Tight Gaussian 4Designs
 Eiichi Bannai,
 Etsuko Bannai
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A Gaussian tdesign is defined as a finite set X in the Euclidean space ℝ^{ n } satisfying the condition: \(\frac{1}{V({\mathbb R}^n)}\int_{{\mathbb R}^n} f(x)e^{\alpha^2x^2}dx=\sum_{u\in X}\omega(u)f(u)\) 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 2edesign in ℝ^{ n }, then \(X\geq {n+e\choose e}\). We call X a tight Gaussian 2edesign in ℝ^{ n } if \(X={n+e\choose e}\) holds. In this paper we study tight Gaussian 2edesigns in ℝ^{ n }. In particular, we classify tight Gaussian 4designs in ℝ^{ n } with constant weight \(\omega=\frac{1}{X}\) or with weight \(\omega(u)=\frac{e^{\alpha^2u^2}} {\sum_{x\in X}e^{\alpha^2x^2}}\). Moreover we classify tight Gaussian 4designs in ℝ^{ n } on 2 concentric spheres (with arbitrary weight functions).
 Title
 Tight Gaussian 4Designs
 Journal

Journal of Algebraic Combinatorics
Volume 22, Issue 1 , pp 3963
 Cover Date
 200508
 DOI
 10.1007/s1080100525053
 Print ISSN
 09259899
 Online ISSN
 15729192
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

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

 Eiichi Bannai ^{(1)}
 Etsuko Bannai ^{(1)}
 Author Affiliations

 1. Faculty of Mathematics, Graduate School, Kyushu University, Japan