Two Classes of Bernstein Type Operators

Radu Păltănea
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Radu Păltănea: Transilvania University, Department of Mathematics

Chapter 5 in Approximation Theory Using Positive Linear Operators, 2004, pp 131-159 from Springer

Abstract: Abstract One of the most natural extensions of the Bernstein operators was made by H. Brass [17]. These operators are of the form 5.1 $$ {{P}_{n}}(f,x): = \sum\limits_{{k = 0}}^{n} {f\left( {\frac{k}{n}} \right){{q}_{{n,k}}}(x),f \in F[0,1],x \in [0,1],n \in \mathbb{N},} $$ where q n ,k are polynomials of degree n that are positive on the interval [0, 1] and are such that the following properties are true: 1) P n is a linear positive operator 2) P n preserves linear functions 3) P n preserves the degree of any polynomial of degree at most n and 4) P n preserves the convexity of higher order k, for any k ≥ -1, (see Definition 1.1.1).

Date: 2004
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DOI: 10.1007/978-1-4612-2058-9_5

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