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Moments and Legendre-Fourier Series for Measures Supported on Curves
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2015, том 11
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Moments and Legendre-Fourier Series for Measures Supported on Curves
Інший ID:
2010 Mathematics Subject Classification: 42C05; 42C10; 42A16; 44A60
DOI:10.3842/SIGMA.2015.077
DOI:10.3842/SIGMA.2015.077
Посилання:
Moments and Legendre-Fourier Series for Measures Supported on Curves / J.B. Lasserre // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 15 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications.
The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html.
Research funded by the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation program (grant agreement ERC-ADG 666981 TAMING).
Дата:
2015
Переглядів:
510
Завантажень:
250
Анотація:
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 μ of the latter solves the former if and only if the measure μ is supported on a ''trajectory'' {(t,x(t)):t∈[0,T]} for some measurable function x(t). We provide necessary and sufficient conditions on moments (γij) of a measure dμ(x,t) on [0,1]² to ensure that μ is supported on a trajectory {(t,x(t)):t∈[0,1]}. Those conditions are stated in terms of Legendre-Fourier coefficients fj=(fj(i)) associated with some functions fj:[0,1]→R, j=1,…, where each fj is obtained from the moments γji, i=0,1,…, of μ.
