  We consider the Buhmann class of compactly supported radial basis functions,
whih includes a wealth of special cases that have been studied in both
numerical analysis and spatial statistics literatures. In particular, the
celebrated Wu, Wendland and Missing Wendland functions are notable special
cases of this class. We propose a very simple difference operator and show the
conditions for which the application of it to Buhmann functions preserves
positive definiteness on $m$-dimensional Euclidean spaces. We also show that
the application of the difference operator increases smoothness at the origin,
whilst keeping positive definiteness in the same $m$-dimensional Euclidean
space, as well as compact support. Thus, our operator is a competitor of the
celebrated Mont{\'e}e operator, which allows to increase the smoothness at the
origin, at the expense of losing positive definiteness in the space where the
radial basis function is originally defined. The proofs of our results
highlight surprising connections with past literatures on celebrated class of
functions. Amongst them, absolute and completely monotone functions.
