Momentum ray transforms

Abstract: The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines with the weight $t^k$: $ (I^k!f)(x,\xi)=\int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\,dt. $ 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 $(I^0!f,I^1!f,\dots, I^m!f)$. In the cases of $m=1$ and $m=2$, we derive the Reshetnyak formula that expresses $|f|{H^s_t({\mathbb{R}}^n)}$ through some norm of $(I^0!f,I^1!f,\dots, I^m!f)$. 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.