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  • Author: Zbigniew J. Jurek x
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Abstract

The Nevalinna transform Ka , ρ(z) of a positive measure ρ and a constant a, plays an important role in complex analysis and – more recently – in the context of the boolean convolution. We show here that its restriction to the imaginary axis, ka , ρ(it), can be expressed as the Laplace transform of the Fourier transform (a characteristic function) of ρ. Consequently, ka ρ is sufficient for the unique identification of the measure ρ and the constant a. Finally, we identify a relation between the free additive Voiculescu ⊞ and boolean ⊎ convolutions.

Abstract

Random integral mappings give isomorphism between the subsemigroups of the classical (I D, *) and the free-infinite divisible (I D, ⊞) probability measures. This allows us to introduce new examples of such measures, more precisely their corresponding characteristic functionals.