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 Ld-1q(S) metric for 1 ≤ q ≤∞, and show that some worst-case asymptotically optimal algorithms are also asymptotically optimal in the average case setting in the Ldq(S-1)metric for 1 ≤ q ≤∞.
We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ(Sd 1)of generalized smoothness on the sphere inL q(Sd 1)for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the exact orders of entropy numbers of Sobolev classesBWr p(Sd 1)inL q(Sd 1)whenpand/orqis equal to 1 or∞.This provides the last piece as far as exact orders of entropy numbers ofBWr p(Sd 1)inL q(Sd 1)are concerned.
