Tight Gaussian 4-designs

Bannai, Eiichi ; Bannai, Etsuko

Journal of Algebraic Combinatorics, Tome 22 (2005) no. 1, pp. 39-63.

Résumé

Summary: A Gaussian $t$-design is defined as a finite set $X$ in the Euclidean space ${\mathbb{R}}^n$ satisfying the condition: $\frac{1}{V}({\mathbb{R}}^n)$ ò $_{\mathbb R $^ n$ f( x) e ^{ - a $^2$||x||$^2$d$x= å ${}_{u\text{~}IX}w(u)f(u)$ frac1$V({\mathbb{R}}^n)$int_$\mathbb{R}$^n $f(x)$e^-alpha^2||x||^2dx=sum_u$\text{in X}omega(u)f(u)}$ for any polynomial $f(x)$ in $n$ variables of degree at most $t$, here $\alpha$is a constant real number and $\omega$is a positive weight function on $X$. It is easy to see that if $X$ is a Gaussian $2e$-design in ${\mathbb{R}}^n$, then $|X{|}^3((n+e)||(e))$ |X|$\ge$n+e$(\genfrac{}{}{0ex}{}{}{e})$ . We call $X$ a tight Gaussian $2e$-design in ${\mathbb{R}}^n$ if $|X|=((n+e)||(e))$ |X|=n+e$(\genfrac{}{}{0ex}{}{}{e})$ holds. In this paper we study tight Gaussian $2e$-designs in ${\mathbb{R}}^n$. In particular, we classify tight Gaussian 4-designs in ${\mathbb{R}}^n$ with constant weight $w=\frac{1}{|}X|$ omega=frac1|X| or with weight $w( u)=\frac e ^{ - a $^2$||u||$^2 å $_{ x \~I X} e ^{ - a $^2$||x||$^2$\omega (u)=$frace^-alpha^2||u||^2 sum_x$\in X$e^-alpha^2||x||^2 . Moreover we classify tight Gaussian 4-designs in ${\mathbb{R}}^n$ on 2 concentric spheres (with arbitrary weight functions).

Détail

Mots-clés : keywords Gaussian design, tight design, spherical design, 2-distance set, Euclidean design, addition formula, quadrature formula