Journal of Algebraic Combinatorics

, Volume 22, Issue 1, pp 39–63 | Cite as

Tight Gaussian 4-Designs

  • Eiichi Bannai
  • Etsuko Bannai
Article

Abstract

A Gaussian t-design is defined as a finite set X in the Euclidean space ℝ n satisfying the condition: \(\frac{1}{V({\mathbb R}^n)}\int_{{\mathbb R}^n} f(x)e^{-\alpha^2||x||^2}dx=\sum_{u\in X}\omega(u)f(u)\) for any polynomial f(x) in n variables of degree at most t, here α is a constant real number and ω is a positive weight function on X. It is easy to see that if X is a Gaussian 2e-design in ℝ n , then \(|X|\geq {n+e\choose e}\). We call X a tight Gaussian 2e-design in ℝ n if \(|X|={n+e\choose e}\) holds. In this paper we study tight Gaussian 2e-designs in ℝ n . In particular, we classify tight Gaussian 4-designs in ℝ n with constant weight \(\omega=\frac{1}{|X|}\) or with weight \(\omega(u)=\frac{e^{-\alpha^2||u||^2}} {\sum_{x\in X}e^{-\alpha^2||x||^2}}\). Moreover we classify tight Gaussian 4-designs in ℝ n on 2 concentric spheres (with arbitrary weight functions).

Keywords

Gaussian design tight design spherical design 2-distance set Euclidean design addition formula quadrature formula 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Bannai, “On extremal finite sets in the sphere and other metric spaces,” in Algebraic, Extremal and Metric Combinatorics, 1986 (Montreal, PQ, 1986), London Math. Soc. Lecture Note Ser., 131, Cambridge Univ. Press, Cambridge, 1988, pp. 13–38.Google Scholar
  2. 2.
    E. Bannai and E. Bannai, Algebraic Combinatorics on Spheres(in Japanese), Springer Tokyo, 1999, pp. xvi + 367.Google Scholar
  3. 3.
    E. Bannai and E. Bannai, “On tight Euclidean 4-designs,” preprint.Google Scholar
  4. 4.
    E. Bannai and R.M. Damerell, “Tight spherical designs I,” J. Math. Soc. Japan 31 (1979), 199–207.Google Scholar
  5. 5.
    E. Bannai and R. M. Damerell, “Tight spherical designs II,” J. London Math. Soc. 21 (1980), 13–30.Google Scholar
  6. 6.
    E. Bannai, K. Kawasaki, Y. Nitamizu, and T. Sato, “An upper bound for the cardinality of an s-distance set in Euclidean space,” Combinatorica 23 (2003), 535–557.CrossRefGoogle Scholar
  7. 7.
    E. Bannai, A. Munemasa, and B. Venkov, “The nonexistence of certain tight spherical designs,” to appear in Algebra i Analiz 16 (2004).Google Scholar
  8. 8.
    G.E.P. Box and J.S. Hunter, “Multi-factor experimental designs for exploring response surfaces,” Ann. Math. Statist. 28 (1957), 195–241.Google Scholar
  9. 9.
    P. Delsarte, J.-M. Goethals and J.J. Seidel, “Spherical codes and designs,” Geom. Dedicata 6 (1977), 363–388.Google Scholar
  10. 10.
    P. Delsarte and J.J. Seidel, “Fisher type inequalities for Euclidean t-designs,” Lin. Algebra and its Appl. 114/115 (1989), 213–230.CrossRefGoogle Scholar
  11. 11.
    P. de la Harpe and C. Pache, “Cubature formulas, geometric designs, reproducing kernels, and Markov operators,” preprint, University of Genève (2004).Google Scholar
  12. 12.
    C.F. Dunkl and Y. Xu, “Orthogonal polynomials of several variables,” Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001, vol. 81, pp. xvi + 390.Google Scholar
  13. 13.
    S.J. Einhorn and I.J. Schoeneberg, “On Euclidean sets having only two distances between points I,” Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 479–488.Google Scholar
  14. 14.
    S.J. Einhorn and I.J. Schoeneberg, “On Euclidean sets having only two distances between points II,” Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 489–504.Google Scholar
  15. 15.
    A. Erdélyi et al. “Higher transcendental Functions, Vol II, (Bateman Manuscript Project),” MacGraw-Hill, 1953.Google Scholar
  16. 16.
    S. Karlin and W.J. Studden, “Tchebycheff Systems with Application in Analysis and Statistics,” Interscience, 1966.Google Scholar
  17. 17.
    J. Kiefer, “Optimum designs V, with applications to systematic and rotatable designs,” Proc. 4th Berkeley Sympos. 1 (1960), 381–405.Google Scholar
  18. 18.
    D.G. Larman, C.A. Rogers and J.J. Seidel, “On two-distance sets in Euclidean space,” Bull London Math. Soc. 9 (1977), 261–267.Google Scholar
  19. 19.
    A. Neumaier and J.J. Seidel, “Discrete measures for spherical designs, eutactic stars and lattices,” Nederl. Akad. Wetensch. Proc. Ser. A 91 = Indag. Math. 50 (1988), 321–334.Google Scholar
  20. 20.
    A. Neumaier and J.J. Seidel, “Measures of strength 2e and optimal designs of degree e,” Sankhya Ser. A 54 (Special Issue), (1992), 299–309.Google Scholar
  21. 21.
    P.D. Seymour and T. Zaslavsky, “Averaging sets: A generalization of mean values and spherical designs,” Adv. in Math. 52(3), (1984), 213–240.CrossRefGoogle Scholar
  22. 22.
    G. Szegö, Orthogonal Polynomials, 4th edition. American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975, pp. xiii + 432.Google Scholar
  23. 23.
    J.L. Ullman, “A class of weight functions that admit Tchebycheff quadrature,” Michigan Math. J. 13 (1966), 417–423.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Eiichi Bannai
    • 1
  • Etsuko Bannai
    • 1
  1. 1.Faculty of Mathematics, Graduate SchoolKyushu UniversityJapan

Personalised recommendations