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On the Cauchy transform vanishing outside a compact

Levin Genadi

Bulletin of the Polish Academy of Sciences. Mathematics

2020

Vol. 68, no. 1

21--40

Motivated by a problem in holomorphic dynamics, we present a certain generalization of the celebrated F. and M. Riesz theorem.

Limit measures related to the conditionally free convolution

Hinz M., Młotkowski W.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 1

75-81

We describe the limit measures for some class of deformations of the free convolution, introduced by A. D. Krystek and Ł. J. Wojakowski. In particular, we provide a counterexample to a conjecture from their paper.

Cauchy transforms of measures viewed as some functionals of Fourier transforms

Jurek Z. J.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

187--200

The Cauchy transform of a positive measure plays an important role in complex analysis and more recently in so-called free probability. We show here that the Cauchy transform restricted to the imaginary axis can be viewed as the Fourier transform of some corresponding measures. Thus this allows the full use of that classical tool. Furthermore, we relate restricted Cauchy transforms to classical com- pound Poisson measures, exponential mixtures, geometric infinite divisibility and free-infinite divisibility. Finally, we illustrate our approach with some examples.