Remarks on Hilbert identities, isometric embeddings, and invariant cubature

Nozaki, H.; Sawa, M.

doi:10.1090/S1061-0022-2014-01310-6

Remarks on Hilbert identities, isometric embeddings, and invariant cubature
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by H. Nozaki and M. Sawa

St. Petersburg Math. J. 25 (2014), 615-646

DOI: https://doi.org/10.1090/S1061-0022-2014-01310-6

Published electronically: June 5, 2014

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Abstract:

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 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.

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