Of Bernstein Operators Lupas Q-analog Of The Convergence Rate And Its Saturation

This paper is concerned with the results of the approximation properties and saturation of convergence for the q-analogue Bernstein operators R_n,1. In 1912, using probability theory Bernstein defined polynomials called nowadays Bernstein Polynomials as follows: Let f : [0,1] â†'R, the Bernstein polynomial of f isLater it was found that Bernstein polynomials possess many remarkable properties, which made them an area of intensive research. Due to che development of q-Calculus, generalization of Bernstein polynomials connected with q-Calculus have energed. The first person to make progress in this ddirection was A. Lupas,who introduced a q-analogue Bernstein operator (see[11]):In the case q = 1, q-analogue Bernstein operators R_n,q(f,x) coincide with the classical ones. They have an advantage of generating positive linear operators for all q > 0. On the one hand, like the classical Bernstein polynomials, q-analogue Bernstein operators share the good properties such as the shape preserving properties and monotonicity, on the other hand, the properties of q-analogue Bernstein operatorsdiffer essentially from those in classical case. For example, Bernstein (see[4]) proved that if f ∈ C[0,1], then the sequence B_nf(x) converges uniformly to f(x) on [0,1]. Sofiya Ostrovaska (see[18]) proved that for all q > 0, and q ≠ 1, R_n,q(f,x) converges uniformly to f(x) if and only if f is linear. Moreover, for q ≠ 1, the operators R_n,q(f,x) are rational functions rather than polynomials. In this paper, we obtain a series of results as follows:(1) Let q ∈ (0,1) be fixed , and let f ∈ C[0,1]. Then||R_n,q(f) - R_∞,q(f)|| ≤C_qω(f,qⁿ),where . This estimate is sharp in the following sense of order :for each a, 0 < α ≤ 1, there exists a function f_α{x) which belongs to the Lipschitz class Lip_α := {f ∈ C[0,1] |ω (f, t) ≤ ct^α} such thatwhere A(n) (?) B(n) means that, there exists c₁., c₂ independent of n, such that c₁B(n) ≤ A(n) ≤ c₂B(n).(2) Let f ∈ C[0,1]. Then for any q ∈ (1, ∞) we get||R_n,q(f)-R_∞,q(f)||≤C'_qω(f, 1/qⁿ),where C'_q is a constant independent of f and n. (3) Let 0 < q < 1. Thenwhere c is an absolute constant. (4) Let q∈ (1,∞). Thenwhere c is an absolute constant.(5) Let q ∈ (0,1) be fixed. For any function f ∈ C¹[0, 1], thenwhereFurthermore the above approximation is uniform for x ∈ [0,1].(6) Let 0 < q < 1, f ∈ C¹[0,1]. Then ||R_n,q(f) - R_∞,q(f)|| = o(qⁿ) if and only if...