A generalized Bernstein approximation theorem

Year, pages: 2011, 99 - 109

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 $f$,

$$ B_n(x)=B_n^f(x)=∑_k=0ⁿ f(((k) / (n)))\binom{n}{k}x^k(1-x)^n-k $$

We shall concern the $m$-dimensional generalization of the Bernstein's polynomials and the Bernstein's approximation theorem by taking an $(m-1)$-dimensional simplex in cube $[0,1]^m$. This is motivated by the fact that in the field of mathematical biology naturally arouse dynamic systems determined by quadratic mappings of ``standard" $ (m-1)$-dimensional simplex $\{x_i ≥ 0$, $i=1,…,m$, $∑_i=1^m x_i=1 \}$ to self. The last condition guarantees saving of the fundamental simplex. Then there are surveyed some other the $m$-dimensional generalizations of the Bernstein's polynomials and the Bernstein's approximation theorem.

Duchoň, M. 2011. A generalized Bernstein approximation theorem. In Tatra Mountains Mathematical Publications, vol. 49, no.2, pp. 99-109. 1210-3195.

Duchoň, M. (2011). A generalized Bernstein approximation theorem. Tatra Mountains Mathematical Publications, 49(2), 99-109. 1210-3195.