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...
Збережено в:
| 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| Резюме: | 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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