Optimal quadrature errors and sampling numbers for Sobolev spaces with logarithmic perturbation on spheres

Abstract

In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in $L_2({\mathbb{S}}^d)$ for Sobolev spaces ${\mathrm{H}}^{\alpha ,\beta }({\mathbb{S}}^d)$ with logarithmic perturbation on the unit sphere ${\mathbb{S}}^d$ in ${\mathbb{R}}^{d+1}$. First we obtain strong equivalences of the approximation numbers for ${\mathrm{H}}^{\alpha ,\beta }({\mathbb{S}}^d)$ with $\alpha >0$, which gives a clue to Open problem 3 as posed by Krieg and Vybíral in \cite{KV}. Second, for the optimal quadrature errors for ${\mathrm{H}}^{\alpha ,\beta }({\mathbb{S}}^d)$, we use the "fooling" function technique to get lower bounds in the case $\alpha >d/2$, and apply Hilbert space structure and Vybíral's theorem about Schur product theory to obtain lower bounds in the case $\alpha =d/2,\beta >1/2$ of small smoothness, which confirms the conjecture as posed by Grabner and Stepanyukin in \cite{GS} and solves Open problem 2 in \cite{KV}. Finally, we employ the weighted least squares operators and the least squares quadrature rules to obtain approximation theorems and quadrature errors for ${\mathrm{H}}^{\alpha ,\beta }({\mathbb{S}}^d)$ with $\alpha >d/2$ or $\alpha =d/2,\beta >1/2$, which are order optimal.

24 pages