Fast evaluation of radial basis functions : methods for four-dimensional polyharmonic splines

摘要

As is now well known for some basic functions ϕ, hierarchical and fast multipole likemethods can greatly reduce the storage and operation counts for fitting and evaluating radialbasis functions. In particular for spline functions of the form[FORM]p a low degree polynomial and certain choices of ⏀, the cost of a single extra evaluation canbe reduced from O(N) to O(log N), or even O(1), operations and the cost of a matrix-vectorproduct (i.e., evaluation at all centres) can be decreased from O(N²) to O(N log N), or evenO(N), operations.This paper develops the mathematics required by methods of these types for polyharmonicsplines in R⁴. That is for splines s built from a basic function from the list ⏀(r) = r⁻² or⏀(r) = r²n ln(r), n = 0, 1, .... We give appropriate far and near field expansions, togetherwith corresponding error estimates, uniqueness theorems, and translation formulae.A significant new feature of the current work is the use of arguments based on the actionof the group of non-zero quaternions, realised as 2 x 2 complex matrices[MATRICES]acting on C² = R⁴. Use of this perspective allows us to give a relatively efficient developmentof the relevant spherical harmonics and their properties.