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 $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm H}^{α,β}(\Bbb S^d)$ with logarithmic perturbation on the unit sphere $\Bbb S^d$ in $\Bbb R^{d+1}$. First we obtain strong equivalences of the approximation numbers for ${\rm H}^{α,β}(\Bbb S^d)$ with $α>0$, which gives a clue to Open problem 3 as posed by Krieg and Vybíral in \cite{KV}. Second, for the optimal quadrature errors for ${\rm H}^{α,β}(\Bbb S^d)$, we use the "fooling" function technique to get lower bounds in the case $α>d/2$, and apply Hilbert space structure and Vybíral's theorem about Schur product theory to obtain lower bounds in the case $α=d/2,\,β>1/2$ 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 ${\rm H}^{α,β}(\Bbb S^d)$ with $α>d/2$ or $α=d/2,\,β>1/2$, which are order optimal.