, Volume 52, Issue 1-2, pp 179-186
Date: 17 Jul 2008

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

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Abstract.

Since for q > 1, the q-Bernstein polynomials B n,q (f;.) are not positive linear operators on C[0,1], their convergence properties are not similar to those in the case 0 < q ≤ 1. It has been known that, in general, \(B_{n,q_{n}} (f; .)\) does not approximate \(f \in C[0, 1]\) if q n → 1+, n → ∞, unlike in the case q n → 1. In this paper, it is shown that if 0 ≤ q n  − 1 = o(n −1 3n ), n → ∞, then for any \(f \in C[0, 1]\) , we have: \(B_{n,q_{n}} (f; x) \rightarrow f(x)\) as n → ∞, uniformly on [0,1].

Received: October 19, 2007. Revised: January 10, 2008.