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

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 𝕁 𝑞 = { 𝑞 − 𝑗 } ∞ 𝑗 = 0 ∪ { 0 } , and that this result is sharp in the sense that the sequence { 𝐵 𝑛 , 𝑞 ( 𝑓 ; 𝑥 ) } ∞ 𝑛 = 1 may be divergent for all 𝑥 ∈ 𝑅 ⧵ 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.