Characterization of measures by potentials

Abstract

Let μ be a measure in a Banach spaceE, f be an even function onR. We consider the potentialg(a)=f E f(‖x−a‖)dμ(x). The question is as follows: For whichf does the potentialg determine μ uniquely? In this article we give answers in the cases whereE=l ∞ and wheref(t)=|t| p andE is a finite dimensional Banach space with symmetric analytic norm. Calculating the Fourier transform of the functionf(‖x‖ ∞) we give a new proof of the J. Misiewicz's result that the functionf(‖x‖ ∞) is positive definite only iff is a constant function.