Anisotropic Radial Basis Functions 199 3

There are multiple reasons why anisotropic basis functions may be needed or be more appropriate. The most obvious is that if the basis function is to be defined on Rn × T then there is no natural norm on this space that would reflect the unique properties of time. A second reason is that function being interpolated or approximated may incorporate a directional dependence. Thirdly, differentiability of the basis function is often critical, i.e., partial differentiability. Separating the differentiability from one dimension to another may be necessary, e.g., differentiability with respect to time as contrasted with differentiability with respect to a space coordinate. Positive definiteness (or conditional positive definiteness) is often dependent on the dimension of the space. Thus construction of non-radially symmetric basis functions which can easily be shown to be strictly positive definite is important, a number of examples and general methods will be given. AMS Subject Classification: 65D15, 65D10