The multidimensional truncated moment problem: Carathéodory numbers

Let A be a finite-dimensional subspace of C(X;R), where X is a locally compact Hausdorff space, and A={f1,…,fm} a basis of A. A sequence s=(sj)j=1m is called a moment sequence if sj=∫fj(x)dμ(x), j=1,…,m, for some positive Radon measure μ on X. Each moment sequence s has a finitely atomic representing measure μ. The smallest possible number of atoms is called the Carathéodory number CA(s). The largest number CA(s) among all moment sequences s is the Carathéodory number CA. In this paper the Carathéodory numbers CA(s) and CA are studied. In the case of differentiable functions methods from differential geometry are used. The main emphasis is on real polynomials. For a large class of spaces of polynomials in one variable the number CA is determined. In the multivariate case we obtain some lower bounds and we use results on zeros of positive polynomials to derive upper bounds for the Carathéodory numbers.