Spherical designs for approximations on spherical caps

Abstract

A spherical t-design is a set of points on the unit sphere, which provides an equal weight quadrature rule integrating exactly all spherical polynomials of degree at most t and has a sharp error bound for approximations on the sphere. This paper introduces a set of points called a spherical cap t-subdesign on a spherical cap C(e3,r) with center e3=(0,0,1)⊤ and radius r∈(0,π) induced by the spherical t-design. We show that the spherical cap t-subdesign provides an equal weight quadrature rule integrating exactly all zonal polynomials of degree at most t and all functions expanded by orthonormal functions on the spherical cap which are defined by shifted Legendre polynomials of degree at most t. We apply the spherical cap t-subdesign and the orthonormal basis functions on the spherical cap to non-polynomial approximation of continuous functions on the spherical cap and present theoretical approximation error bounds. We also apply spherical cap t-subdesigns to sparse signal recovery on the upper hemisphere, which is a spherical cap with r=0.5π. Our theoretical and numerical results show that spherical cap t-subdesigns can provide a good approximation on spherical caps.