave theoretical strength to the results and highlighted the practical value of the operator.

References

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