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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Remarks on Hilbert identities, isometric embeddings, and invariant cubature


Authors: H. Nozaki and M. Sawa
Original publication: Algebra i Analiz, tom 25 (2013), nomer 4.
Journal: St. Petersburg Math. J. 25 (2014), 615-646
MSC (2010): Primary 65D32, 11E76; Secondary 52A21
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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Additional Information

H. Nozaki
Affiliation: Department of Mathematics, Aichi University of Education, Igaya-cho, Kariya-city 448-8542, Japan
Email: hnozaki@auecc.aichi-edu.ac.jp

M. Sawa
Affiliation: Graduate School of Information Sciences, Nagoya University, Chikusa-ku, Nagoya 464-8601. Japan
Email: sawa@is.nagoya-u.ac.jp

DOI: http://dx.doi.org/10.1090/S1061-0022-2014-01310-6
PII: S 1061-0022(2014)01310-6
Keywords: Cubature formula, Hilbert identity, isometric embedding, Victoir method
Received by editor(s): April 5, 2012
Published electronically: June 5, 2014
Additional Notes: The second author was supported in part by Grant-in-Aid for Young Scientists (B) 22740062 and Grant-in-Aid for Challenging Exploratory Research 23654031 by the Japan Society for the Promotion of Science
Article copyright: © Copyright 2014 American Mathematical Society