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 which is 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)): t is an element of [0, T]} for some measurable function x(t). We provide necessary and sufficient conditions on moments (gamma(ij)) of a measure d mu(x,t) on [0,1](2) to ensure that mu is supported on a trajectory {(t, x(t)): t is an element of [0,1]}. Those conditions are stated in terms of Legendre Fourier coefficients f(j) = (f(j)(i)) associated with some functions f(j): [0,1] -> R, j = 1, ... , where each f(j) is obtained from the moments gamma(ji), i = 0, 1, ... , of mu.