On the derivatives of radial positive definite functions

A radial positive definite function phi(parallel to.parallel to) on R-d which is 2k times differentiable at zero has 2k + [(d - 1)/2] continuous derivatives away from zero, and the estimate is sharp. Furthermore, phi((iv))(0) greater than or equal to 3d\phi"(0)\(2)/(d + 2) if the derivatives exist; this proves a conjecture of Paul Switzer and has implications on the smoothness of isotropic random fields and their local averages. The techniques used have evolved in geostatistics, and side results refer to Matheron's turning bands operator and the montee. (C) 1999 Academic Press.