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Jacobi-Piñeiro Multiple Orthogonal Polynomials on the simplex

Lidia Fernández, Ana Foulquié-Moreno, Juan Antonio Villegas

TL;DR

This work develops a Rodrigues-formula-based construction of bivariate Jacobi–Piñeiro-type multiple orthogonal polynomials on the triangle (simplex), extending the univariate theory to multivariate settings and to multiple measures. It establishes degree, symmetry, and orthogonality properties across two, multiple, and multivariate measures, providing a natural framework for bivariate Hermite–Padé approximation with a common denominator given by these polynomials. The HP analysis reveals that the bivariate case yields a non-polynomial numerator Φ^{(j)}(z,w) despite a polynomial denominator, offering explicit constructions and asymptotics for approximants. Numerical simulations illustrate the practical viability of the approach and its potential for accurate bivariate rational approximation on the triangle.

Abstract

It is known that Rodrigues formulas provide a very powerful tool to compute orthogonal polynomials with respect to classical weights. We provide an example of bivariate multiple polynomials on the simplex defined via a Rodrigues formula. This approach offers a natural generalization of Jacobi--Piñeiro polynomials to the multivariate setting. Moreover, we apply these polynomials to the study of the bivariate Hermite--Padé problem on the triangle.

Jacobi-Piñeiro Multiple Orthogonal Polynomials on the simplex

TL;DR

This work develops a Rodrigues-formula-based construction of bivariate Jacobi–Piñeiro-type multiple orthogonal polynomials on the triangle (simplex), extending the univariate theory to multivariate settings and to multiple measures. It establishes degree, symmetry, and orthogonality properties across two, multiple, and multivariate measures, providing a natural framework for bivariate Hermite–Padé approximation with a common denominator given by these polynomials. The HP analysis reveals that the bivariate case yields a non-polynomial numerator Φ^{(j)}(z,w) despite a polynomial denominator, offering explicit constructions and asymptotics for approximants. Numerical simulations illustrate the practical viability of the approach and its potential for accurate bivariate rational approximation on the triangle.

Abstract

It is known that Rodrigues formulas provide a very powerful tool to compute orthogonal polynomials with respect to classical weights. We provide an example of bivariate multiple polynomials on the simplex defined via a Rodrigues formula. This approach offers a natural generalization of Jacobi--Piñeiro polynomials to the multivariate setting. Moreover, we apply these polynomials to the study of the bivariate Hermite--Padé problem on the triangle.
Paper Structure (9 sections, 8 theorems, 133 equations, 1 figure)

This paper contains 9 sections, 8 theorems, 133 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries: Multiple Orthogonal Polynomials
  3. Jacobi--Piñeiro polynomials on the Triangle
  4. The two-measure case
  5. The general multi‑measure case
  6. The multivariate case
  7. Hermite--Padé approximation
  8. Numerical Simulations
  9. Conclusions

Key Result

Lemma 3.3

Given $l,m,n\geq0$ with $n\geq l+m$, consider $x^l y^m (1-x-y)^{n-l-m}$, a polynomial of degree $n$. For a fixed $j$, let $D_j$ be the operator introduced in eq:operator-D_j. If $n>n_j$, then $D_j[x^l y^m (1-x-y)^{n-l-m}]$ is a linear combination of polynomials of degree $n$, therefore a polynomial

Figures (1)

  • Figure 1: First row: plot of $E_j$ (blue) and the approximation $R_j$ (yellow). Second row: plot of the absolute error $\left|E_j-R_j\right|$. Third row: plot of the relative error $\frac{\left|E_j-R_j\right|}{E_j}$.

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 11 more