Existence of Gaussian cubature formulas

Jean-Bernard Lasserre (1)

1 LAAS - Laboratoire d'analyse et d'architecture des systèmes

We provide a necessary and sufficient condition for existence of Gaussian cubature formulas. It consists of checking whether some overdetermined linear system has a solution and so complements Mysovskikh's theorem which requires computing common zeros of orthonormal polynomials. Moreover, the size of the linear system shows that existence of a cubature formula imposes severe restrictions on the associated linear functional. For fixed precision (or degree), the larger the number of variables the worse it gets. And for fixed number of variables, the larger the precision the worse it gets. Finally, we also provide an interpretation of the necessary and sufficient condition in terms of existence of a polynomial with very specific properties.

Mots clés: Orthogonal polynomials; Gaussian cubature formulas

Domaines: Optimisation et contrôle [math.OC]

Jean-Bernard Lasserre. Existence of Gaussian cubature formulas. Journal of Approximation Theory, 2012, 164 (5), p. 572-585. ⟨hal-00595007v2⟩