Chapter

Infinite Groups: Geometric, Combinatorial and Dynamical Aspects

Volume 248 of the series Progress in Mathematics pp 219-267

Cubature Formulas, Geometrical Designs, Reproducing Kernels, and Markov Operators

  • Pierre de la HarpeAffiliated withSection de Mathématiques, Université de Genève
  • , Claude PacheAffiliated withSection de Mathématiques, Université de Genève

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Abstract

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.

Keywords

Cubature formulas spherical designs reproducing kernels Markov operators group representations reflection groups