On the Sets of Convergence for Sequences of the ð ‘ž -Bernstein Polynomials with ð ‘ž > 1

The aim of this paper is to present new results related to the convergence of the sequence of the ð ‘ž -Bernstein polynomials { ð µ ð ‘› , ð ‘ž ( ð ‘“ ; ð ‘¥ ) } in the case ð ‘ž > 1 , where ð ‘“ is a continuous function on [ 0 , 1 ] . It is shown that the polynomials converge to ð ‘“ uniformly on the time scale ð • ð ‘ž = { ð ‘ž − ð ‘— } ∞ ð ‘— = 0 ∪ { 0 } , and that this result is sharp in the sense that the sequence { ð µ ð ‘› , ð ‘ž ( ð ‘“ ; ð ‘¥ ) } ∞ ð ‘› = 1 may be divergent for all ð ‘¥ ∈ ð ‘… ⧵ ð • ð ‘ž . Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.