The Integral Geometry Problem in a Layer for Paraboloidal Surfaces with a Vertex Singularity

Abstract An integral geometry problem in a finite layer is considered for a family of paraboloidal surfaces of revolution possessing a singularity at the vertex. The aim is to reconstruct an unknown function from its surface integrals over this family of surfaces. After parameterizing the surfaces and computing the surface area element, the data operator is reduced to an integral equation of Volterra type. The application of the two-dimensional Fourier transform with respect to the horizontal variables and the Laplace transform with respect to the vertical variable reduces the problem to an algebraic relation containing a special auxiliary integral. The analytic properties of this integral are investigated. It is established that the symbol of the corresponding integral operator does not vanish for large values of the Fourier parameter. On this basis, the theoretical prerequisites for deriving an inversion formula, proving the uniqueness theorem, and obtaining stability estimates in Sobolev spaces are established.

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