Density of Derivatives of Simple Partial Fractions in Hardy Spaces in the Half-Plane

It is proved that the sums
$$ \sum_{k=1}^{n} \frac{1}{(z-a_{k})^{2}}, \qquad \operatorname{Im}a_{k} < 0, \quad n \in \mathbb{N}, $$
are dense in all Hardy spaces $H_{p}$, $1<p< \infty$, in the upper half-plane and in the space of functions analytic in the upper half-plane, continuous in its closure, and tending to zero at infinity.