On positive cubature rules on the simplex and isometric embeddings
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- by Masanori Sawa and Yuan Xu PDF
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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 $5$.References
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Additional Information
- Masanori Sawa
- Affiliation: Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya 464-8601.
- MR Author ID: 776019
- Email: sawa@is.nagoya-u.ac.jp
- Yuan Xu
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222.
- MR Author ID: 227532
- Email: yuan@math.uoregon.edu
- 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
- © Copyright 2013 American Mathematical Society
- Journal: Math. Comp. 83 (2014), 1251-1277
- MSC (2000): Primary 46B04, 65D32
- DOI: https://doi.org/10.1090/S0025-5718-2013-02762-7
- MathSciNet review: 3167458