The resolution of the Gibbs phenomenon for spherical harmonics

Abstract:

Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable oscillations over the entire sphere.

Recently, methods for removing the Gibbs phenomenon for one-dimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first $N$ expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function $f(x)$ is enough to recover an exponentially convergent approximation to the point values of $f(x)$ in any subinterval in which the function is analytic.

Here we take a similar approach, proving that knowledge of the first $N$ spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function $f(\theta ,\phi )$ in any subinterval $[\theta _1,\theta _2], \phi \in [0,2\pi ]$, where the function is analytic. Thus we entirely overcome the Gibbs phenomenon.