Cubature formulas, geometrical designs, reproducing kernels, and Markov operators. (English) Zbl 1124.05019
Bartholdi, Laurent (ed.) et al., Infinite groups: geometric, combinatorial and dynamical aspects. Based on the international conference on group theory: geometric, combinatorial and dynamical aspects of infinite groups, Gaeta, Italy, June 1–6, 2003. Basel: Birkhäuser (ISBN 3-7643-7446-2/hbk). Progress in Mathematics 248, 219-267 (2005).
Authors’ 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 Eucledean 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.
For the entire collection see [Zbl 1083.20500].
For the entire collection see [Zbl 1083.20500].
Reviewer: Rudolf Scherer (Karlsruhe)
MSC:
05B30 | Other designs, configurations |
20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
65D32 | Numerical quadrature and cubature formulas |
46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |