Spherical Designs for Approximations on Spherical Caps

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2506-2528, December 2024.
Abstract. A spherical [math]-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 [math] and has a sharp error bound for approximations on the sphere. This paper introduces a set of points called a spherical cap [math]-subdesign on a spherical cap [math] with center [math] and radius [math] induced by the spherical [math]-design. We show that the spherical cap [math]-subdesign provides an equal weight quadrature rule integrating exactly all zonal polynomials of degree at most [math] and all functions expanded by orthonormal functions on the spherical cap which are defined by shifted Legendre polynomials of degree at most [math]. We apply the spherical cap [math]-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 [math]-subdesigns to sparse signal recovery on the upper hemisphere, which is a spherical cap with [math]. Our theoretical and numerical results show that spherical cap [math]-subdesigns can provide a good approximation on spherical caps.