Abdallah Dhahri | Searcharxiv

arXiv2014

On the multi-dimensionel Favard Lemma

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.

math.FA↗

arXiv2014

Multi-variable orthogonal polynomials

We characterize the atomic probability measure on $\mathbb{R}^d$ which having a finite number of atoms. We further prove that the Jacobi sequences associated to the multiple Hermite (resp. Laguerre, resp. Jacobi) orthogonal polynomials are diagonal matrices. Finally, as a consequence of the multiple Jacobi orthogonal polynomials case, we give the Jacobi sequences of the Gegenbauer, Chebyshev and Legendre orthogonal polynomials.

math.FA↗