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