Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface. Issue 2 (31st December 2021)

• Record Type: Journal Article
• Authors: Charlier, Christophe
• Is Part Of: Transactions of mathematics and its applications. Volume 5:Issue 2(2021)
• Publication Date: 2021-12-31
• Subjects: matrix orthogonal polynomials -- Riemann surfaces -- tiling models; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1093/imatrm/tnab004
• Languages: English
• ISSNs: 2398-4945

Abstract

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.