Isotropic correlation functions ond-dimensional balls

A popular procedure in spatial data analysis is to fit a line segment of the form c ( x ) = 1 - α || x ||, || x || < 1, to observed correlations at (appropriately scaled) spatial lag x in d -dimensional space. We show that such an approach is permissible if and only if the upper bound depending on the spatial dimension d . The proof relies on Matheron's turning bands operator and an extension theorem for positive definite functions due to Rudin. Side results and examples include a general discussion of isotropic correlation functions defined on d -dimensional balls.