, Volume 256, Issue 1, pp 99-112
Date: 08 Nov 2006

SOS approximations of nonnegative polynomials via simple high degree perturbations

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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.