Mollification formulas and implicit smoothing

This paper develops some mollification formulas involving convolutions between popular radial basis function (RBF) basic functions Phi, and suitable mollifiers. Polyharmonic splines, scaled Bessel kernels (Matern functions) and compactly supported basic functions are considered. A typical result is that in R-d the convolution of |.|(beta) and (.(2) + c(2))-((beta+2d)/2) is the generalized multiquadric (.(2) + c(2))(beta/2) up to a multiplicative constant. The constant depends on c > 0, beta, where R(beta) > - d, and d. An application which motivated the development of the formulas is a technique called implicit smoothing. This computationally efficient technique smooths a previously obtained RBF fit by replacing the basic function Phi with a smoother version Psi during evaluation.