On square positive extensions and cubature formulas

We consider linear functionals LdLd defined over Pd⊂P≔R[x1,…,xn]Pd⊂P≔R[x1,…,xn] and study how to extend LdLd to a square positive L:P→R, i.e. to a linear functional L   with L(p2)⩾0L(p2)⩾0 for all p∈Pp∈P. We use the connection of square positive functionals to real ideals, to orthogonal polynomials, and to moment matrices. Finally, we report how such extension techniques are used to construct and analyse cubature formulas.