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 $t^k$, $I_m^kf(x,\xi)=\int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\,\mathrm{d}t$. We compute the normal operator $N_m^k=(I_m^k){}^*I_m^k$ and present an inversion formula recovering a rank $m$ tensor field $f$ from the data $(N_m^0f,\dots,N_m^mf)$.