The eigenfunctions of the Karhunen–Loeve integral equation for a spherical system

The eigenfunctions and eigenvalues of the Karhunen–Loeve integral equation are found for an exponential covariance function in a spherical system. The solution is given in terms of a set of subsidiary integral equations, the kernels of which are the spherical harmonic moments of the covariance function. These are relatively simple and can be solved numerically or analytically. Results are given in the form of a table of eigenvalues λ ̄ n ℓ where n = 0 , 1 , 2 , 3 , 4 and ℓ = 1 , 2 , 3 , 4 . The associated eigenfunctions are given graphically and consistency with certain conservation requirements is demonstrated. A practical example is given, based upon neutron diffusion in a spherical system where these eigenfunctions are required.