Asymptotic Minimax Risk for the White Noise Model on the Sphere

Estimation of an unknown function on the unit sphere of the Euclidean space is considered. The function is observed in Gaussian continuous time white noise. Uniform norm is chosen as a loss function and exact asymptotic minimax risk is derived extending the result of Korostelev (1993). The exact asymptotic minimax risk is also given for the L ₂-loss, applying the result of Pinsker (1980).