Averaging sets: A generalization of mean values and spherical designs. (English) Zbl 0596.05012

Seymour, P. D.; Zaslavsky, Thomas

Adv. Math. 52, 213-240 (1984).

Let F denote a set of continuous \(f: \Omega \to {\mathbb{R}}^ p\), where \(\Omega\) is a path-connected topological space with a suitable measure \(\mu\). An averaging set for F is a finite set \(X\subset \Omega\) such that the average sum of f over X equals the average integral of f over \(\Omega\), for all \(f\in F\). The main theorem states that for given \(\Omega\), \(\mu\), F, there exist averaging sets X of any size, with finitely many exceptions. In the main corollary, F is the space of all polynomials of degree \(\leq t\) on the unit sphere \(\Omega\) in \({\mathbb{R}}^ d\); the theorem then says that for any positive integers d and t, and for sufficiently large n, there exist spherical designs of strength t and of size n in d-space. Previously, this existence had only been known for \(d=1\) and all t, and for \(d\geq 2\) and some values of \(t\leq 19.\)
The main theorem also applies to (moment) t-designs on any bounded region \(\Omega\), to response surface designs, and to rotatable designs. Although nonconstructive, the main theorem is an important advance. The average integral over \(\Omega\) of \(f: \Omega \to {\mathbb{R}}^ p\) may be simplified to a constant, say 0, in the interior of the convex hull of the image under f of certain points \(y_ 0,y_ 1,...,y_ p\in \Omega\). There are several proofs of the main theorem in the paper. The proof for analytic functions uses analytic curves on \(\Omega\) and the implicit function theorem to find points \(x_ 1,x_ 2,...,x_ p\in \Omega\) such that \(n_ 0f(y_ 0)+\sum^{p}_{i=1}n_ if(x_ i)=0\), \(n_ i\in {\mathbb{N}}\), and then makes the averaging multiset into an averaging set. This proof is adapted to the case of continuously differentiable functions. Further proofs are given for continuous functions with (and later without) the restriction that any \(p+1\) distinct points in \(\Omega\) lie on a simple curve. In the final sections questions such as the following are considered: Can averaging sets be found which depend on f in a locally continuous way? Is the minimum size of an averaging set locally bounded? What is the minimum size of an averaging set, or multiset? Can the main theorem be generalized to discontinuous functions? The authors give partial results and make some conjectures about these questions. This is an important and interesting paper.

MSC:

05B30	Other designs, configurations
26B15	Integration of real functions of several variables: length, area, volume

Keywords:

averaging sets; continuous functions

References:

[1]	Abraham, R.; Robbin, J., Transversal Mappings and Flows (1967), Benjamin: Benjamin New York · Zbl 0171.44404
[2]	Box, G. E.P; Draper, N. R., A basis for the selection of a response surface design, J. Amer. Statist. Assoc., 54, 622-654 (1959) · Zbl 0116.36804
[3]	Box, G. E.P; Hunter, J. S., Multi-factor experimental designs for exploring response surfaces, Ann. Math. Statist., 28, 195-241 (1957) · Zbl 0080.35901
[4]	Delsarte, P.; Goethals, J. M.; Seidel, J. J., Spherical codes and designs, Geom. Dedicata, 6, 363-388 (1977) · Zbl 0376.05015
[5]	Greenberg, M., Lectures on Algebraic Topology (1967), Benjamin: Benjamin New York · Zbl 0169.54403
[6]	Hong, Y., On spherical \(t\)-designs in \(R^2\), European J. Combin., 3, 255-258 (1982) · Zbl 0508.05026
[7]	Willard, S., General Topology (1970), Addison-Wesley: Addison-Wesley Reading, Mass · Zbl 0205.26601