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 μ of the latter solves the former if and only if the measure μ is supported on a ''tr...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2015 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут математики НАН України
2015
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/147149 |
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| Zitieren: | 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. |
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Lasserre, J.B. 2019-02-13T17:42:19Z 2019-02-13T17:42:19Z 2015 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. 1815-0659 2010 Mathematics Subject Classification: 42C05; 42C10; 42A16; 44A60 DOI:10.3842/SIGMA.2015.077 https://nasplib.isofts.kiev.ua/handle/123456789/147149 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. 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). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Moments and Legendre-Fourier Series for Measures Supported on Curves Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Moments and Legendre-Fourier Series for Measures Supported on Curves |
| spellingShingle |
Moments and Legendre-Fourier Series for Measures Supported on Curves Lasserre, J.B. |
| title_short |
Moments and Legendre-Fourier Series for Measures Supported on Curves |
| title_full |
Moments and Legendre-Fourier Series for Measures Supported on Curves |
| title_fullStr |
Moments and Legendre-Fourier Series for Measures Supported on Curves |
| title_full_unstemmed |
Moments and Legendre-Fourier Series for Measures Supported on Curves |
| title_sort |
moments and legendre-fourier series for measures supported on curves |
| author |
Lasserre, J.B. |
| author_facet |
Lasserre, J.B. |
| publishDate |
2015 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147149 |
| citation_txt |
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. |
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AT lasserrejb momentsandlegendrefourierseriesformeasuressupportedoncurves |
| first_indexed |
2025-11-27T13:06:54Z |
| last_indexed |
2025-11-27T13:06:54Z |
| _version_ |
1850852288656048128 |