5 papers · 1 filter
Exact Finite Integral Geometry: Directional Kernels and Spherical Designs
Congpei An
We study when an invariant directional average from integral geometry can be replaced exactly by finitely many directions. For an even continuous kernel , let …
Dual Geometry of Spherical Designs: Polarity, Self-Polar Rigidity, and Quadrature Structure
Congpei An
Let $X=\{x_1,\ldots,x_N\}\subset\Sph^{d-1}$ be a spherical -design, , and let be its Minkowski polytope. If , then \[ P_X^\circ=\conv\{…
Minkowski Polytopes of Spherical Designs: High-Order Isotropy and Quantitative Sphericity
Congpei An
Let $X_N=\{x_1,\dots,x_N\}\subset \Sph^2$ be a spherical -design of strength , and let be its empirical measure. Minkowski's theorem associates with a convex…
A Survey on Spherical Designs: Existence, Numerical Constructions, and Applications
Congpei An, Xiaosheng Zhuang
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…
On the role of weak Marcinkiewicz-Zygmund constants in polynomial approximation by orthogonal bases
Congpei An, Alvise Sommariva, Marco Vianello
We compute numerically the Marcinkiewicz-Zygmund constants of cubature rules, with a special attention to their role in polynomial approximation by orthogonal bases. We test…