Fast Evaluation of Radial Basis Functions: Methods for Four-Dimensional Polyharmonic Splines

As is now well known for some basic functions φ, hierarchical and fast multipole-like methods can greatly reduce the storage and operation counts for fitting and evaluating radial basis functions. In particular, for spline functions of the form {equation*} s(x) = p(x) + ∑ₖ₌₁N d_k φ(|x - x_k|), {equation*} where p is a low degree polynomial and with certain choices of φ, the cost of a single extra evaluation can be reduced from {N} to {log N}, or even {1}, operations and the cost of a matrix-vector product (i.e., evaluation at all centers) can be decreased from {N^2} to {Nlog N}, or even {N}, operations. This paper develops the mathematics required by methods of these types for polyharmonic splines in R⁴. That is, for splines s built from a basic function from the list φ(r) = r⁻² or φ(r)= r²ⁿln(r), n = 0,1,…. We give appropriate far and near field expansions, together with corresponding error estimates, uniqueness theorems, and translation formulae. A significant new feature of the current work is the use of arguments based on the action of the group of nonzero quaternions, realized as 2× 2 complex matrices {equation*} {H^1_0} = \{ x={z}{w}{-}{}:\ |z|^2+|w|^2>0 \}, {equation*} acting on C² = R⁴. Use of this perspective allows us to give a relatively efficient development of the relevant spherical harmonics and their properties.