Uniform approximation and Bernstein polynomials with coefficients in the unit interval (Q627934)

Taking into account the authors' research on probabilistic computation with digital circuits, they prove for a given power-form polynomial \(g\) that there exists a Bernstein polynomial of degree \(m\) with coefficients closed to the corresponding values \(g(k/m)\), \(k= 0,1,\dots, m\), where \(m\) is sufficiently large. Further, it is proved that the set of Bernstein polynomials with coefficients in \([0,1]\) is identical with the set of all polynomials which are either identically equal to \(0\) or equal to \(1\), or map \((0,1)\) into \((0,1)\) and the points \(0\) and \(1\) into \([0,1]\).