Bleimann, Butzer, and Hahn Operators Based on the -Integers

Abstract

We give a new generalization of Bleimann, Butzer, and Hahn operators, which includes qintegers. We investigate uniform approximation of these new operators on some subspace of bounded and continuous functions. In Section 3, we show that the rates of convergence of the new operators in uniform norm are better than the classical ones. We also obtain a pointwise estimation in a general Lipschitz-type maximal function space. Finally, we de?fine a generalization of these new operators and study the uniform convergence of them.Copyright © 2007 A. Aral and O. Dogru. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. IntroductionRecently, in 1997, Phillips [1] used the q-integers in approximation theory where it is considered q-based generalization of classical Bernstein polynomials. It was obtained by replacing the binomial expansion with the general one, the q-binomial expansion. Phillips has obtained the rate of convergence and Voronovskaja-type asymptotic formulae for these new Bernstein operators based on q-integers. Later, some results are established in due course by Phillips et al. (see [2,3,1]). In [4], Barbasu gave Stancu-type generalization of these operators and II'inskii and Ostrovska [5] studied their different convergence properties. Also some results on the statistical and ordinary approximation of functions by Meyer-König and Zeller operators based on q-integers can be found in [6,7], respectively.In [8], Bleimann, Butzer, and Hahn introduced the following operators: 1 + t)) ν for ν = 0,1,2. Some generalization of the operators (1.1) were given in [11][12][13].In this paper, we derive a q-integers-type modification of BBH operators that we call q-BBH operators and investigate their Korovkin-type approximation properties by using the test function (t/(1 + t)) ν for ν = 0,1,2. Also, we define a space of generalized Lipschitz-

Some approximation results by (p, q)-analogue of Bernstein–Stancu operators

“…Several generalizations of well-known positive linear operators based on q-integers were introduced and their approximation properties have been studied by several authors. For instance, q-Baskakov-Kantorovich operators in [11]; q -Szász-Mirakjan operators in [25]; q-Bleimann, Butzer and Hahn operators in [3] and [9]; q-analogue of Baskakov and Baskakov-Kantorovich operators in [18]; q-analogue of Szász-Kantorovich operators in [19]; q-analogue of Stancu-Beta operators in [4] and [21]; and q-Lagrange polynomials in [23] were defined and their approximation properties were investigated.…”

On weighted statistical convergence based on (p,q)-integers and related approximation theorems for functions of two variables

“…Based on the development of q-calculus many authors studied some new generalizations of linear positive operators based on q-integers [2,[5][6][7]18,19,24]. First, the concept of q-analogue of Bernstein operators was introduced and investigated corresponding approximating properties by Lupaş [23].…”

On q-Parametric Szász-Mirakjan Operators

“…[1]- [27]). Recently, an intensive research has been conducted on polynomials and operators based on q-integers, see [3], [4], [5], [9], [15], [17], [21]- [14]. The q-Bernstein polynomials B n,q (f ; x), n = 1, 2, .., 0 < q < ∞, were introduced by G. M. Phillips in [24].…”