On the domain of convergence of spherical harmonic expansions

TL;DR

This paper precisely characterizes the domain of convergence for spherical harmonic expansions of planetary gravitational potentials, showing convergence occurs exactly outside the Brillouin sphere with rare exceptions, and relates convergence to analyticity at the highest peak.

Contribution

It provides the first rigorous proof of the convergence domain for SHEs of planets with complex topography, and establishes conditions linking convergence to analyticity at the highest peak.

Findings

SHE converges exactly outside the Brillouin sphere for generic planets.

Convergence below the Brillouin sphere occurs with probability zero.

Analyticity at the highest peak is necessary and sufficient for convergence below the Brillouin sphere.

Abstract

Spherical harmonic expansions (SHEs) play an important role in most of the physical sciences, especially in physical geodesy. Despite many decades of investigation, the large order behavior of the SHE coefficients, and the precise domain of convergence for these expansions, have remained open questions. These questions are settled in the present paper for generic planets, whose shape (topography) may include many local peaks, but just one globally highest peak. We show that regardless of the smoothness of the density and topography, short of outright analyticity, the spherical harmonic expansion of the gravitational potential converges exactly in the closure of the exterior of the Brillouin sphere (The smallest sphere around the center of mass of the planet containing the planet in its interior), and convergence below the Brillouin sphere occurs with probability zero. More precisely, such over-convergence occurs on zero measure sets in the space of parameters. A related result is that, in a natural Banach space, SHE convergence of the potential below the Brillouin sphere occurs for potential functions in a subspace of infinite codimension (while any positive codimension already implies occurrence of probability zero). Provided a certain limit in Fourier space exists, we find the leading order asymptotic behavior of the coefficients of SHEs. We go further by finding a necessary and sufficient condition for convergence below the Brillouin sphere, which requires a form of analyticity at the highest peak, which would not hold for a realistic celestial body. Namely, a longitudinal average of the harmonic measure on the Brillouin sphere would have to be real-analytic at the point of contact with the boundary of the planet. It turns out that only a small neighborhood of the peak is involved in this condition.