The Radon transformation in a band in connection with ray seismic tomography (Q1377929)

The following variant of the Radon transformation is studied: \[ g(q,\eta) = \mathbb{R} u(x,z) \equiv \int^1_{-1} u(x,\eta +qx)dx, \quad (q,\eta) \in{\mathcal R}^2. \tag{1} \] Transformation (1) has been previously investigated in connection with controlled directed reception, fan filtration, and other problems in seismic processing. This transformation is reduced to the classic angular Radon transformation by a change of variables. However, complete identity between the classic and considered transformations is absent: they are connected much as the problems of the approximation of functions on the interval (-1,1) and on a circle.