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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.

##### MSC:
 41A25 Rate of convergence, degree of approximation
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