Abstract:
Let $F(x)$, $x\in\mathbf R$, be a function of bounded variation on the line. This paper investigates whether convolutions of the form $F(x/a_1)*\dots*F(x/a_n)$, $n\geqslant2$, are uniquely determined from their values on the semiaxis $x\in(-\infty,0)$. As a corollary to one of the results a conjecture of Kruglov is proved: if $F(x)$ is a distribution function, $\Phi (x)$ is the standard normal distribution function, and $a_1>0,\dots,a_n>0$, $n\geqslant2$, then the equality
$$
F\biggl(\frac x{a_1}\biggr)*\dots*F\biggl(\frac x{a_n}\biggr)=\Phi(x),\qquad
x\in(-\infty,0),
$$
implies that $F(x)\equiv\Phi((a^2_1+\dots+a^2_n)^{1/2}x)$.
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