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Home > Vol 22, No 1 (2005)

Eiichi Bannai and Etsuko Bannai

Faculty of Mathematics, Graduate School Kyushu University Japan

A Gaussian t-design is defined as a finite set X in the Euclidean space \Bbb R ⁿ satisfying the condition: \frac1 V(\mathbb R ⁿ) ò _{\mathbb R ⁿ} f( x) e ^{- a 2 | | x | | 2} dx= å _{u Ĩ X} w( u) f( u) \frac{1}{V({\mathbb R}^n)}\int_{{\mathbb R}^n} f(x)e^{-α^2||x||^2}dx=\sum_{u\in X}ω(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 2 e-design in \Bbb R ⁿ, then | X | $^{3}$ (( n+ e) || ( e)) |X|\geq {n+e\choose e} . We call X a tight Gaussian 2 e-design in \Bbb R ⁿ if | X |=(( n+ e) || ( e)) |X|={n+e\choose e} holds. In this paper we study tight Gaussian 2 e-designs in \Bbb R ⁿ. In particular, we classify tight Gaussian 4-designs in \Bbb R ⁿ with constant weight w = \frac1 | X | ω=\frac{1}{|X|} or with weight w( u)=\frac e ^{- a 2 | | u | | 2} å _{x Ĩ X} e ^{- a 2 | | x | | 2} ω(u)=\frac{e^{-α^2||u||^2}} {\sum_{x\in X}e^{-α^2||x||^2}} . Moreover we classify tight Gaussian 4-designs in \Bbb R ⁿ on 2 concentric spheres (with arbitrary weight functions).

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