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

<p style='text-indent:20px;'>For an integer <inline-formula><tex-math id="M1">\begin{document}$ r\ge0 $\end{document}</tex-math></inline-formula>, we prove the id="M2">\begin{document}$ r^{\mathrm{th}} $\end{document}</tex-math></inline-formula> order Reshetnyak formula for ray transform of rank id="M3">\begin{document}$ m symmetric tensor fields on id="M4">\begin{document}$ {{\mathbb R}}^n $\end{document}</tex-math></inline-formula>. Roughly speaking, a field id="M5">\begin{document}$ f id="M6">\begin{document}$ r refers to id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-integrability higher derivatives Fourier id="M8">\begin{document}$ \widehat over spheres centered at origin. Certain differential operators id="M9">\begin{document}$ A^{(m,r,l)}\ (0\le l\le r) sphere id="M10">\begin{document}$ S}}^{n-1} are main ingredients formula. The defined by algorithm that can be applied any id="M11">\begin{document}$ although volume calculations grows fast with id="M12">\begin{document}$ is realized small values id="M13">\begin{document}$ and formulas orders id="M14">\begin{document}$ 0,1,2 presented in explicit form.</p>