Handling ray transform of symmetric tensor fields and the Radon transform of differential forms on incomplete data

2014 ◽  
pp. 56-70
Author(s):  
Ziyaudin Medzhidov ◽  
Keyword(s):  
Radon Transform ◽  
Incomplete Data ◽  
Differential Forms ◽  
Symmetric Tensor ◽  
Tensor Fields ◽  
Ray Transform

scholarly journals Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Venkateswaran P. Krishnan ◽  
Vladimir A. Sharafutdinov
Keyword(s):  
Fourier Transform ◽  
Sobolev Spaces ◽  
Tensor Field ◽  
Differential Operators ◽  
Symmetric Tensor ◽  
Higher Order ◽  
Tensor Fields ◽  
Ray Transform ◽  
The Fourier Transform ◽  
Derivatives Of

<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>

scholarly journals Microlocal inversion of a 3-dimensional restricted transverse ray transform on symmetric tensor fields

2021 ◽  
Vol 495 (1) ◽  
pp. 124700
Author(s):  
Venkateswaran P. Krishnan ◽  
Rohit Kumar Mishra ◽  
Suman Kumar Sahoo
Keyword(s):  
Symmetric Tensor ◽  
Tensor Fields ◽  
Ray Transform ◽  
3 Dimensional

scholarly journals A support theorem for the geodesic ray transform of symmetric tensor fields

2009 ◽  
Vol 3 (3) ◽  
pp. 453-464 ◽  
Author(s):  
Venkateswaran P. Krishnan ◽  
◽  
Plamen Stefanov ◽  
Keyword(s):  
Symmetric Tensor ◽  
Tensor Fields ◽  
Support Theorem ◽  
Ray Transform ◽  
Geodesic Ray Transform

scholarly journals The spinor and tensor fields with higher spin on spaces of constant curvature

2021 ◽  
Author(s):  
Yasushi Homma ◽  
Takuma Tomihisa
Keyword(s):  
Constant Curvature ◽  
Differential Operators ◽  
Differential Forms ◽  
Symmetric Tensor ◽  
Tensor Fields ◽  
Spaces Of Constant Curvature ◽  
Standard Sphere ◽  
Spinor Fields ◽  
Spin Manifolds ◽  
Factorization Formula

AbstractIn this article, we give all the Weitzenböck-type formulas among the geometric first-order differential operators on the spinor fields with spin $$j+1/2$$ j + 1 / 2 over Riemannian spin manifolds of constant curvature. Then, we find an explicit factorization formula of the Laplace operator raised to the power $$j+1$$ j + 1 and understand how the spinor fields with spin $$j+1/2$$ j + 1 / 2 are related to the spinors with lower spin. As an application, we calculate the spectra of the operators on the standard sphere and clarify the relation among the spinors from the viewpoint of representation theory. Next we study the case of trace-free symmetric tensor fields with an application to Killing tensor fields. Lastly we discuss the spinor fields coupled with differential forms and give a kind of Hodge–de Rham decomposition on spaces of constant curvature.

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