Squared Functional Systems and Optimization Problems

In this paper we give an explicit description of the cones of polynomials represent able as a sum of squared functions. We prove that such cones can be always seen as a linear image of the cone of positive semidefinite matrices. As a consequence of the result, we get a description of the cones of univariate polynomials, which are non-negative on a ray and on an interval, and a description of non-negative trigonometric polynomials. We discuss some applications of the results to multi-variate polynomials.