Isotropic correlation functions on d-dimensional balls

A popular procedure in spatial data analysis is to fit a line segment of the form c(x) = 1 - alpha parallel to x parallel to, parallel to x parallel to < 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 0 less than or equal to alpha less than or equal to 2 Gamma(d/2)/pi(1/2) Gamma((d + 1)/2), 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.