Phase retrieval for probability measures on cyclic groups

I. P. Il’inskaya

doi:10.15330/ms.46.1.89-95

We construct the classes of probability measures on the groups $\mathbb{Z}_n$ and $\mathbb{Z}$ for which the identification of a measure given modulus of its characteristic function is possible up to a shift and the central symmetry. For groups $\mathbb{Z}_3$, $\mathbb{Z}_4$ we give the completely description of such classes.

1. Rosenblatt J., Phase retrieval, Comm. Math. Phys., 95 (1984), 317–343.

2. Carnal H., Dozzi M., On a decomposition problem for multivariate probability measures, J. Multivar. Anal., 31 (1989), 165–177.

3. Carnal H., Fel’dman G.M., Phase retrieval for probability measures on abelian groups, I. J.Theor. Probab., 8 (1995), №3, 717–725.

4. Carnal H., Fel’dman G.M., Phase retrieval for probability measures on abelian groups, II. J.Theor. Probab., 10 (1997), №4, 1065–1074.

5. Carnal H., Feldman G.M., On a property of entire characteristic functions of finite order with real zeros, Doklady RAN, 366 (1999), 162–163. (in Russian)

6. Carnal H., Feldman G.M., A stability property for probability measures on Abelian groups, Statistics and probability Letters, 49 (2000) 39–44.

7. Il’inskaya I.P., Phase retrieval for probability measures on the character group of Cantor-Walsh group, Dopovidi NAN Ukraine, (2003), №8, 11–14. (in Russian)

8. Il’inskaya I.P., Neguritsa D.S., On probability measures on the group of Walsh functions with a trivial equivalence class, Ukrainian Mathematical Journal, 65 (2013), №5, 717–721. (in Russian)

9. Fel’dman G.M., Arithmetic of Probability Distributions and Characterization Problems on Abelian Groups, AMS, Providence, RI, 1993.

	Phase retrieval for probability measures on cyclic groups
Author	I. P. Il’inskaya iljinskii@univer.kharkov.ua Kharkov Karazin National University
Abstract	We construct the classes of probability measures on the groups $\mathbb{Z}_n$ and $\mathbb{Z}$ for which the identification of a measure given modulus of its characteristic function is possible up to a shift and the central symmetry. For groups $\mathbb{Z}_3$, $\mathbb{Z}_4$ we give the completely description of such classes.
Keywords	probability measure on locally compact abelian group; characteristic function; trivial equivalence class; phase retrieval
DOI	doi:10.15330/ms.46.1.89-95
Reference	1. Rosenblatt J., Phase retrieval, Comm. Math. Phys., 95 (1984), 317–343. 2. Carnal H., Dozzi M., On a decomposition problem for multivariate probability measures, J. Multivar. Anal., 31 (1989), 165–177. 3. Carnal H., Fel’dman G.M., Phase retrieval for probability measures on abelian groups, I. J.Theor. Probab., 8 (1995), №3, 717–725. 4. Carnal H., Fel’dman G.M., Phase retrieval for probability measures on abelian groups, II. J.Theor. Probab., 10 (1997), №4, 1065–1074. 5. Carnal H., Feldman G.M., On a property of entire characteristic functions of finite order with real zeros, Doklady RAN, 366 (1999), 162–163. (in Russian) 6. Carnal H., Feldman G.M., A stability property for probability measures on Abelian groups, Statistics and probability Letters, 49 (2000) 39–44. 7. Il’inskaya I.P., Phase retrieval for probability measures on the character group of Cantor-Walsh group, Dopovidi NAN Ukraine, (2003), №8, 11–14. (in Russian) 8. Il’inskaya I.P., Neguritsa D.S., On probability measures on the group of Walsh functions with a trivial equivalence class, Ukrainian Mathematical Journal, 65 (2013), №5, 717–721. (in Russian) 9. Fel’dman G.M., Arithmetic of Probability Distributions and Characterization Problems on Abelian Groups, AMS, Providence, RI, 1993.
Pages	89-95
Volume	46
Issue	1
Year	2016
Journal	Matematychni Studii
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