Results in Mathematics

, Volume 52, Issue 1, pp 179–186

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

Article

DOI: 10.1007/s00025-008-0288-2

Cite this article as:
Ostrovska, S. Result. Math. (2008) 52: 179. doi:10.1007/s00025-008-0288-2

Abstract.

Since for q > 1, the q-Bernstein polynomials Bn,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 qn → 1+, n → ∞, unlike in the case qn → 1. In this paper, it is shown that if 0 ≤ qn − 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].

Mathematics Subject Classification (2000).

Primary 41A10 Secondary 30A10 

Keywords.

q-Bernstein polynomials q-integers uniform convergence maximum modulus principle 

Copyright information

© Birkhaueser 2008

Authors and Affiliations

  1. 1.Department of MathematicsAtilim UniversityIncek, AnkaraTurkey

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