Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Orthogonal Laurent polynomials of two real variables

Ruymán Cruz-Barroso, Lidia Fernández

Abstract

In this paper we consider an appropriate ordering of the Laurent monomials $x^{i}y^{j}$, $i,j \in \mathbb{Z}$ that allows us to study sequences of orthogonal Laurent polynomials of the real variables $x$ and $y$ with respect to a positive Borel measure $μ$ defined on $\mathbb{R}^2$ such that $\{ x=0 \}\cup \{ y=0 \} \not\in \textrm{supp}(μ)$. This ordering is suitable for considering the {\em multiplication plus inverse multiplication operator} on each varibale $\left( x+\frac{1}{x}\right.$ and $\left. y+\frac{1}{y}\right)$, 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.

Orthogonal Laurent polynomials of two real variables

Abstract

In this paper we consider an appropriate ordering of the Laurent monomials , that allows us to study sequences of orthogonal Laurent polynomials of the real variables and with respect to a positive Borel measure defined on such that . This ordering is suitable for considering the {\em multiplication plus inverse multiplication operator} on each varibale and , 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.
Paper Structure (9 sections, 14 theorems, 113 equations)

This paper contains 9 sections, 14 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Orthogonal Laurent polynomials of two real variables. Five-term relations
  3. Favard's theorem and Christoffel-Darboux formula
  4. A connection with the one variable case
  5. Applications for future research
  6. Applications to cubature formulas
  7. Multivariate strong moment problems
  8. Conclusions
  9. Acknowledgments

Key Result

Theorem 1.1

Let $\omega$ be a positive Borel measure on $\mathbb{R}^{+}$ and let $\{ \psi_k \}_{k \geq 0}$ be the sequence of orthonormal Laurent polynomials induced by the inner product $\langle f,g \rangle_{\omega}=\int_0^{\infty} f(x)g(x)d\omega(x)$ and the balanced ordering (balanced1v). Then, setting $\psi Furthermore, $\psi_0 \equiv \frac{1}{\sqrt{m_0}}$, $\Omega_0=\frac{m_0}{m_1}$ and $C_0=\frac{\sqrt{

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • Lemma 2.7
  • ...and 20 more