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This article is cited in 1 scientific paper (total in 1 paper)
Moments and Legendre–Fourier Series for Measures Supported on Curves
Jean B. Lasserre LAAS-CNRS and Institute of Mathematics, University of Toulouse,
7 Avenue du Colonel Roche, BP 54 200, 31031 Toulouse Cédex 4, France
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 Legendre–Fourier 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 $\mu$.
Keywords:
moment problem; Legendre polynomials; Legendre–Fourier series.
Received: August 28, 2015; in final form September 26, 2015; Published online September 29, 2015
Citation:
Jean B. Lasserre, “Moments and Legendre–Fourier Series for Measures Supported on Curves”, SIGMA, 11 (2015), 077, 10 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1058 https://www.mathnet.ru/eng/sigma/v11/p77
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