Q-operator Approximation

This paper is concerned with the results of the approximation properties of the q-Bernstein polynomials B_n,q and q-Meyer Konig and Zeller operators M_n,q In 1912, using probability theory Bernstein defined polynomials called nowadays Bernstein Polynomials as follows: Let f : [0,1]â†'R, the Bernstein polynomial of / isLater it was found that Bernstein polynomials possess many remarkable properties, which made them an area of intensive research. Generalized Bernstein polynomials based on the 9-integers, or 9-Bernstein polynomials isIn the case 9 = 1, 9-Bernstein polynomials coincide with the classical ones. For 0 < q < 1, on the one hand, like the classical Bernstein polynomials, q-Bernstein polynomials share the good properties such as the shape preserving properties and monotonicity, on the other hand, the properties of 9-Bernstein polynomials differ essentially from those in classical case. For example, H.oruc and Neciber Tuncer ([8]) proved that for fixed 9, 0 < q < 1, B_n,q(f, x) converges uniformly to f(x) if and only if f is linear. Bernstein ([3]) proved that if f ∈ C[0,1], then the sequence B_nf(x) converges uniformly to f(x) on [0,1].In 2000, Tiberiu Trif ([12]) introduced the following generalization of Meyer Konig and Zeller operators, based on 9-integers. For each positive integer n, f∈ C[0,1], we defineIn this paper, we obtain a series of results as follows: where and the above estimate is sharp in the sense of order.(2) For any f∈ C[0,1], and for all x ∈ [0,1], q ∈ (0,1], M_n,q(f; x) converges uniformly (3) Let 0 < q < 1, f∈ C[0,1], thenwhere (4) For 0 < q ≤ r ≤ 1 and for f convex on [0,1], then M_n.r f≤ M_n.q f.