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On positive cubature rules on the simplex and isometric embeddings

Authors: Masanori Sawa and Yuan Xu
Journal: Math. Comp. 83 (2014), 1251-1277
MSC (2000): Primary 46B04, 65D32
Published electronically: August 13, 2013
MathSciNet review: 3167458
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Abstract: Positive cubature rules of degree $ 4$ and $ 5$ on the $ d$-dimensional simplex are constructed for a range of dimensions $ d$ and used to construct cubature rules of index $ 8$ or degree $ 9$ on the unit sphere. The latter ones lead to explicit isometric embedding among the classical Banach spaces. Among other things, our results include several explicit representations of $ (x_1^2+ \cdots + x_d^2)^t$ in terms of linear forms of degree $ 2t$ with rational coefficients for $ t = 4$ and $ 5$.

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Additional Information

Masanori Sawa
Affiliation: Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya 464-8601.

Yuan Xu
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222.

Keywords: Cubature rule, simplex, sphere, isometric embedding, spherical design
Received by editor(s): July 9, 2011
Received by editor(s) in revised form: July 17, 2012, and October 1, 2012
Published electronically: August 13, 2013
Additional Notes: The work was supported in part by NSF Grant DMS-1106113
Article copyright: © Copyright 2013 American Mathematical Society