Cubature formulas and Sobolev inequalities

Abstract

We study a problem in the theory of cubature formulas on the sphere: given $\theta \in (0, 1)$, determine the infimum of $\|\nu\|_\theta = \sum_{i = 1}^n \nu_i^\theta$ over cubature formulas $\nu$ of strength $t$, where $\nu_i$ are the weights of the formula $\nu$. This problem, which generalizes the classical problem of bounding the minimal cardinality of a cubature formula -- the case $\theta = 0$ -- was introduced in recent work of Hang and Wang (arXiv:2010.10654), who showed the problem to be related to optimal constants in Sobolev inequalities. Using the elementary theory of reproducing kernel Hilbert spaces on $S^{n - 1}$, we extend the best known upper and lower bounds for the minimal cardinality of strength-$t$ cubature formulas to bounds for the infimum of $\|\cdot\|_\theta$ for any $\theta \in (0, 1)$. In particular, we completely characterize the cubature measures of strength $3$ minimizing $\|\cdot\|_\theta$, showing that these are precisely the tight spherical $3$-designs.