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

As is now well known for some basic functions phi, hierarchical and fast mu ltipole-like methods can greatly reduce the storage and operation counts fo r fitting and evaluating radial basis functions. In particular, for spline functions of the form [GRAPHICS] where p is a low degree polynomial and with certain choices of phi, the cos t of a single extra evaluation can be reduced from O(N) to O(log N), or eve n O(1), operations and the cost of a matrix-vector product (i.e., evaluatio n at all centers) can be decreased from O(N-2) to O(N log N), or even O(N), operations. This paper develops the mathematics required by methods of these types for polyharmonic splines in R-4. That is, for splines s built from abasic funct ion from the list phi (r) = r(-2) or phi (r) = r(2n) ln(r), n = 0, 1,.... W e give appropriate far and near field expansions, together with correspondi ng 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 x 2 compl ex matrices [GRAPHICS] acting on C-2 = R-4. Use of this perspective allows us to give a relatively efficient development of the relevant spherical harmonics and their proper ties.