On expansion of analytical functions in ultraspherical polynomials on the sphere

An algorithm is proposed for computing two-dimensional Fourier series in a spherical system of coordinates over a set of orthogonal ultraspherical polynomials whose special cases are the Legendre polynomials and Chebyshev polynomials of the first and second kind. The series uniformly converge at all points of the sphere including the poles. Unlike traditional spectral expansions on the sphere, they explicitly contain additional terms that characterize an odd component of a desired analytical function relative to the poles. It is shown that the expansion in the small vicinity of the poles (polar caps) is simplified because of the closeness to zero of the Fourier series terms responsible for the approximation of the components of the function that are odd relative to the poles. As the equatorial zone is approached, the value of the components of the desired function unsymmetrical relative to the poles increases and becomes comparable to the contribution of the symmetrical components. The new method is used for a special case of the spectral approximation of a continuous scalar analytical function with spherical harmonics used as the orthogonal basis. It is shown that double Fourier series in this case give an extension of the traditional spectral method. An alternative is to construct double Fourier series in associated Chebyshev polynomials of the first and second kind. An example of the spectral approximation of an analytical function on the sphere is presented.