Tight Gaussian 4-Designs

A Gaussian t-design is defined as a finite set X in the Euclidean space Rn satisfying the condition: 1 V (Rn ) ∫ Rn f (x)e −α2||x ||2 dx = u∈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 2e-design in Rn , then |X | ≥ (n+e e ) . We call X a tight Gaussian 2e-design in Rn if |X | = (n+e e ) holds. In this paper we study tight Gaussian 2e-designs in Rn . In particular, we classify tight Gaussian 4-designs in Rn with constant weight ω = 1 |X | or with weight ω(u) = e −α2 ||u||2 ∑ x∈X e−α 2 ||x ||2 . Moreover we classify tight Gaussian 4-designs in R n on 2 concentric spheres (with arbitrary weight functions).