Moments and Legendre-Fourier Series for Measures Supported on Curves

Some important problems (e.g., in optimal transport and optimal control) have a relaxed (or weak) formulation in a space of appropriate measures whichis much easier to solve. However, an optimal solution $\mu$ of the latter solves the former if and only if the measure $\mu$ is supported on a “trajectory” $\{(t,x(t))\text{\hspace{0.17em}⁣}:t\in [0,T]\}$ for some measurable function $x(t)$. We provide necessary and sufficient conditions on moments $(\gamma \mathrm{\_}ij)$ of a measure $d\mu (x,t)$ on $[0,1{]}^2$ to ensure that $\mu$ is supported on a trajectory $\{(t,x(t))\text{\hspace{0.17em}⁣}:t\in [0,1]\}$. Those conditions are stated in terms of Legendre-Fourier coefficients $\mathbf{f}\mathrm{\_}j=(\mathbf{f}\mathrm{\_}j(i))$ associated with some functions $f\mathrm{\_}j\text{\hspace{0.17em}⁣}:[0,1]\to \mathbb{R}$, $j=1,\dots$, where each $\mathbf{f}\mathrm{\_}j$ is obtained from the moments $\gamma \mathrm{\_}ji$, $i=0,1,\dots$, of $\mu$.