On the uniqueness conditions for functions of bounded variation and for distribution functions with given values on a halfline

We prove some uniqueness theorems for functions $F(x)$ of bounded variation (and for distribution functions) with given values on a halfline. The uniqueness is proved for functions belonging to the classes of functions $F(x)$ such that the characteristic function
$$ \varphi(t;F)=\int_{-\infty}^\infty e^{itx}\,dF(x) $$
is analytic in the strip $0<\operatorname{Im}t<H<\infty$, $\varphi(t;F)\ne 0$ ($0<\operatorname{Im}t<H$) and $\varphi(t;F)$ grows rather quickly when $\operatorname{Im}t\uparrow H$.