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English

Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

PublisherBasel : Birkhäuser
Collection
  • Progress in Mathematics; 248
Publication date2005
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 $(Omega,sigma)$ and appropriate spaces of functions inside $L^2(Omega,sigma)$. 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
Classification
  • arxiv : math.CO
Citation (ISO format)
DE LA HARPE, Pierre, PACHE, Claude. Cubature formulas, geometrical designs, reproducing kernels, and Markov operators. In: Infinite groups: geometric, combinatorial and dynamical aspects. Basel : Birkhäuser, 2005. p. 219–267. (Progress in Mathematics) doi: 10.1007/3-7643-7447-0_6
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