Isotropic correlation functions on d-dimensional balls

Abstract

A popular procedure in spatial data analysis is to fit a line segment of the formc(x) = 1 - α ||x||, ||x|| < 1, to observed correlations at (appropriately scaled) spatial lagxind-dimensional space. We show that such an approach is permissible if and only ifthe upper bound depending on the spatial dimensiond. 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 ond-dimensional balls.