Momentum ray transforms

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines with the weight $t^k$: $(I^k\text{\hspace{0.17em}⁣}f)(x,\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{t}^k⟨f(x+t\xi ),{\xi }^m⟩dt\mathrm{.}$ In particular, the ray transform $I=I^0$ was studied by several authors since it had many tomographic applications. We present an algorithm for recovering $f$ from the data $(I^0\text{\hspace{0.17em}⁣}f,I^1\text{\hspace{0.17em}⁣}f,\dots ,I^m\text{\hspace{0.17em}⁣}f)$. In the cases of $m=1$ and $m=2$, we derive the Reshetnyak formula that expresses $\mathrm{\parallel }f{\mathrm{\parallel }}_{H_t^s({\mathbb{R}}^n)}$ through some norm of $(I^0!f,I^1!f,\dots, I^m!f)$\mathrm{.}The$H^{s}_{t}$-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.