Journal of Complexity· 2026Q1· Review
A survey on spherical designs: Existence, numerical constructions and applications
- 0citations
- Q1SCImago
- 2026year
Short summary
This survey reviews spherical designs, finite sets on a sphere that enable exact integration of polynomials up to a certain degree, focusing on numerical analysis aspects like construction and stability.
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Key points
- Spherical designs are finite sets on a sphere enabling exact integration of spherical polynomials up to degree t.
- The survey covers cardinality bounds, existence, and nonexistence of spherical designs.
- It details certified and large-scale numerical construction techniques.
- Connections between polynomial exactness, covering/separation geometry, conditioning, and numerical stability are emphasized.
- Applications include numerical integration, interpolation, and spherical signal processing.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Journal of Complexity, 2026 · DOI ↗
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Field: Numerical Analysis
Numerical AnalysisMathematics