A class of negative-definite functions

Abstract

Let G be a finite-dimensional real vector space.A proper negativedefinite function defined on G is a complex-valued function \p with the property that for each t > 0 exp{-#(*)} = f exp(if*)P(J, do where P(t, •) is a (Radon) probability measure on the dual space, G'.We shall be concerned with real, homogeneous, negative-definite functions, i.e., those negative-definite functions for which there exists a positive constant a such that for each scalar X and each vector xEG we haveThe associated probability measures here correspond to the symmetric stable laws of Paul Levy [ô].It is very easy to see that one must have a ^2.We shall restrict our attention to the range l^a^2 and treat \plla instead of \\i.Let us make a definition and alter the notation slightly.Definition.Suppose 1 ^p á 2. A continuous non-negative function ip defined on G is an ¿"-norm if (i) ^(Xx) = |X|^(x) for each real X and each xEG, and (ii) \¡/p is a proper negative-definite function.The concern of this paper is what are the Lp-norms on G.In one sense the question has been answered by Levy; see Theorem 1 below.It does not appear to me, however, that the connection between normed vector spaces and symmetric stable laws is obvious from Levy's presentation, and I think the connection is an illuminating one.The central idea of the present article is that the terminology uLp-norm" is apt.It is well known (cf.[2]) that if 0 <ß^ 1 then \p? is a proper negative-definite function whenever \p is.A fortiori, an L^-norm is an Lr-norm for lúr^p.The simplest example of an L2-norm is a function \p of the form ^(x) = | £x| where t;EG'.Since sums and positive multiples of negative-definite functions are negative-definite, the Lvnorms on G form a cone which contains, in particular, functions of the type

Locations

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