Disk polynomials and the one-dimensional wave equation

There is a remarkable connection between orthogonal polynomials on the unit disk and the one dimensional wave equation in the case where the wave speed is piecewise constant. The Green's function for the wave equation has an explicit formula in which Zernike polynomials and scattering polynomials serve as building blocks. Two special cases of the Green's function, corresponding to reflection and transmission, respectively, have been studied in previous work where scattering polynomials initially came to light. The present article analyzes the full Green's function, revealing the interplay between Zernike and scattering polynomials in wave propagation. An explicit formula for the resolvent of an almost-CMV matrix is obtained as a byproduct of the analysis. (C) 2019 Elsevier Inc. All rights reserved.