caa a22 4500 605508488 CHVBK 20210128100636.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00010-014-0254-x doi (NATIONALLICENCE)springer-10.1007/s00010-014-0254-x On a functional equation appearing in characterization of distributions by the optimality of an estimate [Elektronische Daten] [Gennadiy Feldman, Piotr Graczyk] Let X be a second countable locally compact Abelian group containing no subgroup topologically isomorphic to the circle group $${\mathbb{T}}$$ T , Y be its character group. Letμ be a probability distribution on X such that its characteristic function $${\widehat\mu(y)}$$ μ ^ ( y ) does not vanish and $${\widehat\mu(y)}$$ μ ^ ( y ) for some $${n \geq 3}$$ n ≥ 3 satisfies the equation $$\prod_{j=1}^{n} \hat\mu(y_j + y) = \prod_{j=1}^{n}\hat\mu(y_j - y), \quad \sum_{j=1}^{n} y_j = 0, \quad y_1,\dots,y_n,y \in Y.$$ ∏ j = 1 n μ ^ ( y j + y ) = ∏ j = 1 n μ ^ ( y j - y ) , ∑ j = 1 n y j = 0 , y 1 , ⋯ , y n , y ∈ Y . Thenμ is a convolution of a Gaussian distribution and a distribution supported in the subgroup of X generated by elements of order 2. Springer Basel, 2014 Difference equation nationallicence Gaussian distribution nationallicence optimal estimate nationallicence Feldman Gennadiy B. Verkin Institute for Low Temperature, Physics and Engineering, 47, Lenin ave, 61103, Kharkiv, Ukraine aut Graczyk Piotr Laboratoire LAREMA UMR 6093-CNRS, Universite a'Angers, 2 bd. Lavoisier, 49045, Angers, France aut Aequationes mathematicae Springer Basel 89/3(2015-06-01), 663-671 0001-9054 89:3<663 2015 89 10 https://doi.org/10.1007/s00010-014-0254-x text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0254-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Feldman Gennadiy B. Verkin Institute for Low Temperature, Physics and Engineering, 47, Lenin ave, 61103, Kharkiv, Ukraine aut NATIONALLICENCE 700 1- Graczyk Piotr Laboratoire LAREMA UMR 6093-CNRS, Universite a'Angers, 2 bd. Lavoisier, 49045, Angers, France aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/3(2015-06-01), 663-671 0001-9054 89:3<663 2015 89 10