Range conditions for the exponential Radon transform
The exponential Radon transform is defined by \[ {\mathfrak R}_\mu(f)(p,\omega)=\int_{x\cdot\omega=p} f(x)e^{\mu x\cdot\omega^\perp}dx \] where \(f\) is a compactly supported function on the plane \(\mathbb{R}^2\), \(\omega=(\omega_1,\omega_2)\in S^1\subset\mathbb{R}^2\) is a unit vector, \(\omega^\perp=(-\omega_2,\omega_1)\), \(\mu\) is a real number, \(x\cdot\omega=x_1\omega_1+x_2\omega_2\), and \(p\) is a real number, in the case when \(\mu=0\) we get the standard Radon transform. \textit{O. Y. Tretiak} and \textit{C. Metz} [SIAM J. Appl. Math. 39, 341-354 (1980; Zbl 0459.44003)] and \textit{P. Kuchment} and \textit{S. L'vin} [Acta Appl. Math. 18, No. 3, 251-260 (1990; Zbl 0705.44001)]\ and Dokl. Akad. Nauk SSSR 313, No. 6, 1329-1330 (1990; Zbl 0748.44002)]\ have given range conditions for the transform \({\mathfrak R}_\mu\). In this paper the authors give a new form of range conditions for \({\mathfrak R}_\mu\) which has two advantages: First, it has some simple geometric meaning, which disappears when \(\mu=0\), secondly it permits to give some necessary range conditions in the case of any dimension.
scientific article; zbMATH DE number 927192