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 Mat. Sb. (N.S.), 1987, Volume 132(174), Number 3, Pages 434–443 (Mi msb1878)

On functions of bounded variation that are determined by restriction to a semiaxi

A. M. Ulanovskii

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 $F(x/a_1)*…*F(x/a_n)$, $n\geqslant2$, 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 $a_1>0,…,a_n>0$, $n\geqslant2$, then the equality
$$F(\frac x{a_1})*…*F(\frac x{a_n})=\Phi(x),\qquad x\in(-\infty,0),$$
implies that $F(x)\equiv\Phi((a^2_1+…+a^2_n)^{1/2}x)$.
Bibliography: 10 titles.

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English version:
Mathematics of the USSR-Sbornik, 1988, 60:2, 427–436

Bibliographic databases:

UDC: 517.44+519.21
MSC: Primary 26A45, 60E99; Secondary 26A42, 60E07

Citation: A. M. Ulanovskii, “On functions of bounded variation that are determined by restriction to a semiaxi”, Mat. Sb. (N.S.), 132(174):3 (1987), 434–443; Math. USSR-Sb., 60:2 (1988), 427–436

Citation in format AMSBIB
\Bibitem{Ula87} \by A.~M.~Ulanovskii \paper On~functions of bounded variation that are determined by restriction to a~semiaxi \jour Mat. Sb. (N.S.) \yr 1987 \vol 132(174) \issue 3 \pages 434--443 \mathnet{http://mi.mathnet.ru/msb1878} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=889603} \zmath{https://zbmath.org/?q=an:0663.30003|0631.30003} \transl \jour Math. USSR-Sb. \yr 1988 \vol 60 \issue 2 \pages 427--436 \crossref{https://doi.org/10.1070/SM1988v060n02ABEH003179}