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Construction of spherical cubature formulas using lattices


Authors: P. de la Harpe, C. Pache and B. Venkov
Original publication: Algebra i Analiz, tom 18 (2006), nomer 1.
Journal: St. Petersburg Math. J. 18 (2007), 119-139
MSC (2000): Primary 65D32, 05B30; Secondary 11F11, 11H06
DOI: https://doi.org/10.1090/S1061-0022-07-00946-6
Published electronically: January 19, 2007
MathSciNet review: 2225217
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Abstract: We construct cubature formulas on spheres supported by homothetic images of shells in some Euclidean lattices. Our analysis of these cubature formulas uses results from the theory of modular forms. Examples are worked out on $ \mathbb{S}^{n-1}$ for $ n=4$, $ 8$, $ 12$, $ 14$, $ 16$, $ 20$, $ 23$, and $ 24$, and the sizes of the cubature formulas we obtain are compared with the lower bounds given by Linear Programming.


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Additional Information

P. de la Harpe
Affiliation: Section de Mathématiques, Université de Genève, C.P. 64, 1211 Genève 4, Switzerland
Email: Pierre.delaHarpe@math.unige.ch

C. Pache
Affiliation: Section de Mathématiques, Université de Genève, C.P. 64, 1211 Genève 4, Switzerland
Email: Claude.Pache@math.unige.ch

B. Venkov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: bbvenkov@yahoo.com

DOI: https://doi.org/10.1090/S1061-0022-07-00946-6
Keywords: Cubature formula, modular lattice, modular form, spherical $t$-design
Received by editor(s): June 3, 2005
Published electronically: January 19, 2007
Additional Notes: The authors acknowledge support from the Swiss National Science Foundation
Article copyright: © Copyright 2007 American Mathematical Society

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