Buhmann covariance functions, their compact supports, and their smoothness

We consider the Buhmann class of radially symmetric and compactly supported covariance functions, that includes the most prominent classes used in both numerical analysis and spatial statistics literature. We discuss a very simple difference operator and report the conditions for which the application of it to Buhmann functions preserves positive definiteness on m-dimensional Euclidean spaces. We also show a relation between the Zastavnyi-Porcu problem [39] for Buhmann functions and a class of completely monotone functions.