Approximation of Functions by Modified Phillips q−Bernstein Operators of Order Two

ABSTRACT In this paper, we introduce modified Phillips Bernstein operators of order two, which provide better uniform approximation than Phillips Bernstein operators and Phillips Durrmeyer Bernstein operators. In this study, we consider functions belonging to the class and show that the modified Phillips Bernstein operators converge uniformly to all functions in this class using a strengthened version of the Korovkin theorem. Also, we prove a Voronovskaya‐type theorem for modified Phillips Bernstein operators, which show better approximation than the Phillips Bernstein operators. Using the modulus of continuity, we show that the order of approximation of modified Phillips Bernstein operators is better than that of Phillips Bernstein operators and Phillips Durrmeyer Bernstein operators. Finally, we compare the convergence rate of Phillips Bernstein operators, modified Phillips Bernstein operators, and Phillips Durrmeyer Bernstein operators by considering the functions and belonging to .