Journal of Algebraic Combinatorics

, Volume 22, Issue 1, pp 39–63

Tight Gaussian 4-Designs

  • Eiichi Bannai
  • Etsuko Bannai
Article

DOI: 10.1007/s10801-005-2505-3

Cite this article as:
Bannai, E. & Bannai, E. J Algebr Comb (2005) 22: 39. doi:10.1007/s10801-005-2505-3

Abstract

A Gaussian t-design is defined as a finite set X in the Euclidean space ℝn 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 ℝn, then \(|X|\geq {n+e\choose e}\). We call X a tight Gaussian 2e-design in ℝn if \(|X|={n+e\choose e}\) holds. In this paper we study tight Gaussian 2e-designs in ℝn. In particular, we classify tight Gaussian 4-designs in ℝn 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 ℝn on 2 concentric spheres (with arbitrary weight functions).

Keywords

Gaussian designtight designspherical design2-distance setEuclidean designaddition formulaquadrature formula

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