Space-Time Radial Basis Functions

Radial basis functions are "isotropic"; i.e., under a rotation, the basis function is left unchanged and is obtained as a function of a distance on the space. For Euclidean space this is not a problem since there is a natural metric. To extend radial basis functions to space-time, i.e., R-m x T, either a zonal anisotropy has to be incorporated or a metric must be defined on space-time. While the sum of two valid radial basis functions defined on different dimensional spaces is generally only semidefinite on the product space, the product of two positive definite functions on lower dimensional spaces is positive definite on the product space. This construction can be extended in several ways including a product-sum, integrated product, and the integrated product-sum. Examples are given for each construction and an application is given. The constructions are equally applicable to extending from space to space-time or for splitting higher-dimensional Euclidean spaces into the product of several lower-dimensional spaces.