INV

Abstract

For a given smooth, positive function $m(x, \xi, \eta)$ we consider a weighted Radon transform $R$ defined by $Rf(\xi, \eta) = \int f(x, \xi x + \eta) m(\xi, \eta, x) dx$ for functions $f(x, y)$ that are defined in some neighborhood of the origin and are supported in $y\ge x^2$. The question is for which $m(x, \xi, \eta)$ it is true that $R$ is injective. A similar problem when the family of lines $y = \xi x + \eta$ is replaced by a family of curves is also considered.