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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| Дата: | 2015 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2015
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/147149 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1471492019-02-14T01:26:24Z Moments and Legendre-Fourier Series for Measures Supported on Curves Lasserre, J.B. 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. 2015 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/147149 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Lasserre, J.B. |
| spellingShingle |
Lasserre, J.B. Moments and Legendre-Fourier Series for Measures Supported on Curves Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Lasserre, J.B. |
| author_sort |
Lasserre, J.B. |
| title |
Moments and Legendre-Fourier Series for Measures Supported on Curves |
| 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 |
| publisher |
Інститут математики НАН України |
| publishDate |
2015 |
| url |
http://dspace.nbuv.gov.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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT lasserrejb momentsandlegendrefourierseriesformeasuressupportedoncurves |
| first_indexed |
2023-05-20T17:26:43Z |
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2023-05-20T17:26:43Z |
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1796153310764335104 |