Heun Polynomials in the Construction of Vector Valued Slepian Functions on a Spherical Cap

Abstract

I summarize the existing work on the problem of finding vector valued Slepian functions on the unit sphere: separable vector fields whose energy is concentrated within a compact region; in this case, a spherical cap. The radial and tangential components are independent for an appropriate choice of basis, and for each component the problem is recast as that of finding real eigenfunctions of an integral operator. There exist Sturm-Liouville differential operators that commute with these integral operators and hence share their eigenfunctions. Therefore, the radial and tangential eigenfunctions are solutions to second order linear ODEs. After introducing the Heun differential equation and some of its basic properties, I show how our equations can be put into Heun form by a change of variables, at which point the Slepian functions can be expressed in terms of Heun polynomials: polynomial solutions to a Heun equation.