Structure of non-negative polynomials and optimization problems

In this paper we give an explicit description of the cone of univariate non-negative polynomials in terms of the cone of positive semidefinite matrices. As a consequence of that result, we get a description of the cones of polynomials, which are non-negative on a ray and on an interval, and the description of non-negative trigonometric polynomials. We present some extension of these results on multi-dimensional case.