Journal of Algebraic Combinatorics

, Volume 22, Issue 1, pp 39–63

# 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

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## 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