On the multi-dimensionel Favard Lemma

Abstract

We prove that of 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, $d \geq 1$) Favard Lemmas. Moreover, we investigate the dependence of the Jacobi sequences on the linear change of basis of $\mathbb{C}^d$. Finally we prove that the Jacobi sequences associated to the probability measure product on $\mathbb{R}^d$ are diagonal matrices in the basis introduced by the tensor product of the orthogonal polynomials of the factor measures.