Year: 2024
Type: article
Abstract: . A spherical [Formula: see text]-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 [Formula: see text] and has a sharp error bound for approximations on the sphere. This paper introduces a set of points called a spherical cap [Formula: see text]-subdesign on a spherical cap [Formula: see text] with center [Formula: see text] and radius [Formula: see text] induced by the spherical [Formula: see text]-design. We show that the spherical cap [Formula: see text]-subdesign provides an equal weight quadrature rule integrating exactly all zonal polynomials of degree at most [Formula: see text] and all functions expanded by orthonormal functions on the spherical cap which are defined by shifted Legendre polynomials of degree at most [Formula: see text]. We apply the spherical cap [Formula: see text]-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 [Formula: see text]-subdesigns to sparse signal recovery on the upper hemisphere, which is a spherical cap with [Formula: see text]. Our theoretical and numerical results show that spherical cap [Formula: see text]-subdesigns can provide a good approximation on spherical caps.
DOI: 10.1137/23m1555417
Authors: Chao Li, Xiaojun Chen
Institutions: Hong Kong Polytechnic University, Taiyuan Normal University
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G276400187
OpenAlex ID: W4404224009