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On the convergence and iterates of $$q$$-Bernstein polynomials. (English) Zbl 1015.33012
Three theorems are given, concerning the convergence of the $$q$$-Bernstein operator $${\mathcal B}_n$$, its iterates, and Boolean sums. The first theorem asserts that if $$q\geq 1$$ and $$p$$ is any polynomial, then $$\lim_{n\to \infty} {\mathcal B}_n(p(x))= p(x)$$; and if $$0<q<1$$ and $$f$$ is continuous on $$[0, 1]$$, then $$\lim_{n\to\infty} {\mathcal B}_n(f(x)) =f(x)$$ if and only if $$f$$ is linear.
The second theorem asserts that the $$m$$th iterate of $${\mathcal B}_n$$ applied to $$f$$ converges to the linear interpolating polynomial for $$f$$ at the end points of $$[0,1]$$, for any fixed $$q>0$$, as $$m\to\infty$$.
The third theorem asserts that the $$m$$-fold Boolean sum of $${\mathcal B}_n$$ with itself, applied to a function $$f\in C[0,1]$$, converges to an interpolating polynomial for $$f$$ of degree $$\leq n$$, and $$m\to\infty$$, and the interpolating points are explicitly given.

##### MSC:
 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 41A25 Rate of convergence, degree of approximation 41A35 Approximation by operators (in particular, by integral operators)
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