Fast evaluation of radial basis functions. I (Q1207553)

For the calculation of the potential of many-body systems several authors have introduced the technique of hierarchical and multipole expansions. In this paper the authors report about this technique in detail and describe its application for the rapid evaluation and fitting of radial basis functions. In particular, this is performed for the \(N\) term thin- plate spline \(s(x)=\sum_{j=1}^ N d_ j\varphi(x-x_ j)\), where \(\varphi(u)=\| u\|_ 2^ 2\log\| u\|_ 2\) in 2- dimensions. Lemmata are presented on series expansions for \(\varphi(x)\) which help to reduce the computational expense and provide error bounds for the truncated series expansions.