Zastavnyi Operators and Positive Definite Radial Functions

Positive definite functions are fundamental to many areas of applied mathematics, probability theory, spatial statistics and machine learning, amogst others. Motivated by a problem coming from the maximum likelihood estimation under fixed domain asymptotics, we consider a new operator acting on rescaled weighted differences between two members of the class $\Phi_d$ of positive definite radial functions,. In particular, we study the positive definiteness of the operator for the Mat\'ern, Generalized Cauchy and Generalized Wendland families. It turns out that proposed operator allows to govern differentiability at the origin, and to attain negative correlations.