In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ 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 ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with $\alpha>0$, 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 ${\rm H}^{\alpha,\beta}(\Bbb S^d)$, we use the "fooling" function technique to get lower bounds in the case $\alpha>d/2$, and apply Hilbert space structure and Vyb\'iral's theorem about Schur product theory to obtain lower bounds in the case $\alpha=d/2,\,\beta>1/2$ 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 ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with $\alpha>d/2$ or $\alpha=d/2,\,\beta>1/2$, which are order optimal.

翻译：本文研究$\Bbb R^{d+1}$中单位球面$\Bbb S^d$上带有对数扰动的Sobolev空间${\rm H}^{\alpha,\beta}(\Bbb S^d)$在$L_2(\Bbb S^d)$中的最优求积误差、逼近数与采样数。首先，我们建立了$\alpha>0$时${\rm H}^{\alpha,\beta}(\Bbb S^d)$逼近数的强等价关系，这为Krieg和Vybíral在文献\cite{KV}中提出的问题3提供了解决线索。其次，针对${\rm H}^{\alpha,\beta}(\Bbb S^d)$的最优求积误差，我们采用"欺骗"函数技术推导了$\alpha>d/2$情形下的下界，并利用Hilbert空间结构以及Vybíral关于Schur积理论的定理，获得了小光滑性情形$\alpha=d/2,\,\beta>1/2$的下界，从而证实了Grabner和Stepanyuk在文献\cite{GS}中提出的猜想，并解决了文献\cite{KV}中的问题2。最后，我们运用加权最小二乘算子与最小二乘求积规则，建立了$\alpha>d/2$或$\alpha=d/2,\,\beta>1/2$时${\rm H}^{\alpha,\beta}(\Bbb S^d)$的逼近定理与求积误差，这些结果在阶的意义上是最优的。