# 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 $R×T$ and in the $a$-adic solenoid $∑_{a}$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 $R×T$ and $∑_{a}×T$ 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 $n≥3$pp. 153–172 |

15 | Random variables with values in the $a$-adic solenoid $∑_{a}$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 |