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Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions
Yu. A. Neretinabcd a University of Vienna
b State Scientific Center of the Russian Federation - Institute for Theoretical and Experimental Physics, Moscow
c Lomonosov Moscow State University
d Institute for Information Transmission Problems, Russian Academy of Sciences
Abstract:
We consider the space $\mathcal C$ of conjugacy classes of the unitary group
$\mathrm U(n+m)$ with respect to a smaller unitary group $\mathrm U(m)$.
It is known that to every element of $\mathcal C$ we can canonically assign
a rational matrix-valued function (the Livshits characteristic function)
on the Riemann sphere. We find an explicit expression for the natural measure
on $\mathcal C$ obtained as the push-forward of the Haar measure
of $\mathrm U(n+m)$ in terms of characteristic functions.
Keywords:
inner functions, characteristic functions, Haar measure, Cayley transform, random functions.
Funding Agency |
Grant Number |
Russian Science Foundation  |
14-50-00150 |
This work was supported by the Russian Science Foundation under grant no. 14-50-00150. |
DOI:
https://doi.org/10.4213/im8467
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English version:
Izvestiya: Mathematics, 2016, 80:6, 1118–1130
Bibliographic databases:
UDC:
512.546.32+517.547.5+517.548.5
MSC: 47A48, 28C10, 15B52 Received: 28.10.2015
Citation:
Yu. A. Neretin, “Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions”, Izv. RAN. Ser. Mat., 80:6 (2016), 127–140; Izv. Math., 80:6 (2016), 1118–1130
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/izv8467https://doi.org/10.4213/im8467 http://mi.mathnet.ru/eng/izv/v80/i6/p127
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