A generalization of the Bernstein polynomials (Q1359559)

The author generalizes the Bernstein polynomials by taking averages of point evaluations of the function \(f\) instead of the usual values \(f(i/n)\). The number of point evaluations \(s_n\) taken for the \(n\)th polynomial, may grow to \(\infty\) as \(n\to\infty\), but it is necessary and sufficient for insuring approximation of continuous functions, that \(s_n/n\to 0\). An estimate on the degree of approximation involving the modulus of continuity of \(f\) is also given.