The Approximation of Power Function by the Q-Bernstein Polynomials in the Case Q > 1

Abstract

Since for q > 1, q-Bernstein polynomials are not positive linear operators on C[0, 1], the investigation of their convergence properties turns out to be much more difficult than that in the case 0 < q < 1. It is known that, in the case q > 1, the q-Bernstein polynomials approximate the entire functions and, in particular, polynomials uniformly on any compact set in ℂ. In this paper, the possibility of the approximation for the function (z + a)α, a ≥ 0, with a non-integer α > -1 is studied. It is proved that for a > 0, the function is uniformly approximated on any compact set in {z: \z| < a}, while on any Jordan arc in {z: \z\ > a}, the uniform approximation is impossible. In the case a = 0, the results of the paper reveal the following interesting phenomenon: the power function zα, α > 0, is approximated by its q-Bernstein polynomials either on any (when α ∈ ℕ) or no (when α ∉ ℕ) Jordan arc in ℂ.