On the Approximation Order of Splines on Spherical Triangulations

Bounds are provided on how well functions in Sobolev spaces on the sphere can be approximated by spherical splines, where a spherical spline of degree d is a C r function whose pieces are the restrictions of homogoneous polynomials of degree d to the sphere. The bounds are expressed in terms of appropriate seminorms deened with the help of radial projection, and are obtained using appropriate quasi-interpolation operators. x1. Introduction In a series of papers 2{4], together with P. Alfeld we have developed a theory of spherical splines on triangulations of the sphere in IR 3. Such splines closely resemble the classical piecewise polynomials on planar triangulations. Whereas splines in the plane are ordinary piecewise polynomials, spherical splines are formed by piecing together so-called spherical polynomials which are deened as the restrictions of homogeneous trivariate polynomials to the sphere. Since spherical polynomials are combinations of spherical harmonics, they are good candidates for constructing spaces of splines on the sphere. In 4] we presented a variety of methods for tting scattered data on the sphere using spherical splines. These included analogs of several classical bivariate methods , such as interpolation, least squares, and minimum-energy methods. That paper also presented the results of extensive numerical experimentation, and connrmed our expectations that spherical splines perform very well numerically. However, in contrast to the planar case, to date no theory has been developed to justify these experimental ndings. In this paper we ll this gap by providing bounds on the error of approximation of smooth functions by spherical splines. As in the planar case, the error bounds will be expressed in terms of the mesh size (the diameter of the largest triangle in a given triangulation), and also in terms of appropriate seminorms measuring the smoothness of the approximated functions. The error estimates will show that for spherical splines of suuciently high degree d, namely d 3r + 2, where r is the