Multivariate orthogonal polynomials: quantum decomposition, deficiency rank and support of measure

TL;DR

This paper explores multivariate orthogonal polynomials through quantum operator decompositions, linking measure support to the deficiency rank of Jacobi operators, and extends Favard's theorem in this framework.

Contribution

It introduces a quantum operator approach to multivariate orthogonal polynomials, establishing measure reconstruction and support characterization via Jacobi operator rank.

Findings

Deficiency rank of Jacobi operators indicates measure support on algebraic surfaces.

Develops a Favard-type theorem in the quantum operator setting.

Provides examples illustrating the measure support and operator properties.

Abstract

In this paper we investigate the multivariate orthogonal polynomials based on the theory of interacting Fock spaces. Our framework is on the same stream line of the recent paper by Accardi, Barhoumi, and Dhahri \cite{ABD}. The (classical) coordinate variables are decomposed into non-commuting (quantum) operators called creation, annihilation, and preservation operators, in the interacting Fock spaces. Getting the commutation relations, which follow from the commuting property of the coordinate variables between themselves, we can develop the reconstruction theory of the measure, namely the Favard's theorem. We then further develop some related problems including the marginal distributions and the rank theory of the Jacobi operators. We will see that the deficiency rank of the Jacobi operator implies that the underlying measure is supported on some algebraic surface and vice versa. We will provide with some examples.