| Abstrakt: |
As is now well known for some basic functions φ, hierarchical and fast multipole-like methods can greatly reduce the storage and operation counts forfitting and evaluating radial basis functions. In particular, for spline functions of the form [Multiple line equation(s) cannot be represented in ASCII text] where p is a low degree polynomial and with certain choices of φ, the cost of a single extra evaluation can be reduced from O(N)to O(log N), or even O(1), operations and the cost of a matrix-vector product (i.e., evaluation at all centers) can be decreased from O(N²) 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 R4. That is, for splines s built from a basic function from the list φ(r) = r-2 or φ(r) = r2n ln(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 x 2 complex matrices H0¹ = {x = [z/-w w/z]: |z|² + |w|² > 0}, acting on C² = R4. Use of this perspective allows us to give a relatively efficient development of the relevant spherical harmonics and their properties. [ABSTRACT FROM AUTHOR] |
