On the Domain of Convergence of Spherical Harmonic Expansions

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