Informácia o dokumente



In: Tatra Mountains Mathematical Publications, vol. 49, no. 2
Miloslav Duchoň

A generalized Bernstein approximation theorem

Detaily:

Rok, strany: 2011, 99 - 109
Kľúčové slová:
Bernstein polynomial, Bernstein approximation theorem, generalized

O článku:

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 $f$,

$$ Bn(x)=Bnf(x)=∑k=0n f(((k) / (n)))\binom{n}{k}xk(1-x)n-k $$

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 $\{xi ≥ 0$, $i=1,…,m$, $∑i=1m xi=1 \}$ 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.

Ako citovať:

ISO 690:
Duchoň, M. 2011. A generalized Bernstein approximation theorem. In Tatra Mountains Mathematical Publications, vol. 49, no.2, pp. 99-109. 1210-3195.

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