Optimal quadrature errors and sampling numbers for Sobolev spaces with logarithmic perturbation on spheres
Jiaxin Geng, Yun Ling, Jiansong Li, Heping Wang
TL;DR
The paper analyzes quadrature, approximation, and sampling in L2 on the unit sphere for Sobolev spaces with logarithmic perturbation H^{α,β}(S^d). It proves sharp, strong asymptotics for approximation numbers a_n and establishes matching lower bounds for optimal quadrature errors e_n across several smoothness regimes, including the challenging borderline α= d/2, β>1/2. It provides constructive algorithms based on weighted least squares (L_N^S) and least-squares quadrature (I_N^S) that achieve order-optimal rates for g_n and e_n, offering practical, scalable schemes for high-dimensional spheres. The results resolve open problems in Krieg and Vybíral and in Grabner and Stepanyukin, and illuminate the impact of logarithmic perturbation on generalized smoothness, with explicit rate constants reflected in C_{α,β}(N).
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.