Treffer: Fast Evaluation of Radial Basis Functions: Methods for Four-Dimensional Polyharmonic Splines: Fast evaluation of radial basis functions: Methods for four-dimensional polyharmonic splines

This paper is organized as follows. One section deals with some of the properties of polyharmonic functions on \(\mathbb{R}^4\), including realizations of \(\mathbb{R}^4\). A significant technique is the use of a group action perspective, in particular, of arguments based on the action of the group of non-zero quaternions, realized as \(2\times 2\) complex matrices \[ H_0'=\left\{x=\left[\begin{matrix} \r & \quad \r\\ z & w\\ -\overline w & \overline z\end{matrix}\right]:|z|^2+|w|^2>0\right\} \] acting on \(\mathbb{C}^2=\mathbb{R}^4\). The authors also introduce the inner and outer functions that form the basis of the far field expansions. The next section develops a number of properties of these functions that can be applied to far field expansions. These include recurrence formulae, derivative formula, and symmetries. The middle section contains the main results on the far field expansions themselves and the associated error bounds. The last two sections contain uniqueness results and outer-to-inner and inner-to-inner translation formulae needed to approximate far field series by local Taylor series.