A survey on spherical designs: Existence, numerical constructions and applications

This survey reviews spherical designs from a numerical-analysis perspective while retaining their connections with algebraic combinatorics and discrete geometry. A finite set X N ⊂ S d is a spherical t -design when equal-weight integration over X N is exact for all spherical polynomials of degree at most t . We organize the subject around three themes: cardinality bounds, existence and nonexistence; certified and large-scale numerical construction; and approximation and computational applications. Particular attention is paid to the relations among polynomial exactness, covering and separation geometry, conditioning, and numerical stability. We also discuss applications to numerical integration, interpolation, hyperinterpolation, differential and integral equations, spherical signal and image processing.