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

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

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