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A generalization of the Bernstein polynomials. (English) Zbl 0838.41017
Summary: For the functions \(f\in C^r [0,1 ]\), \(r=0, 1,2, \dots\) the polynomials \[ B_{n,r} (f; x)= \sum^n_{k=0} \sum^r_{i=0} f^{(i)} {{(k/n)} \over {1!}} (x- k/n)^i {n \choose k} x^k (1-x)^{n-k} \] are introduced. For \(r=0\) they coincide with the classical Bernstein polynomials, but for \(r\geq 1\) in contrast with the last ones, they are sensitive to the degree of smoothness of the function \(f\) as approximations to \(f\).

MSC:
41A36 Approximation by positive operators
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