Robust interpolation of random fields homogeneous in time and isotropic on a sphere, which are observed with noise

Abstract

We study the problem of optimal linear estimation of the functional $$A_N \xi = \sum\limits_{k = 0}^{\rm N} {\int\limits_{S_n } {a(k,x)\xi (k,x)m_n (dx),} }$$ , which depends on unknown values of a random field ξ(k, x),k∃Z,x∃S _n homogeneous in time and isotropic on a sphereS _n, by observations of the field ξ(k,x)+η(k,x) with k∃ Z{0, 1, ...,N},x∃S_n (here, η (k, x) is a random field uncorrelated with ξ(k, x), homogeneous in time, and isotropic on a sphere S_n). We obtain formulas for calculation of the mean square error and spectral characteristic of the optimal estimate of the functionalA _Nξ. The least favorable spectral densities and minimax (robust) spectral characteristics are found for optimal estimates of the functionalA _Nξ.