A short proof of Schoenberg’s theorem

Using positive semidefiniteness of Laplace transforms, we give a short and simple proof of Schoenberg's theorem characterising radially symmetric positive semidefinite functions on a Hilbert space. A slight generalisation of this theorem is also given. In his paper Metric spaces and completely monotone functions [3], I. J. Schoenberg raises the question about the connection between the class of Fourier transforms of (finite, nonnegative) measures in Euclidean spaces and the class of Laplace transforms of (finite, nonnegative) measures on the halfline R+ = [0, oo). He states: "In spite of the entirely different analytical character of these two classes, a certain kinship was to be expected for the following two reasons: 1. In both classes the defining kernel is the exponential function. 2. The less formal reason of the similarity of the closure properties of both classes, for both classes are convex, i.e. alf, + a2f2 (a, > 0,a2 > 0) belongs to the class if f1 and f2 belong to it, multiplicative, i.e., also f1 . f2 belongs to the class, and finally closed with respect to ordinary convergence to a continuous limit function." The answer Schoenberg could give to the above question was the remarkable result that to each continuous function f: R+ -C with the property that f o I ln is positive semidefinite on Rn for all n (I n denoting the Euclidean norm) there exists a finite nonnegative measure on R+ with Laplace transform f(V/t) [3, Theorem 2]. Positive semidefiniteness of a mapping g: Rn -, C has the meaning that the kernel K(x,y) = g(x y) is positive semidefinite. The proof of this theorem, even that given in the more recent book of Donoghue [1, pp. 201-206], however is rather complicated and technical in nature. But there is a further common feature of Fourier and Laplace transforms seemingly unknown until quite recently: Laplace transforms, too, are characterised essentially by positive semidefiniteness. More precisely, a functionf: RP -+ C is the Laplace transform of a finite nonnegative measure on the Borel sets of RP if and only if / is continuous, bounded and positive semidefinite in the sense that = Ej=1 ai ajf (ti + tj) > 0 for all (a1, . . .,ak) E Rk, (tl, , tk) E (RP))k, and k E N [2, Satz 1]. Using this we give a new proof of THEOREM 1 (SCHOENBERG). A continuous function f: R+ -C has the property that f o I In is positive semidefinite on Rn for all n E N if and only if there exists a finite nonnegative measure A on R+ such that Received by the editors July 17, 1975. AMS (MOS) subject classifications (1970). Primary 43A35; Secondary 44A 10. ? American Mathematical Society 1976