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On the sets of convergence for sequences of the \(q\)-Bernstein polynomials with \(q > 1\). (English) Zbl 1254.41013
Summary: The aim of this paper is to present new results related to the convergence of the sequence of the \(q\)-Bernstein polynomials \(\{B_{n, q}(f; x)\}\) in the case \(q > 1\), where \(f\) is a continuous function on \([0 ,1]\). It is shown that the polynomials converge to \(f\) uniformly on the time scale \(\mathbb J_q = \{q^{-j}\}^\infty_{j=0} \cup \{0\}\), and that this result is sharp in the sense that the sequence \(\{B_{n, q}(f; x)\}^\infty_{n=1}\) may be divergent for all \(x \in R \setminus \mathbb J_q\). Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.

41A25 Rate of convergence, degree of approximation
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