Revisiting the Concentration Problem of Vector Fields within a Spherical Cap: A Commuting Differential Operator Solution

Abstract

We propose a novel basis of vector functions, the mixed vector spherical harmonics that are closely related to the functions of Sheppard and Török and help us reduce the concentration problem of tangential vector fields within a spherical cap to an equivalent scalar problem. Exploiting an analogy with previous results published by Grünbaum and his colleagues, we construct a differential operator that commutes with the concentration operator of this scalar problem and propose a stable and convenient method to obtain its eigenfunctions. Having obtained the scalar eigenfunctions, the calculation of tangential vector Slepian functions is straightforward.

Time and Band Limiting for Matrix Valued Functions, an Example

“…For a very recent account of several computational issues see [15,11] and [1]. For new areas of applications involving (sometimes) vector-valued quantities on the sphere, see [10,20,23,24].…”

A Unified Approach to Scalar, Vector, and Tensor Slepian Functions on the Sphere and Their Construction by a Commuting Operator

“…To reap the benefits of the spatiospectral analysis previously developed for Euclidean spaces, Albertella, Sansò, and Sneeuw [1], and later Simons, Dahlen, and Wieczorek [35,37] developed corresponding scalar-valued functions by spatiospectrally optimizing linear combinations of spherical harmonics and named the resulting orthogonal basis "Slepian functions". These Slepian functions found a wide range of applications in fields such as geodesy and geophysics, gravimetry, geodynamics, cosmology, planetary science, biomedical science, and in computer science (see [20,31] and the references therein).…”

“…Depending on the region of interest and the bandlimit, the underlying matrix can become ill-conditioned, leading to an eigenvalue problem, which is numerically unstable to solve. For some special regions, alternative eigenvalue problems based on commuting operators were discovered for the scalar and the vectorial case on the sphere (see [20,35,37]). The alternative problems have the same eigenvectors but are numerically stable.…”

A Further Look at Time-and-Band Limiting for Matrix Orthogonal Polynomials

“…For more details on computational issues see [2,18,22]. For applications involving (sometimes) vector-valued quantities on the sphere, see [17,26,29,30].…”