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arXiv · 2009.13098

Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface

Christophe Charlier
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Abstract

We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel--Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus $0$, then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.

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Christophe Charlier. 2020-09-28. Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface. https://arxiv.org/abs/2009.13098

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