A Survey on Spherical Designs: Existence, Numerical Constructions, and Applications

Abstract: This paper provides a survey of spherical designs and their applications, with a particular emphasis on the perspective of ``numerical analysis''. A set (X_N) of (N) points on the unit sphere (\mathbb{S}^d) is called a \textit{spherical (t)-design} if the average value of any polynomial of degree at most (t) over (X_N) equals its average over the entire sphere. Spherical designs represent one of the most significant topics in the study of point distributions on spheres. They are deeply connected to algebraic combinatorics, discrete geometry, differential geometry, approximation theory, optimization, coding theory, quantum physics, and other fields, which have led to the development of profound and elegant mathematical theories. This article reviews fundamental theoretical results, numerical construction methods, and applied outcomes related to spherical designs. Key topics covered include existence proofs, optimization-based construction techniques, fast computational algorithms, and applications in interpolation, numerical integration, hyperinterpolation, signal and image processing, as well as numerical solutions to partial differential and integral equations.