A class of isotropic distributions inRnand their characteristic functions

Let r(t) be a characteristic function.Suppose that there is an integer n ^ 2 such that r{{t\ + + tl) 1/2 ) is, as a function of n variables, also the characteristic function of some distribution in R n .Then, as is known, the distribution is necessarily rotationally invariant, and r has a canonical form as a certain Bessel transform of a bounded nondecreasing function.A certain subclass of the class of such characteristic functions was defined and studied by Mittal, who furnished an analytic characterization of functions in the subclass.The purposes of this paper are (i) to present an alternative probabilistic characterization of these functions, and (ii) to characterize, for this subclass, the bounded nondecreasing function appearing in the Bessel transform.