Approximation Properties of rth Order Generalized Bernstein Polynomials Based on $q$-Calculus (Analysis in Theory and Applications, 2018)
In this paper we introduce a generalization of Bernstein polynomials based on $q$ calculus. With the help of Bohman-Korovkin type theorem, we obtain $A$−statistical approximation properties of these operators. Also, by using the Modulus of continuity and Lipschitz class, the statistical rate of convergence is established. We also gives the rate of $A$−statistical convergence by means of Peetre’s type $K$−functional. At last, approximation properties of a rth order generalization of these operators is discussed.
Keywords: $q$−integers; $q$−Bernstein polynomials; $A$−statistical convergence; modulus of continuity; Lipschitz class; Peetre’s type $K$−functional.