Random Oper. & Stock. Ec/u., Vol. 1, No. 2, pp. 193-203 (1993) © VSP 1993 A problem of minimax smoothing for homogeneous isotropic on a sphere random fields M. P. MOKLYACHUK Department of Mechanics and Mathematics, Kyjiv University, Kyjiv, 252017, Ukraine Received for ROSE 2 December 1991 Abstract—The problem of the least in a square-mean linear estimation for the transformation of a homogeneous isotropic on a sphere Sn random field £(j, x), j £ N, x 6 Sn, using observations of. £(i>®) Ti{ h x ) f°r j ^ 0, x € Sn, where rj(j, x) is a homogeneous isotropic on a sphere Sn random field uncorrelated with x), is considered. The least favourable spectral densities and the minimax (robust) spectral characteristics are determined for some classes of spectral densities. 1. Let Sn be the unit sphere in the n-dimensional Euclidean space, mn(*) be the Lebesgue measure on 5n, x € Sn, m = 0,1,..., 1 = 1 ,2 , h(m,n), be the orthonormal spherical harmonics of degree m, h(m, n) = (2m + n — 2)(m + n —3)!((n —2)!m!)_1 being the number of linearly independent spherical harmonics of degree m (for properties of spherical harmonics, see [1-3]). Let £(j, x) be a continuous in a square-mean random field on N x S„. We call the random field <((j, x) homogeneous isotropic on a sphere if E£2(j, x) < oo, E i(j, x) = 0 and E £(j, x)£(k, y) = B(j - k, cos (x, y)), where cos (x, y) — (x, y) is the “angular” distance between the points x, y 6 Sn. A homogeneous isotropic on a sphere random field has the form [4] oo h(m,n ) £C7»=5Z s!n(x)(L(j), (i) m=0 1=1 dm(j)= f £(ii x)Sl m(x)mn(dx), Jsn Translated by the author