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

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. Let (Ω, σ) be a finite measure space and let F be a vector space of integrable real-valued functions on Ω. It is a natural question to ask when and how integrals ∫ Ω φ(ω)dσ(ω) can be computed, or approximated, by sums ∑ x∈X W (x)φ(x), where X is a subset of Ω and W : X −→ R+ a weight function, for all φ ∈ F . When Ω is an interval of the real line, this is a basic problem of numerical integration with a glorious list of contributors: Newton (1671), Cotes (1711), Simpson (1743), Gauss (1814), Chebyshev (1874), Christoffel (1877), and Bernstein (1937), to quote but a few; see [Gauts–97] for some historical notes, [Gauts–76], and [DavRa–84]. The theory is inseparable of that of orthogonal polynomials [Szegö–39]. When Ω is a space of larger dimension, the problems involved are often of geometrical and combinatorial interest. One important case is that of spheres in Euclidean spaces, with rotation-invariant probability measure, as in [DeGoS–77] and [GoeSe–79]. Work related to integrations domains with dim(Ω) > 1 goes back to [Maxwe–77] and [Appel–90]; see also the result of Voronoi (1908) recalled in Item 1.15 below. Examples which have been considered include various domains in Euclidean spaces (hypercubes, simplices, ..., Item 1.20), surfaces (e.g. tori), and Euclidean spaces with Gaussian measures (Item 3.11). There are also interesting cases where Ω itself is a finite set [Delsa–78]. Section 1 collects the relevant definitions for the general case (Ω, σ). It reviews several known examples on intervals and spheres. Our main point is to show that there are two notions which are convenient for the study of cubature formulas, even if they are rarely explicitely introduced in papers of this subject. First, we introduce in Section 2 the formalism of reproducing kernel Hilbert spaces ([Arons–50], [Krein–63], [BeHaG–03]), which is appropriate for generalizing to other spaces 1991 Mathematics Subject Classification. Primary 05B30, 65D32. Secondary 46E22.