Extremal measures with prescribed moments

TL;DR

This paper explores the characterization of extremal measures with given moments, focusing on symmetric measures with four prescribed moments, and investigates their connection to extremal quadratures within convex sets.

Contribution

It provides a description of extremal symmetric measures with four prescribed moments, linking extremal measures to extremal quadratures in the context of convex analysis.

Findings

Characterization of extremal symmetric measures with four moments.

Connection between extremal measures and extremal quadratures.

Partial description of extremal measures beyond symmetric cases.

Abstract

In the approximate integration some inequalities between the quadratures and the integrals approximated by them are called \emph{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 \wrt~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.