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 consider Jackson inequality in L^2(B^d × T,W_(κ,μ)~B(x)),where the weight function W_(κ,μ)~B(x) is defined on the ball B^d and related to reflection group,and obtain the sharp Jackson inequality E_(n- 1,m-1)(f)2 ≤k_(n,m)(r,r)ω_r(f,t)_2,T ≥ 2τ_(n,λ),where τ_(n,λ) is the first positive zero of the Gegenbauer cosine polynomial C_n~λ(cosθ)(n ∈ N).
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.
