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 $\Vert \nu {\Vert }_{\theta }={\sum }_{i=1}^n{\nu }_i^{\theta }$ 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 $\Vert \cdot {\Vert }_{\theta }$ for any $\theta \in (0,1)$. In particular, we completely characterize the cubature measures of strength $3$ minimizing $\Vert \cdot {\Vert }_{\theta }$, showing that these are precisely the tight spherical $3$-designs.

18 pages