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.

31 pages

References in corpus (8)

• The nearest neighbor recurrence coefficients for multiple orthogonal polynomials
• Interlacing properties of zeros of multiple orthogonal polynomials
• Discrete integrable systems generated by Hermite-Padé approximants
• Multiple orthogonal polynomials: Pearson equations and Christoffel formulas
• Inverse Spectral Problems for Tridiagonal N by N Complex Hamiltonians
• Multidimensional Toda Lattices: Continuous and Discrete Time
• Computing recurrence coefficients of multiple orthogonal polynomials
• Interlacing and monotonicity of zeros of Angelesco-Jacobi polynomials

Cited by in corpus (1)

• Zeros of Multiple Orthogonal Polynomials: Location and Interlacing