Abstract:
We consider classes of integral operators on the spaces of square-integrable
functions on the sphere and of locally integrable functions on Lobachevsky
space. The kernels of these operators depend only on the distance
between points in the spherical and hyperbolic geometry, respectively. These
operators are intertwining for the quasi-regular representation of the
corresponding Lie group, and this enables us to evaluate their spectra and
diagonalize the operators themselves. As applications, we take the Minkowski
problem and the Funk–Hecke theorem for Euclidean space $\mathbb R^n$.
A generalization is obtained of the Funk–Hecke theorem in the case of
hyperbolic space $\mathbb R^{n-1,1}$ with indefinite inner product.