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 n∞ 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.
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Part of this work was done during the author's stay at the Courant Institute of Mathematical Sciences, New York University.
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Koldobsky, A. Characterization of measures by potentials. J Theor Probab 7, 135–145 (1994). https://doi.org/10.1007/BF02213364
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DOI: https://doi.org/10.1007/BF02213364