SOS approximations of nonnegative polynomials via simple high degree perturbations

Abstract

We show that every real polynomial f nonnegative on [−1,1]ⁿ can be approximated in the l ₁-norm of coefficients, by a sequence of polynomials ${\{f_{\epsilon r}\}}$ that are sums of squares (s.o.s). This complements the existence of s.o.s. approximations in the denseness result of Berg, Christensen and Ressel, as we provide a very simple and explicit approximation sequence. Then we show that if the moment problem holds for a basic closed semi-algebraic set ${K_{S} \subset \mathbb{R}^n}$ with nonempty interior, then every polynomial nonnegative on K _S can be approximated in a similar fashion by elements from the corresponding preordering. Finally, we show that the degree of the perturbation in the approximating sequence depends on ${\epsilon}$ as well as the degree and the size of coefficients of the nonnegative polynomial f, but not on the specific values of its coefficients.