New classes of integral geometry problems of Volterra type in three-dimensional space.

During the past decade, our society has become dependent on advanced mathematics for many of our daily needs. Mathematics is at the heart of the 21st century technologies and more specifically the emerging imaging technologies from thermoacoustic tomography and ultrasound computed tomography to nondestructive testing. All of these applications reconstruct the internal structure of an object from external measurements without damaging the entity under investigation. Very often the basic mathematical idea common to such reconstruction problems is based upon integral geometry. In this paper considers the problem of recovering a function from families of spheres in space. The uniqueness of the solution of the problem is proved by reducing it to the Volterra integral equation of the first and then the second kind. The methods of the theory of partial differential equations are applied. The proof of the uniqueness theorem is based on the researching of boundary value problems for auxiliary functions. Fourier transform methods are also used. Uniqueness theorems are proved for some new classes of operator equations of Volterra type in three-dimensional space. [ABSTRACT FROM AUTHOR]