FaVeST: Fast Vector Spherical Harmonic Transforms

Abstract: Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have broad applications in geophysics, quantum mechanics and astrophysics. In the representation of a tangent vector field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of vector spherical harmonics with a degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated tangent fields illustrate the accuracy, efficiency and stability of FaVeST.