**"Numerical integration, comparison of probabilistic and deterministic point sets"**
#### Stepaniuk, TetianaWe study the worst-case error of numerical integration on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, for certain spaces of continuous functions on $\mathbb{S}^{d}$. For the classical Sobolev spaces $\mathbb{H}^s(\mathbb{S}^d)$ ($s>\frac d2$) upper and lower bounds for the worst case integration error have been obtained by Brauchart, Hesse and Sloan.
We analyze energy integrals with regard to area-regular partitions of the sphere and compare obtained estimates with discrete energy sums. In particular the asymptotic equalities for the discrete Riesz $s$-energy of $N$-point sequence of well separated $t$-designs on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$ are found. |