Christoffel Transform and Multiple Orthogonal Polynomials

TL;DR

This paper explores how Christoffel transforms affect multiple orthogonal polynomials, providing algorithms, formulas, and interlacing properties that deepen understanding of their zeros and recurrence relations.

Contribution

It introduces a new algorithm for computing transformed recurrence coefficients and establishes interlacing properties of zeros for transformed multiple orthogonal polynomials.

Findings

Zeros of transformed polynomials interlace with original zeros

New algorithm for Jacobi coefficient computation

Interlacing properties extend classical orthogonal polynomial results

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.