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

2024/01/29 by Jiaxin Geng, Yun Ling, Geng, Jiaxin +5
Mathematics · #41A63 #65C05 #65D15 #65Y20 #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2401.16199

openalex publication_date 2024/01/29 · openalex created_date 2024/01/31 · openalex updated_date 2026/07/28

Abstract

In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in L₂(\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 \citeKV. 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 \citeGS and solves Open problem 2 in \citeKV. 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.

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