How well is a measure characterized by its moments is an old question
which appear in many contexts and applications. In polynomial
optimization, it is the basis for so-called moment relaxation
hierarchies, which allow to compute global optima of polynomial
functions on (compact) basic semi-algebraic sets. Computing the optimal
moment sequence(s), positive on the quadratic module of the
semi-algebraic set, by convex optimization, one can approximate the
global solution(s) of the non-linear optimization problem.
In this talk, we will discuss conditions for which this approach gives
an exact moment representation of a measure. We will then consider the
properties of approximation of moment sequences, give an Effective
Putinar Positivstellensatz and present a quantitative analysis of the
approximation of measures by positive moment sequences, with new
polynomial bounds in the intrinsic parameters of the problem. This
presentation is based on a join work with Lorenzo Baldi.