Section: New Results

Approximation theory

In [40], M. Herda et al. propose a new iterative algorithm for the calculation of sum of squares decompositions of polynomials, reformulated as positive interpolation. The method is based on the definition of a dual functional $G$ from values at interpolation points. The domain of $G$, the boundary of the domain and the behavior of $G$ at infinity are analyzed in details. In the general case, $G$ is closed convex. For univariate polynomials in the context of the Lukacs representation, $G$ is coercive and strictly convex which yields a unique critical point, corresponding to a sum of squares decomposition of $G$. Various descent algorithms are evoked. Numerical examples are provided, for univariate and bivariate polynomials.