## Numerical integration errors, probabilistic and deterministic point sets on sphere

For the classical Sobolev spaces $\mathbb{H}^s(\mathbb{S}^d)$ ($s>\frac d2$) upper and lower bounds for the worst case integration error of numerical integration on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, have been obtained by Brauchart, Hesse and Sloan. We investigate the case when $s\to\frac d2$ and introduce the spaces $\mathbb{H}^{\frac d2,\gamma}(\mathbb{S}^d)$ of continuous functions on $\mathbb{S}^{d}$ with an extra logarithmic weight. For these spaces we obtain estimates for the worst case integration error.

We make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (spherical $t$-designs, minimizing point-sets) are better or as good as probabilistic ones. 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.