Remarks on Hilbert identities, isometric embeddings, and invariant cubature

H. Nozaki, M. Sawa
2014 St. Petersburg Mathematical Journal  
In 2004, Victoir developed a method to construct cubature formulas with various combinatorial objects. Motivated by this, the authors generalize Victoir's method with yet another combinatorial object, called the regular t-wise balanced design. Many cubature formulas of small indices with few points are provided, which are used to update Shatalov's table (2001) of isometric embeddings in small-dimensional Banach spaces, as well as to improve some classical Hilbert identities. A famous theorem of
more » ... Bajnok (2007) on Euclidean designs invariant under the Weyl group of Lie type B is extended to all finite irreducible reflection groups. A short proof of the Bajnok theorem is presented in terms of Hilbert identities. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 616 H. NOZAKI AND M. SAWA But most of them, including the original proof by Hilbert, involve nonconstructive arguments in analysis, and do not give any explicit constructions of embeddings. 1 Thus, publications with explicit embeddings continued to appear. Isometric embeddings are also related to a certain object in numerical analysis. Let Ω be a subset of R m on which a normalized measure μ is defined. A finite subset X of Ω with a positive weight w is called a cubature formula of index q if for every f ∈ Hom q (Ω), where Hom q (Ω) is the space of all homogeneous polynomials of degree q restricted to Ω. Lyubich and Vaserstein [22] and Reznick [27] proved the equivalence between an embedding l m 2 → l n q and an n-point cubature formula of index q for the surface measure ρ on the (m − 1)-dimensional unit sphere S m−1 . Many papers are devoted to the construction of spherical cubature formulas. There are two classical approaches. One employs orbits of finite subgroups of the orthogonal group O(m) acting on S m−1 [33] , and the other takes a "product" of several lower-dimensional cubatures [34] . Cubature formulas that are studied in the context of numerical analysis and related areas, are often of degree type. Victoir [35] developed a novel technique to construct degree-type cubature for integrals with special symmetry. His idea is as follows. Given a cubature formula invariant under the Weyl group of Lie type B, one eliminates some specified points of the formula by using combinatorial objects such as t-designs and orthogonal arrays. With this method, Victoir found many cubature formulas of small degrees with few points in general dimensional spaces. In this paper, we have several important aims. First, we generalize the Victoir method to a special class of block designs, called regular t-wise balanced designs. The concept of a regular t-wise balanced design has been substantiated by applications in statistics [8, 10, 18] , however, it seems that there is insufficient evidence to support it from other mathematical aspects. To find a new meaning of this concept, as well as to let researchers in many areas of mathematics know it, are among our important aims in this paper. On the other hand, Bajnok [1, Theorem 3] proved that Euclidean designs, a generalization of the spherical cubatures, that are invariant under the Weyl group of Lie type B have degree at most 7. We further discuss the Bajnok theorem both from a combinatorial and analytic point of view. This paper is organized as follows. In §2 we review some basic facts and notions, and explain the Victoir method in detail. In §3 we generalize the Victoir method to regular t-wise balanced designs. In §4, we give general-dimensional index-four and -six cubature formulas, together with some additional examples of index-six cubature formulas that improve Shatalov's table [32, Theorem 4.7.20] of isometric embeddings l m 2 → l n 6 . In §5, we generalize the Bajnok theorem for all finite irreducible reflection groups, and thereby classify the spherical cubature formulas with a certain geometric meaning. In §6, some of the cubatures constructed in §4 and §5 are translated into Hilbert identities, in order to improve classical identities as those by Schur [6] and Reznick [27] . An extremely short proof of the Bajnok theorem is given in terms of Hilbert identities. §2. Preliminaries Isometric embeddings and Hilbert identities. Lyubich and Vaserstein [22] and Reznick [27] observed a close relationship between Hilbert identities, isometric embeddings, and spherical cubature formulas.
doi:10.1090/s1061-0022-2014-01310-6 fatcat:egisjcnmkjdrniesiulvsaocwi