On the Spectral Approximation of Discrete Scalar and Vector Functions on the Sphere

Several topics are discussed which concern the representation of a discrete function on the sphere in terms of surface spherical harmonics. Several methods are reviewed, with particular attention given to the recent work of Machenhauer and Daley. The aliasing of the spherical harmonics is discussed and for a given grid a finite number of spherical harmonics are chosen as the discrete basis. A harmonic is included in the basis if and only if it does not alias on the grid. The number of basis functions is about half the number of grid points, with the result that the approximation may not fit all the function values. Nevertheless, it is shown that a weighted least-squares approximation is obtained. This approximation has the property that waves are resolved uniformly on the sphere. That is, if the discrete function is replaced by a tabulation of its spectral representation, then the high frequencies which are artificially induced by the closeness of the grid points near the pole are removed. The approximation of discrete vector functions is also discussed. A vector function which is smooth in cartesian coordinates will likely be discontinuous or have a discontinuous derivative in spherical coordinates. Hence, an approximation in terms of spherical harmonics is unsatisfactory due to the undesirable convergence characteristics. A method is given in which the convergence characteristics are determined by the smoothness of the vector function in cartesian coordinates.