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: | Englisch |
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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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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| 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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862595469585154048 |
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| author | Lasserre, J.B. |
| author_facet | Lasserre, J.B. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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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| first_indexed | 2025-11-27T13:06:54Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147149 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-27T13:06:54Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
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| spelling | 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 |
| spellingShingle | Moments and Legendre-Fourier Series for Measures Supported on Curves Lasserre, J.B. |
| title | 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_short | Moments and Legendre-Fourier Series for Measures Supported on Curves |
| title_sort | moments and legendre-fourier series for measures supported on curves |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147149 |
| work_keys_str_mv | AT lasserrejb momentsandlegendrefourierseriesformeasuressupportedoncurves |