On the Sets of Convergence for Sequences of the 𝑞-Bernstein Polynomials with 𝑞>1

Abstract

The aim of this paper is to present new results related to the convergence of the sequence of the 𝑞-Bernstein polynomials {𝐵𝑛,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 𝕁𝑞={𝑞−𝑗}∞𝑗=0∪{0}, and that this result is sharp in the sense that the sequence {𝐵𝑛,𝑞(𝑓;𝑥)}∞𝑛=1 may be divergent for all 𝑥∈𝑅⧵𝕁𝑞. Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.