Summation-Integral-Phillips for a Sequence of -Bernstein Type Operators

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.