Radial Basis Functions, Discrete Differences, and Bell-Shaped Bases

In this paper, we introduce the notion of a normalized radial basis function. In the univariate case, taking these basis functions in combinations determined by certain discrete differences leads to the B-spline basis. In the bivariate case, these combinations lead to a generalization of the B-spline basis to the surface case. Subdivision rules for the resulting basis functions can easily be derived.