On the multi-dimensional Favard lemma

Abstract

We prove that the creator operators, on the d commuting indeterminates polynomial algebra, are linearly independent. We further study the connection between the classical (one dimensional) and the multi-dimensional (d-dimensional, d ≥ 1) Favard Lemmas. Moreover, we investigate the dependence of the Jacobi sequences on the linear change of basis of C d . Finally we prove that the Jacobi sequences associated to a product probability measure on R d , with finite moments of any order, are diagonal matrices in the basis introduced by the tensor product of the orthogonal polynomials of the factor measures (see page 21).