Optimum Linear Estimates for the Regression Coefficients of Isotropic Random Fields

Isotropic random fields (fields whose correlation functions are invariant under rotations about a fixed point) are discussed. Linear unbiased estimates optimum in the sense of minimizing the mean-square error are given for the regression coefficients when the random field is observed on a sphere. It turns out, in particular, that the optimum unbiased estimate of an unknown expectation coincides with the average over the sphere. Optimum nonlinear estimates for random regression coefficients are investigated.