A generalized Bernstein approximation theorem

Abstract

The present paper is concerned with some generalizations of Bernstein's approximation theorem.
One of the most elegant and elementary proofs of the classic result, for a function $f(x)$ defined on the closed interval $[0,1]$, uses the Bernstein's polynomials of $f$,
$$ %\begin{equation}
B_n(x)=B_n^f(x)=\sum_{k=0}^n f\left(\frac {k}{n}\right)\binom{n}{k}x^k(1-x)^{n-k}
$$ %\end{equation}
We shall concern the $k$-dimensional generalization of the Bernstein's polynomials and the
Bernstein's approximation theorem by taking a $(k-1)$-dimensional simplex in cube $[0,1]^k$.
This is motivated by the fact that in the field of mathematical biology naturally arouse dynamic systems determined by quadratic mappings of "standard" $(k-1)$-dimensional simplex $\{ x_i \ge 0, i=1,\dots,n, \sum_{i=1}^n x_i=1 \}$
to self. The last condition guarantees saving of the fundamental simplex. Then there are surveyed some
other the $k$-dimensional generalizations of the Bernstein's polynomials and the
Bernstein's approximation theorem.