2018/11/07 by Philip Greengard, Greengard, Philip, Kirill Serkh +1 · 1 citation
Mathematics · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1811.02720
openalex publication_date 2018/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. They arise in a number of applications including optics and atmospheric sciences. In this paper, we provide a self-contained reference on Zernike polynomials, algorithms for evaluating them, and what appear to be new numerical schemes for quadrature and interpolation. We also introduce new properties of Zernike polynomials in higher dimensions. The quadrature rule and interpolation scheme use a tensor product of equispaced nodes in the angular direction and roots of certain Jacobi polynomials in the radial direction. An algorithm for finding the roots of these Jacobi polynomials is also described. The performance of the interpolation and quadrature schemes is illustrated through numerical experiments. Discussions of higher dimensional Zernike polynomials are included in appendices.