A Multiresolution Tensor Spline Method for Fitting Functions on the Sphere

We present the details of a multiresolution method which we proposed at the Taormina Wavelet Conference in 1993 (see "L-spline wavelets" in Wavelets: Theory, Algorithms, and Applications, C. Chui, L. Montefusco, and L. Puccio, eds., Academic Press, New York, pp. 197--212) which is suitable for fitting functions or data on the sphere. The method is based on tensor products of polynomial splines and trigonometric splines and can be adapted to produce surfaces that are either tangent plane continuous or almost tangent plane continuous. The result is a convenient compression algorithm for dealing with large amounts of data on the sphere. We give full details of a computer implementation that is highly efficient with respect to both storage and computational cost. We also demonstrate the performance of the method on several test examples.