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

Abstract

The paper focuses at the estimates for the rate of convergence of the q-Bernstein polynomials (0 < q < 1) in the complex plane. In particular, a generalization of previously known results on the possibility of analytic continuation of the limit function and an elaboration of the theorem by Wang and Meng is presented.

The moments for q-Bernstein operators in the case 0 < q < 1

“…[2][3][4][5][6][7][8][9][10][11][12][13][14]; Mahmudov and Sabancıgil, unpublished manuscript). It is known that the central moments and their estimates are important for studying the approximation properties of the linear positive operators.…”

Approximation by q-Bernstein type operators

“…The convergence properties of B n,q (0 < q < 1) in the complex plane were studied in [3]. The goal of the paper is to construct a new non-trivial sequence {L n } of bounded positive linear operators in C[0, 1] using the q-Bernstein basic polynomials (1.1), such that L n has different properties from (1.3), (1.4) and (1.5), but L n (f, x) converges uniformly to f (x) on [0, 1] as n → ∞.…”

The Approximation of All Continuous Functions on [0, 1] by q-Bernstein Polynomials in the Case q → 1+

“…The subject is currently under study and new papers are constantly coming out (see, for example, [10,20], and [22]). A two-parametric generalization of q-Bernstein polynomials has been studied in [7] and [21], while a version of the Bernstein-Durrmeyer operator related to q-Bernstein polynomials has been considered in [3].…”

“…For the time being, the convergence of q-Bernstein polynomials in the case 0 < q < 1, has been investigated in detail including the rate of convergence, saturation theorems and the properties of the limit operator, see [6,10,11,[15][16][17][18][19].…”