Convergence of Generalized Bernstein Polynomials

Abstract

Let f ¥ C [0,1], q ¥ (0, 1), and B n (f, q; x) be generalized Bernstein polynomials based on the q-integers. These polynomials were introduced by G. M. Phillips in 1997. We study convergence properties of the sequence {B n (f, q; x)} . n=1. It is shown that in general these properties are essentially different from those in the classical case q=1. © 2002 Elsevier Science (USA)

On the analyticity of functions approximated by their q-Bernstein polynomials when q>1

“…This absence of similarity is caused by the fact that, for 0 < q < 1, B n,q are positive linear operators on C[0, 1], while for q > 1, the positivity does not hold any longer. It should be pointed out that in terms of the convergence properties, the similarity between the classical Bernstein and q-Bernstein polynomials ceases to be true even in the case 0 < q < 1, see, e.g., [15,16]. This is because, for 0 < q < 1, the q-Bernstein polynomials -despite being positive linear operators -do not satisfy the conditions of Korovkin's Theorem.…”

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.…”

On a New Type of q-Baskakov Operators

“…In 1997, G. M. Phillips [6] proposed another q-version offers. For interesting properties and different versions this polynomials we refer to [7], [8], [9], [10] [11], [12] and, [13]. Firstly, let us recall the Baskakov Operators [14]: for any function f continuous on [0, ∞),…”