An application of intertwining operators in functional analysis

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.