THE RESOLUTION OF THE GIBBS PHENOMENON FOR SPHERICAL-HARMONICS

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 G ibbs phenomenon was already observed by Weyl for spherical harmonic ex pansion approximations to functions with discontinuities, causing unde sirable oscillations over the entire sphere. Recently, methods for rem oving the Gibbs phenomenon for one-dimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved tha t the knowledge of the first N expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function f(x) is enough to reco ver 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 h armonic coefficients yield an exponentially convergent approximation t o a spherical piecewise smooth function f(theta, phi) in any subinterv al [theta(1), theta(2)], phi is an element of [0,2 pi], where the func tion is analytic. Thus we entirely overcome the Gibbs phenomenon.