On the improvement of analytic properties under the limit q-Bernstein operator

Abstract

Let B n (f, q; x), n = 1, 2, . . . be the q-Bernstein polynomials of a function f ∈ C[0, 1]. In the case 0 < q < 1, a sequence {B n (f, q; x)} generates a positive linear operator B ∞ = B ∞,q on C[0, 1], which is called the limit q-Bernstein operator. In this paper, a connection between the smoothness of a function f and the analytic properties of its image under B ∞ is studied.

The convergence of q ‐Bernstein polynomials (0 < q < 1) in the complex plane

“…In particular, we present an elaboration of the result by Wang and Meng, and a generalization of results on an analytic continuation of the limit function obtained in [10].…”

“…Surveys of results on the q-Bernstein polynomials together with comprehensive lists of references on the subject are given in [13] (results obtained before 2000) and [9] (results obtained in [2000][2001][2002][2003][2004]. The subject is currently under study and new papers are constantly coming out (see, for example, [2], [10], and [19]). …”

“…has been considered in [10] and [11]. In particular, it has been proved that if f is m (m ≥ 0) times differentiable from the left at 1 and f (m) satisfies the Lipschitz condition of order…”

Generalized q-Baskakov operators

“…Research results show that q-Bernstein operators possess good convergence and approximation properties in C [0, 1]. These operators have been studied by a number of authors, we mention the some due to II'inskii and Ostrovska [16], Oruc and Tuncer [21], Ostrovska [22], [23] and Videnskii [28] etc. Heping [14], Heping and Fanjun [15] discussed Voronovskaya-type formulas and saturation of convergence for q-Bernstein polynomials for arbitrary fixed q, 0 < q < 1.…”

The sharpness of convergence results for q-Bernstein polynomials in the case q > 1

“…As a result, the convergence of q-Bernstein polynomials in the case 0 < q < 1 has been investigated in detail, including a Korovkin type theorem, the properties of the limit operator, the rate of convergence, and the saturation phenomenon (cf. [10], [15], [16], [18] - [21]). In contrast, there are only two papers, namely [14] and [22], dealing systematically with the convergence in the case q > 1.…”