Uniform approximation and Bernstein polynomials with coefficients in the unit interval

This paper presents two main results. The first result pertains to uniform approximation with Bernstein polynomials. We show that, given a power-form polynomial gg, we can obtain a Bernstein polynomial of degree mm with coefficients that are as close as desired to the corresponding values of gg evaluated at the points 0,1m,2m,…,1, provided that mm is sufficiently large. The second result pertains to a subset of Bernstein polynomials: those with coefficients that are all in the unit interval. We show that polynomials in this subset map the open interval (0,1)(0,1) into the open interval (0,1)(0,1) and map the points 0 and 1 into the closed interval [0,1][0,1]. The motivation for this work is our research on probabilistic computation with digital circuits. Our design methodology, called stochastic logic, is based on Bernstein polynomials with coefficients that correspond to probability values; accordingly, the coefficients must be values in the unit interval. The mathematics presented here provides a necessary and sufficient test for deciding whether polynomial operations can be implemented with stochastic logic.