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-
