Radial positive definite functions are of importance both as the characteristic functions of spherically symmetric probability distributions, and as the correlation functions of isotropic random fields. The Euclid's hat functionhn(||x||),x[set membership, variant]n, is the self-convolution of an indicator function supported on the unit ball in n. This function is evidently radial and positive definite, and so are its scale mixtures that form the classHn. Our main results characterize the classesHn,n[greater-or-equal, slanted]1, andH[infinity]=[intersection]n[greater-or-equal, slanted]1 Hn. This leads to an analogue of Pólya's criterion for radial functions on n,n[greater-or-equal, slanted]2: If[phi]: [0, [infinity])--> is such that[phi](0)=1,[phi](t) is continuous, limt-->[infinity] [phi](t)=0, andis convex fork=[(n-2)/2], the greatest integer less than or equal to (n-2)/2, then[phi](||x||) is a characteristic function in n. Along the way, side results on multiply monotone and completely monotone functions occur. We discuss the relations ofHnto classes of radial positive definite functions studied by Askey (Technical Report No. 1262, Math. Res. Center, Univ. of Wisconsin-Madison), Mittal (Pacific J. Math.64(1976), 517-538), and Berman (Pacific J. Math.78(1978), 1-9), and close with hints at applications in geostatistics.
