, Volume 22, Issue 1, pp 39-63

Tight Gaussian 4-Designs

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Abstract

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