Spherical fractional integrals and their application to a problem
We first present some facts from the Spherical Harmonic Analysis,
related to decompositions of functions into series of spherical
harmonics and spherical convolution operators invariant with
respect to rotations. Then we use some properties of spherical
convolution operators to solve an integral equation over semishere
in the n-dimensional Euclidean space which arises in a certain
problem of aerodynamics. In this problem there is considered a
rarefied medium of non-interacting point masses moving at unit
velocity in all directions. Given the density of the velocity
distribution, one easily calculates the pressure created by the
medium in any direction. We consider the inverse problem: given the
pressure distribution, determine the density. This leads to the
problem of solving the above mentioned integral equation. In the
"application part" the talk is based on a joint paper with
Alexander Plakhov (Universidade de Aveiro).
