Orthogonal Laurent Polynomials of Two Real Variables

Abstract

In this paper, we consider an appropriate ordering of the Laurent monomials $x^i{y}^j$, $i,j\in Z$ that allows us to study sequences of orthogonal Laurent polynomials of the real variables $x$ and $y$ with respect to a positive Borel measure $\mu$ defined on $R^2$ such that $\left(\right\{x=0\}\cup \{y=0\left\}\right)\cap \mathrm{supp}\left(\mu \right)=\varnothing$. This ordering is suitable for considering the multiplication plus inverse multiplication operator on each variable $(x+\frac{1}{x}$ and $y+\frac{1}{y})$, and as a result we obtain five-term recurrence relations, Christoffel–Darboux and confluent formulas for the reproducing kernel and a related Favard's theorem. A connection with the one-variable case is also presented, along with some applications for future research.