Asmaa J. Barak1,*, Zahra Abdullah Shawi2, Mayada Marid Abdul Hussain3,*
1 Ministry of Education,Haifa Preparatory School for Girl, Basrah, Iraq
2 Ministry of Education, General Directorate of Education,AL-Qurna Preparatory School for Girls
3 Department of Mathematics, Southern Technical University, Basra Technical Institute
Email: Asmaa J.Barak@gmail.com,
Received 24 Sep. 2025, Accepted 3 Nov. 2025, published 30 Dec. 2025.
AbstractKey wordsDOI
This paper introduces a new hybrid operator based on combining the Phillips concept with a sequence of lambda-Bernstein operators. This operator represents a qualitative improvement over classical Bernstein-Durrmeyer operators, which faced significant limitations in controlling the behavior of functions at critical points such as the zero point and suffered from a significantly slow rate of convergence. The developed operator overcomes these challenges, achieving a substantial improvement in the quality and accuracy of convergence. To demonstrate the effectiveness of this operator, the study proves a set of basic theoretical results. First, the paper proves the regular convergence theorem for the operator. This is followed by establishing the error estimation theorem using a continuum measure, which in turn confirms the achievement of first-order convergence. Finally, the study presents a precise Voronovskaya-type asymptotic formula that reveals the detailed behavior of the operator’s approximation rate when studying functions regularly.
λ-Bernstein operators, Bernstein-Durrmeyer Operators, convergence, M-th order moment, The Voronovskaja formula.
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