<abstract xmlns="http://www.w3.org/1999/xhtml"><p> 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, </p><p>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 <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-011-0029-x#0008_graphic2.png"></inline-graphic></inline-formula> to self. The last condition guarantees saving of the fundamental simplex. Then there are surveyed some other the m-dimensional generalizations of the Bern- stein’s polynomials and the Bernstein’s approximation theorem.</p></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, 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  to self. The last condition guarantees saving of the fundamental simplex. Then there are surveyed some other the m-dimensional generalizations of the Bern- stein’s polynomials and the Bernstein’s approximation theorem.

A generalized Bernstein approximation theorem

Mathematical Institute, Slovak Academy of Sciences, ˇStef´anikova 49, SK–814-73 Bratislava, SLOVAKIA

Tatra Mountains Mathematical Publications

{"article-title":"A generalized Bernstein approximation theorem"}

The present paper is concerned with some generalizations of Bernstein’s approximation theorem. One of the most elegant and elementary proofs of the classic...

A generalized Bernstein approximation theorem

Published Online: Nov 13, 2012

Page range: 99 - 109

DOI: https://doi.org/10.2478/v10127-011-0029-x

Keywords
Bernstein polynomial, Bernstein approximation theorem, generalized simplex

This content is open access.

A generalized Bernstein approximation theorem

Published Online: Nov 13, 2012

Page range: 99 - 109

DOI: https://doi.org/10.2478/v10127-011-0029-x

KeywordsBernstein polynomial, Bernstein approximation theorem, generalized simplex

This content is open access.

Keywords
Bernstein polynomial, Bernstein approximation theorem, generalized simplex