Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different Settings

This article primarily examines the approximation properties of a weighted Sobolev space W2,κr defined on a sphere Sd−1 equipped with Gaussian measures. Specifically, this study focuses on both the average case and probabilistic case settings. The exact asymptotic orders of the Gel’fand n-width and the linear n-width of W2,κr on Sd−1 are derived for these settings, providing a comprehensive understanding of their approximation characteristics.