Abstract
We give a new generalization of Bleimann, Butzer, and Hahn operators, which includes -integers. We investigate uniform approximation of these new operators on some subspace of bounded and continuous functions. In Section, 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 define a generalization of these new operators and study the uniform convergence of them.
Highlights
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
Summary
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. In [4], Barbasu gave Stancu-type generalization of these operators and II‘inskii and Ostrovska [5] studied their different convergence properties. Some results on the statistical and ordinary approximation of functions by Meyer-Konig and Zeller operators based on q-integers can be found in [6, 7], respectively. 1 (1 + x)n n k=0 f k n−k+1 n k xk, x > 0, n ∈ N
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