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.