An example of non-uniqueness for Radon transforms with continuous positive rotation invariant weights

Abstract

We consider weighted Radon transforms $R_W$ along hyperplanes in $R^3$ with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel $\mathrm{Ker}R_W$ in the space of infinitely smooth compactly supported functions and with continuous weight. Moreover, in this example the weight $W$ is rotation invariant. In particular, by this result we continue studies of Quinto (1983), Markoe, Quinto (1985), Boman (1993) and Goncharov, Novikov (2017). We also extend our example to the case of weighted Radon transforms along two-dimensional planes in $R^d$ , $d \geq 3$.