Optimal Recovery of Functions on the Sphere on a Sobolev Spaces with a Gaussian Measure in the Average Case Setting (Analysis in Theory and Applications, 2017)
In this paper, we study optimal recovery (reconstruction) of functions on the sphere in the average case setting. We obtain the asymptotic orders of average sampling numbers of a Sobolev space on the sphere with a Gaussian measure in the $L_q({\mathbb{S}^{d-1}})$ metric for $1\le q\le ∞$, and show that some worst-case asymptotically optimal algorithms are also asymptotically optimal in the average case setting in the $L_q(\mathbb{S}^{d-1})$ metric for $1\le q\le ∞$.
Keywords: Optimal recovery on the sphere; average sampling numbers; optimal algorithm; Gaussian measure.