The multidimensional truncated Moment Problem: Carath\'eodory Numbers

Let $\mathcal{A}$ be a finite-dimensional subspace of $C(\mathcal{X};\mathbb{R})$, where $\mathcal{X}$ is a locally compact Hausdorff space, and $\mathsf{A}=\{f_1,\dots,f_m\}$ a basis of $\mathcal{A}$. A sequence $s=(s_j)_{j=1}^m$ is called a moment sequence if $s_j=\int f_j(x) \, d\mu(x)$, $j=1,\dots,m$, for some positive Radon measure $\mu$ on $\mathcal{X}$. Each moment sequence $s$ has a finitely atomic representing measure $\mu$. The smallest possible number of atoms is called the Carath\'eodory number $\mathcal{C}_{\mathsf{A}}(s)$. The largest number $\mathcal{C}_{\mathsf{A}}(s)$ among all moment sequences $s$ is the Carath\'eodory number $\mathcal{C}_{\mathsf{A}}$. In this paper the Carath\'eodory numbers $\mathcal{C}_{\mathsf{A}}(s)$ and $\mathcal{C}_{\mathsf{A}}$ 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 $\mathcal{C}_{\mathsf{A}}$ 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 Carath\'eodory numbers.