Surface-Fitting and Analysis of Scattered Data via Radial and Related Basis Functions with Applications to Neural Networks

摘要

The research described below was carried out during the period 1 February 1998 to 30 September 2001. We have made significant progress on several fronts. We have obtained positive-weight quadrature rules that are exact for spherical harmonics of prescribed order mid that allow function evaluations at scattered points, and we have given algorithms for obtaining these weights. Based on these rules, we were able to construct neural networks for spheres using zonal activation functions. We also made progress on the difficult problem of locating multiple sources with neural networks. On another front, we provided error estimates for interpolating less smooth functions via networks with smooth activation functions; this is the first result of its kind. In addition, we provided a class of functions to which our error estimates apply; these functions are both easy to use and locally supported, so that interpolation matrices arising from them will be banded.