2005/02/15 by Pierre De La Harpe, De La Harpe, Pierre, Claude Pache +1
Mathematics · #65D32. Secondary 46E22 #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Point processes and geometric inequalities #Primary 05B30 #math.CO #msc:05B30 #msc:46E22 #msc:65D32.
paper · pdf · doi:10.48550/arxiv.math/0502312
arxiv created 2005/02/15 · openalex publication_date 2005/02/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces (Ω,σ) and appropriate spaces of functions inside L2(Ω,σ). The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups.