Adaptive Least Squares Fitting with Radial Basis Functions on the Sphere

We investigate adaptive least squares approximation to scattered data given over the surface of the unit sphere in IR 3. Two basic algorithms (as well as some modiications) are described and evaluated: knot insertion and knot removal. x1. Introduction Since their introduction by R. L. Hardy in the 1970s, radial basis functions have become a standard tool in CAGD and approximation theory for the construction of surfaces interpolating and approximating scattered data. Most of the work on scattered data approximation with radial basis functions has been in the Euclidean setting (see e.g. 4], and some references therein). On the sphere interpolation was treated e.g. in 1,3,7,8]. So far there has been no discussion of the use of radial basis functions for approximation over the sphere. This, however, is a problem of great interest in many areas of the applied sciences, in particular geodesy and meteorology. The purpose of this paper is to investigate adaptive approximation to scattered data deened over the surface of the sphere. Our main goal is to use only as few knots as necessary (selected according to some quality criterion) to achieve a desired accuracy for the t. We describe two algorithms { one based on adaptively inserting knots, the other on knot removal { and compare the results. The paper is organized as follows. We formulate the general approximation problem in Section 2, and specialize the discussion to adaptive least squares approximation using knot insertion and knot removal in Sections 3 and 4, respectively. In Section 5 we brieey mention some possible modiica-tions of the two basic algorithms and our experience with them. We close the paper with a few examples in Section 6.