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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
Posted:
January 19, 2007
MathSciNet review:
2225217
Full-text PDF Free Access
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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  for  ,  ,  ,  ,  ,  ,  , and  , and the sizes of the cubature formulas we obtain are compared with the lower bounds given by Linear Programming.
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de la Harpe and Claude
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and Markov operators, Infinite groups: geometric, combinatorial and
dynamical aspects, Progr. Math., vol. 248, Birkhäuser, Basel,
2005, pp. 219–267. MR 2195455
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H. Lehmer, The vanishing of Ramanujan’s function
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(9,12b)
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- N. N. Andreev, A minimal design of order 11 on the three-dimensional sphere, Mat. Zametki 67 (2000), no. 4, 489-497; English transl., Math. Notes 67 (2000), no. 3-4, 417-424. MR 1769895 (2001g:05020)
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- Ch. Bachoc and B. B. Venkov, Modular forms, lattices and spherical designs, Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math., no 37, Enseign. Math., Geneva, 2001, pp. 87-111. MR 1878746 (2003d:11096)
- [BanDam79]
- E. Bannai and R. Damerell, Tight spherical designs. I, J. Math. Soc. Japan 31 (1979), 199-207. MR 0519045 (80b:05014)
- [BanDam80]
- -, Tight spherical designs. II, J. London Math. Soc. (2) 21 (1980), 13-30. MR 0576179 (81g:05024)
- [BaMuVe02]
- E. Bannai, A. Munemasa, and B. Venkov, The nonexistence of certain tight spherical designs, Algebra i Analiz 16 (2004), no. 4, 1-23; English transl., St. Petersburg Math. J. 16 (2005), no. 4, 609-625. MR 2090848 (2005e:05022)
- [Bajn91]
- B. Bajnok, Chebyshev-type quadrature formulas on the sphere, Proceedings of the Twenty-Second Southeastern Conference on Combinatorics, Graph Theory, and Computing (Baton Rouge, LA, 1991), Congr. Numer. 85 (1991), 214-218. MR 1152134 (93a:05038)
- [ConSlo99]
- J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren Math. Wiss., vol. 290, Springer-Verlag, New York, 1999. MR 1662447 (2000b:11077)
- [DeGoSe77]
- P. Delsarte, J.-M. Goethals, and J. J. Seidel, Spherical codes and designs, Geom. Dedicata 6 (1977), 363-388. MR 0485471 (58:5302)
- [Dick19]
- L. E. Dickson, History of the theory of numbers. Vol. II: Diophantine analysis, Carnegie Inst., Washington, 1919; reprinted by Chelsea Publ. Co., New York, 1966. MR 0245500 (39:6807b)
- [Ebel94]
- W. Ebeling, Lattices and codes. A course partially based on lectures by F. Hirzebruch, Friedr. Vieweg and Sohn, Braunschweig, 1994; 2nd revised ed., 2002. MR 1280458 (95c:11084); MR 1938666 (2003i:11093)
- [GoeSei79]
- J.-M. Goethals and J. J. Seidel, Spherical designs, Relations Between Combinatorics and Other Parts of Mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978), Proc. Sympos. Pure Math., vol. 34, Amer. Math. Soc., Providence, RI, 1979, pp. 255-272. MR 0525330 (82h:05014)
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- -, Cubature formulae, polytopes, and spherical designs, The Geometric Vein, The Coxeter Festschrift, Springer, New York-Berlin, 1981, pp. 203-218. MR 0661779 (83k:05033)
- [HadSlo94]
- R. H. Hardin and N. J. A. Sloane, Expressing
 as a sum of 23 sixth powers, J. Combin. Theory Ser. A 68 (1994), 481-485. MR 1297185 (96e:11048)
- [HadSlo96]
- -, McLaren's improved snub cube and other new spherical designs in three dimensions, Discrete Comput. Geom. 15 (1996), 429-441. MR 1384885 (97b:52013)
- [HarPac04]
- P. de la Harpe and C. Pache, Spherical designs and finite group representations (some results of E. Bannai), European J. Combin. 25 (2004), 213-227. MR 2070542 (2005f:05031)
- [HarPac05]
- -, Cubature formulas, geometrical designs, reproducing kernels, and Markov operators, Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Progr. Math., vol. 248, Birkhäuser, Basel, 2005, pp. 219-267. MR 2195455
- [Kupe06a]
- G. Kuperberg, Numerical cubature using error-correcting codes, SIAM J. Numer. Anal. 44 (2006), no. 3, 897-907 (electronic).
- [Kupe06b]
- -, Numerical cubature from Archimedes' hat-box theorem, SIAM J. Numer. Anal. 44 (2006), no. 3, 908-935 (electronic).
- [Lehm47]
- D. H. Lehmer, The vanishing of Ramanujan's function
 , Duke Math. J. 14 (1947), 429-433. MR 0021027 (9:12b)
- [MartV01]
- J. Martinet (ed.), Réseaux euclidiens, designs sphériques et formes modulaires. Autour des travaux de B. Venkov, Monogr. Enseign. Math., no 37, Enseign. Math., Geneva, 2001. MR 1881618 (2002h:11062)
- [NikNik02]
- S. Nikova and V. Nikov, Improvement of the Delsarte bound for
 -designs when it is not the best bound possible, Des. Codes Cryptogr. 28 (2003), no. 2, 201-222. MR 1962806 (2003m:05046)
- [Pach05]
- C. Pache, Shells of selfdual lattices viewed as spherical designs, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 1085-1127. MR 2197823
- [Queb95]
- H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory 54 (1995), no. 2, 190-202. MR 1354045 (96i:11072)
- [Rank77]
- R. A. Rankin, Modular forms and functions, Cambridge Univ. Press, Cambridge, 1977. MR 0498390 (58:16518)
- [Sali75]
- G. N. Salikhov, Cubature formulae for a hypersphere that are invariant with respect to the group of the regular
 -gon, Dokl. Akad. Nauk SSSR 223 (1975), no. 5, 1075-1078; English transl., Soviet Math. Dokl. 16 (1975), no. 4, 1046-1050. MR 0408213 (53:11978)
- [Serr85]
- J.-P. Serre, Sur la lacunarité des puissances de
 , Glasgow Math. J. 27 (1985), 203-221 =  uvres, Collected Papers IV. 1985-1998, Springer-Verlag, Berlin, 2000, pp. 66-84, 640. MR 0819840 (87e:11066); MR 1730973 (2001e:01037)
- [SeyZas84]
- P. D. Seymour and T. Zaslavsky, Averaging sets: A generalization of mean values and spherical designs, Adv. in Math. 52 (1984), 213-240. MR 0744857 (85m:05031)
- [Smit95]
- L. Smith, Polynomial invariants of finite groups, Res. Notes Math., No. 6, A. K. Peters, Wellesley, MA, 1995. MR 1328644 (96f:13008)
- [Sobo62]
- S. L. Sobolev, Cubature formulas on the sphere which are invariant under transformations of finite rotation groups, Dokl. Akad. Nauk SSSR 146 (1962), no. 2, 310-313; English transl. in Soviet Math. Dokl. 3 (1962). MR 0141225 (25:4635)
- [Sobo96]
- S. L. Sobolev and V. Z. Vaskevich, Cubature formulas, Akad. Nauk Sibirsk. Otdel. Inst. Mat., Novosibirsk, 1996; English transl., Math. Appl., vol. 415, Kluwer Acad. Publishers Group, Dordrecht, 1997. MR 1463155 (99g:65029a); MR 1462617 (99g:65029b)
- [Venk84]
- B. B. Venkov, Even unimodular extremal lattices, Trudy Mat. Inst. Steklov. 165 (1984), 43-48; English transl., Proc. Steklov Inst. Math. 1985, no. 3, 47-52. MR 0752931 (85i:05044)
- [VenMar01]
- B. B. Venkov (notes by J. Martinet), Réseaux et designs sphériques, Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math., no 37, Enseign. Math., Geneva, 2001, pp. 10-86. MR 1878745 (2002m:11061)
- [Vile68]
- N. Ya. Vilenkin, Special functions and the theory of group representations, ``Nauka'', Moscow, 1965; English transl., Transl. Math. Monogr., vol. 22, Amer. Math. Soc., Providence, RI, 1968. MR 0209523 (35:420); MR 0229863 (37:5429)
- [Yudi97]
- V. A. Yudin, Lower bounds for spherical designs, Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 3, 213-223; English transl., Izv. Math. 61 (1997), no. 3, 673-683. MR 1478566 (98i:05042)
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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:
http://dx.doi.org/10.1090/S1061-0022-07-00946-6
PII:
S 1061-0022(07)00946-6
Keywords:
Cubature formula,
modular lattice,
modular form,
spherical $t$-design
Received by editor(s):
June 3, 2005
Posted:
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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