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The Skitovich–Darmois equation in the classes of continuous and measurable functions

Summary.

Let Y be a locally compact abelian group, Aut(Y) be the group of topological automorphisms of the group Y and ɛ ∈ Aut(Y). We study the continuous and measurable solutions to the Skitovich–Darmois equation

$$f_1(u + v)f_2(u + {\varepsilon}v) = f_1(u)f_1(v)f_2(u)f_2({\varepsilon}v),\,\, u, v \in Y,$$

which appears in the problem of characterization of a Gaussian distribution by the independence of linear forms.

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Correspondence to Gennadiy Feldman.

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This research was supported by part by the Ukrainian–French program “DNIPRO”.

Manuscript received: May 25, 2006 and, in final form, December 11, 2006.

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Feldman, G., Myronyuk, M. The Skitovich–Darmois equation in the classes of continuous and measurable functions. Aequ. math. 75, 75–92 (2008). https://doi.org/10.1007/s00010-007-2873-y

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  • DOI: https://doi.org/10.1007/s00010-007-2873-y

Mathematics Subject Classification (2000).

  • 39B52 (60B15, 62E10)

Keywords.

  • Functional equation
  • Skitovich–Darmois equation
  • locally compact abelian group