Normal Operators for Momentum Ray Transforms, I: The Inversion Formula

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