Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Christoffel Transform and Multiple Orthogonal Polynomials

Rostyslav Kozhan, Marcus Vaktnäs

Abstract

We investigate multiple orthogonal polynomials associated with the system of measures obtained by applying a Christoffel transform to each of the orthogonality measures. We present an algorithm for computing the transformed recurrence coefficients and determinantal formulas for the transformed multiple orthogonal polynomials of type I and type II. We apply these results to show that zeros of multiple orthogonal polynomials of an Angelesco or an AT system interlace with the zeros of the polynomials corresponding to its one-step Christoffel transform. This allows us to prove a number of interlacing properties satisfied by the multiple orthogonality analogues of classical orthogonal polynomials. For the discrete polynomials, this also produces an estimate on the smallest distance between consecutive zeros. We also identify a connection between the Christoffel transform of orthogonal polynomials and multiple orthogonality systems containing a finitely supported measure. In consequence, the compatibility relations for the nearest neighbour recurrence coefficients provide a new algorithm for the computation of the Jacobi coefficients of the one-step or multi-step Christoffel transforms.

Christoffel Transform and Multiple Orthogonal Polynomials

Abstract

We investigate multiple orthogonal polynomials associated with the system of measures obtained by applying a Christoffel transform to each of the orthogonality measures. We present an algorithm for computing the transformed recurrence coefficients and determinantal formulas for the transformed multiple orthogonal polynomials of type I and type II. We apply these results to show that zeros of multiple orthogonal polynomials of an Angelesco or an AT system interlace with the zeros of the polynomials corresponding to its one-step Christoffel transform. This allows us to prove a number of interlacing properties satisfied by the multiple orthogonality analogues of classical orthogonal polynomials. For the discrete polynomials, this also produces an estimate on the smallest distance between consecutive zeros. We also identify a connection between the Christoffel transform of orthogonal polynomials and multiple orthogonality systems containing a finitely supported measure. In consequence, the compatibility relations for the nearest neighbour recurrence coefficients provide a new algorithm for the computation of the Jacobi coefficients of the one-step or multi-step Christoffel transforms.
Paper Structure (33 sections, 29 theorems, 131 equations, 2 figures)

This paper contains 33 sections, 29 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Orthogonal polynomials with respect to moment functionals
  4. $\mu$ associated with finite (complex) Jacobi matrices
  5. Christoffel transform
  6. Basics of multiple orthogonal polynomials on the real line (MOPRL)
  7. MOPRL: nearest neighbour recurrence relations
  8. Normality criteria
  9. Appendix: Proof of Theorem \ref{['thm:nnrTypeI']}
  10. Main results
  11. Christoffel Transforms of Type II Polynomials
  12. Nearest Neighbour Recurrence Coefficients for the Christoffel Transforms
  13. Two examples
  14. An example on the step-line
  15. Christoffel Transforms of Type I polynomials
  16. ...and 18 more sections

Key Result

Lemma 2.1

Suppose $\mu\in\mathcal{L}_N$ for some $N\in\mathbb{Z}_{+}$. Then Moreover, we have if and only if $P(x)$ is divisible by $P_N(x)$.

Figures (2)

  • Figure 1: The recurrence coefficients of $(\mu_1,\mu_2)$ can be computed via the CC algorithm (Theorem \ref{['thm:NNCC']}). On the upper boundary we find the recurrence coefficients of ${\widehat{\mu}}_1$.
  • Figure 2: Along the blue vertical arrows we find the Christoffel transforms $\Phi_1\mu,(\Phi_1\Phi_2)\mu,(\Phi_1\Phi_2\Phi_3)\mu,\dots$. The Jacobi coefficients of each transform can be computed via the NNCC algorithm.

Theorems & Definitions (72)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • ...and 62 more