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Zbl 0838.41017
Kirov, G.H.
A generalization of the Bernstein polynomials. (English)
[J] Math. Balk., New Ser. 6, No.2, 147-153 (1992). ISSN 0205-3217

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 2000:: *41A36 Approximation by positive operators

Keywords: Bernstein polynomials; degree of smoothness

Cited in: Zbl 1079.41022 Zbl 1058.41007 Zbl 1103.41304

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