
SIGMA 11 (2015), 077, 10 pages arXiv:1508.06884
https://doi.org/10.3842/SIGMA.2015.077
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications
Moments and LegendreFourier Series for Measures Supported on Curves
Jean B. Lasserre
LAASCNRS and Institute of Mathematics, University of Toulouse, 7 Avenue du Colonel Roche, BP 54 200, 31031 Toulouse Cédex 4, France
Received August 28, 2015, in final form September 26, 2015; Published online September 29, 2015
Abstract
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'' $\{(t,x(t))\colon t\in [0,T]\}$ 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 $\{(t,x(t))\colon t\in [0,1]\}$. Those conditions are stated in terms of LegendreFourier coefficients ${\mathbf f}_j=({\mathbf f}_j(i))$ associated with some functions $f_j\colon [0,1]\to {\mathbb R}$, $j=1,\ldots$, where each ${\mathbf f}_j$ is obtained from the moments $\gamma_{ji}$, $i=0,1,\ldots$, of $\mu$.
Key words:
moment problem; Legendre polynomials; LegendreFourier series.
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