A generalized Bernstein approximation theorem

Miloslav Duchoň

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.

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DOI: https://doi.org/10.2478/tatra.v49i0.114