On Minimal Cubature Formulae of Small Degree for Spherically Symmetric Integrals

The aim of this paper is to develop the existence and nonexistence problem of a cubature formula of degree $4k+1$ for a spherically symmetric integral which attains the Moller lower bound. For this purpose we bring together the theory of Euclidean design in combinatorics and that of reproducing kernels in numerical analysis. We show that if there exists a minimal formula of degree $4k+1$, then the nodes are distributed into $k+1$ concentric spheres including the origin, and the weights in the formula are a constant on each concentric sphere. We particularly focus on the cases of degrees 5 and 9. We show the equivalence between a minimal formula of degrees 5 and 4, and a spherical tight 5- and 4-design. We also prove that there exists no minimal formula of degree 9 for some classical weights such as the Gaussian weight on $\mathbb{R}^d$, the radial exponential weight on $\mathbb{R}^d$, and the Jacobi weight on $B^d$.

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