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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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