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 $\Bbb R ^{ n}$ satisfying the condition: $\frac1 V(\mathbb R ^{ n})$ ò $_{\mathbb R $^ n$ f( x) e ^{ - a $^2$ | | x | | $^2$ d$x= å $_{ u \~I X} w( u) f( u)$ frac1${V({\mathbb R}^n)}$int_$\mathbb R$^n $f(x)$e^-alpha^2||x||^2dx=sum_u$\in X}\omega(u)f(u)$ 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 $2 e$-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 $w = \frac1 | X |$ 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 $\Bbb R ^{ n}$ 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