Extremal measures with prescribed moments

In the approximate integration some inequalities between the quadratures and the integrals approximated by them are called extremalities. On the other hand, the set of all quadratures is convex. We are trying to find possible connections between extremalities and extremal quadratures (in the sense of extreme points of a convex set). Of course, the quadratures are the integrals with respect to discrete measures and, moreover, a quadrature is extremal if and only if the associated measure is extremal. Hence the natural problem arises to give some description of extremal measures with prescribed moments in the general (not only discrete) case. In this paper we deal with symmetric measures with prescribed first four moments. The full description (with no symmetry assumptions, and/or not only four moments are prescribed and so on) is far to be done.