{"title":"对称张量场的Sobolev空间上的射线变换I:高阶Reshetnyak公式","authors":"Venky Krishnan, V. Sharafutdinov","doi":"10.3934/ipi.2021076","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>For an integer <inline-formula><tex-math id=\"M1\">\\begin{document}$ r\\ge0 $\\end{document}</tex-math></inline-formula>, we prove the <inline-formula><tex-math id=\"M2\">\\begin{document}$ r^{\\mathrm{th}} $\\end{document}</tex-math></inline-formula> order Reshetnyak formula for the ray transform of rank <inline-formula><tex-math id=\"M3\">\\begin{document}$ m $\\end{document}</tex-math></inline-formula> symmetric tensor fields on <inline-formula><tex-math id=\"M4\">\\begin{document}$ {{\\mathbb R}}^n $\\end{document}</tex-math></inline-formula>. Roughly speaking, for a tensor field <inline-formula><tex-math id=\"M5\">\\begin{document}$ f $\\end{document}</tex-math></inline-formula>, the order <inline-formula><tex-math id=\"M6\">\\begin{document}$ r $\\end{document}</tex-math></inline-formula> refers to <inline-formula><tex-math id=\"M7\">\\begin{document}$ L^2 $\\end{document}</tex-math></inline-formula>-integrability of higher order derivatives of the Fourier transform <inline-formula><tex-math id=\"M8\">\\begin{document}$ \\widehat f $\\end{document}</tex-math></inline-formula> over spheres centered at the origin. Certain differential operators <inline-formula><tex-math id=\"M9\">\\begin{document}$ A^{(m,r,l)}\\ (0\\le l\\le r) $\\end{document}</tex-math></inline-formula> on the sphere <inline-formula><tex-math id=\"M10\">\\begin{document}$ {{\\mathbb S}}^{n-1} $\\end{document}</tex-math></inline-formula> are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any <inline-formula><tex-math id=\"M11\">\\begin{document}$ r $\\end{document}</tex-math></inline-formula> although the volume of calculations grows fast with <inline-formula><tex-math id=\"M12\">\\begin{document}$ r $\\end{document}</tex-math></inline-formula>. The algorithm is realized for small values of <inline-formula><tex-math id=\"M13\">\\begin{document}$ r $\\end{document}</tex-math></inline-formula> and Reshetnyak formulas of orders <inline-formula><tex-math id=\"M14\">\\begin{document}$ 0,1,2 $\\end{document}</tex-math></inline-formula> are presented in an explicit form.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"51 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2021-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas\",\"authors\":\"Venky Krishnan, V. 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Roughly speaking, for a tensor field <inline-formula><tex-math id=\\\"M5\\\">\\\\begin{document}$ f $\\\\end{document}</tex-math></inline-formula>, the order <inline-formula><tex-math id=\\\"M6\\\">\\\\begin{document}$ r $\\\\end{document}</tex-math></inline-formula> refers to <inline-formula><tex-math id=\\\"M7\\\">\\\\begin{document}$ L^2 $\\\\end{document}</tex-math></inline-formula>-integrability of higher order derivatives of the Fourier transform <inline-formula><tex-math id=\\\"M8\\\">\\\\begin{document}$ \\\\widehat f $\\\\end{document}</tex-math></inline-formula> over spheres centered at the origin. Certain differential operators <inline-formula><tex-math id=\\\"M9\\\">\\\\begin{document}$ A^{(m,r,l)}\\\\ (0\\\\le l\\\\le r) $\\\\end{document}</tex-math></inline-formula> on the sphere <inline-formula><tex-math id=\\\"M10\\\">\\\\begin{document}$ {{\\\\mathbb S}}^{n-1} $\\\\end{document}</tex-math></inline-formula> are main ingredients of the formula. 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引用次数: 2
摘要
For an integer \begin{document}$ r\ge0 $\end{document}, we prove the \begin{document}$ r^{\mathrm{th}} $\end{document} order Reshetnyak formula for the ray transform of rank \begin{document}$ m $\end{document} symmetric tensor fields on \begin{document}$ {{\mathbb R}}^n $\end{document}. Roughly speaking, for a tensor field \begin{document}$ f $\end{document}, the order \begin{document}$ r $\end{document} refers to \begin{document}$ L^2 $\end{document}-integrability of higher order derivatives of the Fourier transform \begin{document}$ \widehat f $\end{document} over spheres centered at the origin. Certain differential operators \begin{document}$ A^{(m,r,l)}\ (0\le l\le r) $\end{document} on the sphere \begin{document}$ {{\mathbb S}}^{n-1} $\end{document} are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any \begin{document}$ r $\end{document} although the volume of calculations grows fast with \begin{document}$ r $\end{document}. The algorithm is realized for small values of \begin{document}$ r $\end{document} and Reshetnyak formulas of orders \begin{document}$ 0,1,2 $\end{document} are presented in an explicit form.
Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas
For an integer \begin{document}$ r\ge0 $\end{document}, we prove the \begin{document}$ r^{\mathrm{th}} $\end{document} order Reshetnyak formula for the ray transform of rank \begin{document}$ m $\end{document} symmetric tensor fields on \begin{document}$ {{\mathbb R}}^n $\end{document}. Roughly speaking, for a tensor field \begin{document}$ f $\end{document}, the order \begin{document}$ r $\end{document} refers to \begin{document}$ L^2 $\end{document}-integrability of higher order derivatives of the Fourier transform \begin{document}$ \widehat f $\end{document} over spheres centered at the origin. Certain differential operators \begin{document}$ A^{(m,r,l)}\ (0\le l\le r) $\end{document} on the sphere \begin{document}$ {{\mathbb S}}^{n-1} $\end{document} are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any \begin{document}$ r $\end{document} although the volume of calculations grows fast with \begin{document}$ r $\end{document}. The algorithm is realized for small values of \begin{document}$ r $\end{document} and Reshetnyak formulas of orders \begin{document}$ 0,1,2 $\end{document} are presented in an explicit form.
期刊介绍:
Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing.
This journal is committed to recording important new results in its field and will maintain the highest standards of innovation and quality. To be published in this journal, a paper must be correct, novel, nontrivial and of interest to a substantial number of researchers and readers.