Speaker
Description
Bernstein operators are associated with Bernoulli scheme as follows: $$B_n(f,x)=\mathbb{E}\left(f\circ Z(n,x)\right),$$ where $Z(n,x)=\frac{1}{n}\sum_{i=1}^n Y(i,x),$ $Y(i,x)$ is the sequence of independent Bernoulli random variables with parameters $P{Y(i,x)=1}=x$ and $P{Y(i,x)=0}=1-x.$ V.S. Videnskii in a series of papers studied generalizations of Bernstein operators to the case of rational functions. They can be written in the form $$V_n(f,x)=\mathbb{E}\left(f\circ(\mathbb{E}Z(n,x))^{-1}\circ Z(n,x)\right),$$ where $Y(i,n)$ has now parameters $p_{in}(x)=\frac{\rho_{in}x}{1+\rho_{in}-x},$ $ \rho_{i,n}>0, $ instead of $x.$
We give a survey of results to compare approximation properties of Bernstein and Videnskii type generalizations for one or several intervals, and for the semi-axis.
