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Let f ¥ C [0,1], q ¥ (0, 1), and B n (f, q; x) be generalized Bernstein polynomials based on the q-integers. These polynomials were introduced by G. M. Phillips in 1997. We study convergence properties of the sequence {B n (f, q; x)} . n=1. It is shown that in general these properties are essentially different from those in the classical case q=1. © 2002 Elsevier Science (USA)

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Let B n (f, q; x), n = 1, 2, . . . be the q-Bernstein polynomials of a function f ∈ C[0, 1]. In the case 0 < q < 1, a sequence {B n (f, q; x)} generates a positive linear operator B ∞ = B ∞,q on C[0, 1], which is called the limit q-Bernstein operator. In this paper, a connection between the smoothness of a function f and the analytic properties of its image under B ∞ is studied.