Characterization of measures by potentials |
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Authors: | Alexander Koldobsky |
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Institution: | (1) Division of Mathematics, Computer Science and Statistics, University of Texas at San Antonio, 78229, Texas |
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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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Keywords: | Fourier transform convolution measures on Banach spaces analytic norms positive definite functions |
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