Radial positive definite functions generated by Euclid's hat

Radial positive definite unctions are of importance both as the characteris tic functions of spherically symmetric probability distributions, and as th e correlation functions of isotropic random fields. The Euclid's hat functi on h(n)(\\x\\), x is an element of R-n, is the self-convolution of an indic ator function supported on the unit ball in R-n. This function is evidently radial and positive definite, and so are its scale mixtures that form the class H-n. Our main results characterize the classes H-n, n greater than or equal to l, and H-infinity = boolean AND(n greater than or equal to 1) H-n . This leads to an analogue of Polya's criterion for radial functions on R- n, n greater than or equal to 2: If phi:[0, infinity) --> R is such that ph i(0) = 1, phi(t) is continuous, lim(t-->infinity) phi(t) = 0, and (-1)(k) d(k)/dt(k) [-phi'(root t)] is convex for k = [(n-)/2], the greatest integer less than or equal to (n- 2)/2, then phi(\\x\\) is a characteristic function in R-n. Along the way, s ide results on multiply monotone and completely monotone functions occur. W e discuss the relations of H-n to classes of radial positive definite funct ions studied by Askey (Technical Report No. 1262, Math. Res. Center, Univ. of Wisconsin-Madison), Mittal (Pacific J. Math. 64 (1976, 517-538), and Ber man (Pacific J. Math. 78 (1978), 1-9), and close with hints at applications in geostatistics. (C) 1999 Academic Press.