On square positive extensions and cubature formulas

We consider linear functionals L d defined over P d ⊂ P ≔ R [ x 1 , … , x n ] and study how to extend L d to a square positive L : P → R , i.e. to a linear functional L with L ( p 2 ) ⩾ 0 for all p ∈ 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.