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

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

2005-02-15

arxiv.org/abs/math/0502312v1

Mathematics

Combinatorics

Scientific paper

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.

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