Functional Equations and Characterization Problems on Locally Compact Abelian Groups

Gennadiy Feldman

doi:10.4171/045

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Functional Equations and Characterization Problems on Locally Compact Abelian Groups

Gennadiy Feldman
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences, Kharkov, Ukraine

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	FrontmatterDownload pp. i–iv
	PrefaceDownload pp. v–ix
	ContentsDownload pp. xi–xii
I	Chapter I Preliminariesp. 1
1	Locally compact Abelian groupspp. 1–11
2	Probability distributions on locally compact Abelian groupspp. 11–18
II	Chapter II Gaussian distributions on locally compact Abelian groupsp. 19
3	Properties of Gaussian distributionspp. 19–32
4	Cramér’s theorem on the decomposition of a Gaussian distribution on locally compact Abelian groupspp. 32–38
5	Polynomials on locally compact Abelian groups and the Marcinkiewicz theorempp. 38–49
6	Gaussian distributions in the sense of Urbanikpp. 49–55
III	Chapter III The Kac–Bernstein theorem for locally compact Abelian groupsp. 56
7	Locally compact Abelian groups for which the Kac–Bernstein theorem holdspp. 56–68
8	Random variables with values in the group $R \times T$ and in the $a$ -adic solenoid $\sum_{a}$ pp. 69–81
9	Gaussian distributions in the sense of Bernsteinpp. 81–91
IV	Chapter IV The Skitovich–Darmois theorem for locally compact Abelian groups (the characteristic functions of random variables do not vanish)p. 92
10	Locally compact Abelian groups for which the Skitovich–Darmois theorem holdspp. 92–107
11	Random variables with values in the two-dimensional torus $T^{2}$ pp. 107–121
12	Random variables with values in the groups $R \times T$ and $\sum_{a} \times T$ pp. 121–132
V	Chapter V The Skitovich–Darmois theorem for locally compact Abelian groups (the general case)p. 133
13	The number of random variables $n = 2$ pp. 133–153
14	The number of random variables $n \geq 3$ pp. 153–172
15	Random variables with values in the $a$ -adic solenoid $\sum_{a}$ pp. 172–185
VI	Chapter VI The Heyde theorem for locally compact Abelian groupsp. 186
16	The characteristic functions of random variables do not vanishpp. 186–197
17	Random variables with values in finite and discrete Abelian groupspp. 197–221
	The Kac–Bernstein and Skitovich–Darmois functional equations on locally compact Abelian groupspp. 223–235
	Comments and unsolved problemspp. 237–245
	Bibliographypp. 247–252
	Symbol indexpp. 253–254
	Subject indexpp. 255–256

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