Normal operators for momentum ray transforms, I: The inversion formula

Abstract

The momentum ray transform $I_m^k$ integrates a rank $m$ symmetric tensor field $f$ on ${\mathbb{R}}^n$ over lines with the weight $t^k$, $I_m^{k}f(x,\hat{I}¾)={\int }_{-\infty }^{\infty }{t}^k⟨f(x+t\hat{I}¾),\hat{I}{¾}^m⟩\mathrm{d}t$. We compute the normal operator $N_m^k=(I_m^k){}^{\ast }{I}_m^k$ and present an inversion formula recovering a rank $m$ tensor field $f$ from the data $(N_m^{0}f,\dots ,N_m^{m}f)$.

29 pages. This manuscript has been accepted for publication in the Journal of Fourier Analysis and Applications