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

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.

Радиопоглощающий метаматериал и экран антенны на его основе

Metamaterial absorber located near solid metal surface was investigated and antenna design was developed for receiving satellite navigation signals using absorptive ground plane comprises proposed material. The metamaterial structure has been developed, frequency characteristics have been analyzed by finite element method in frequency domain, operating dimensions of elements have been established. Modeling and analysis of radiation pattern for antenna with absorptive ground plane versus the antenna with the classic high impedance ground plane were carried out. Conclusions were drawn on the applicability of the proposed metamaterial structure.