A generalized Bernstein approximation theorem

Duchoň, Miloslav

Tatra Mountains Mathematical Publications, Tome 49 (2011), / Harvested from Mathematical Institute

Résumé

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 theBernstein'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 someother the $k$-dimensional generalizations of the Bernstein's polynomials and theBernstein's approximation theorem.

Détail

Publié le : 2011-01-01
DOI : https://doi.org/10.2478/tatra.v49i0.114