Abstract:
Let F(x), x∈R, be a function of bounded variation on the line. This paper investigates whether convolutions of the form F(x/a1)∗⋯∗F(x/an), n⩾2, are uniquely determined from their values on the semiaxis x∈(−∞,0). As a corollary to one of the results a conjecture of Kruglov is proved: if F(x) is a distribution function, Φ(x) is the standard normal distribution function, and a1>0,…,an>0, n⩾2, then the equality
F(xa1)∗⋯∗F(xan)=Φ(x),x∈(−∞,0),
implies that F(x)≡Φ((a21+⋯+a2n)1/2x).
Bibliography: 10 titles.
\Bibitem{Ula87}
\by A.~M.~Ulanovskii
\paper On~functions of bounded variation that are determined by restriction to a~semiaxi
\jour Math. USSR-Sb.
\yr 1988
\vol 60
\issue 2
\pages 427--436
\mathnet{http://mi.mathnet.ru/eng/sm1878}
\crossref{https://doi.org/10.1070/SM1988v060n02ABEH003179}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=889603}
\zmath{https://zbmath.org/?q=an:0663.30003|0631.30003}
Linking options:
https://www.mathnet.ru/eng/sm1878
https://doi.org/10.1070/SM1988v060n02ABEH003179
https://www.mathnet.ru/eng/sm/v174/i3/p434
This publication is cited in the following 1 articles:
Alexander Ulanovskii, “On a Uniqueness Property of n-th Convolutions and Extensions of Titchmarsh Convolution Theorem”, Z. mat. fiz. anal. geom., 20:4 (2024), 525