Moscow Mathematical Journal

Volume 26, Issue 3, July–September 2026 pp. 389–409.

Generalized Eigenfunctions of Measure Preserving Transformations and Riesz Products

Authors: Yury A. Neretin (1)
Author institution:(1) University of Graz, Department of Mathematics and Scientific Computing;
High School of Modern Mathematics MIPT;
Moscow State University, MechMath Faculty;
Univesity of Vienna, Faculty of Mathematics

Summary:

We consider rank one measure preserving transformations $g$ and the corresponding Koopman unitary operators $U(g)$. It is known that a generic (in the sense of Baire category) measure preserving transformation has rank one, spectral measure of $U(g)$ is purely singular and is given by a Riesz product. For such transformations we write explicitly spectral decompositions of operators $U(g)$ in generalized eigenfunctions.

2020 Math. Subj. Class. 47A35, 28D05, 37A40, 42A55, 47B15.

Keywords: Riesz products, Koopman operator, spectral decomposition, singular measure, lacunary series.

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