Momentum ray transforms

The momentum ray transform I ^k integrates a rank m symmetric tensor field f over lines in R ⁿ with the weight t ^k : (I ^k f)(x,ξ)=∫ ^∞ _-∞ t ^k 〈 f(x+tξ),ξ ^m〉 dt. In particular, the ray transform I=I0 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,…,I ^m f). In the cases of m=1 and m=2, we derive the Reshetnyak formula that expresses ∥f∥Hst(ℝn) through some norm of (I ⁰ f,I ¹ f,…,I ^m f). The Hst-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.