Density of Sums of Shifts of a Single Function in Hardy Spaces on the Half-Plane

It is proved that there exists a function defined in the closed upper half-plane for which the sums of its real shifts are dense in all Hardy spaces $H_{p}$ for $2 \le p < \infty$, as well as in the space of functions analytic in the upper half-plane, continuous on its closure, and tending to zero at infinity.