We review the current state of the spectral theory of random functions of several variables created by Professor M. I. Yadrenko at the end of 1950s. It turns out that the spectral expansions of multi-dimensional homogeneous and isotropic random fields are governed by a pair of convex compacts and are especially simple when these compacts are simplexes. Our new result gives necessary and sufficient conditions for such a situation in terms of the group representation that defines the field.
Special Issue by the Department of Probability, Statistics and Actuarial Mathematics at TSNU of Kyiv