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

A. M. Ulanovskii

Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 427-436

Résumé

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.