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 [FORM] p a low degree polynomial and certain choices of ⏀, 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 centres) 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 ⏀(r) = r⁻² or ⏀(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 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 development of the relevant spherical harmonics and their properties.