On the Convergence and Iterates of q-Bernstein Polynomials

The convergence properties of q-Bernstein polynomials are investigated. When q≥1 is fixed the generalized Bernstein polynomials Bnf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0 0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of Bnf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0, 1].

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