  We determine the optimal majorant $M^+$ and minorant $M^-$ of exponential
type for the truncation of $x\mapsto (x^2+a^2)^{-1}$ with respect to general de
Branges measures. We prove that \[ \int_\mathbb{R} (M^+ - M^-) |E(x)|^{-2}dx =
\frac{1}{a^2 K(0,0)} \] where $K$ is the reproducing kernel for
$\mathcal{H}(E)$. As an application we determine the optimal majorant and
minorant for the Heaviside function that vanish at a fixed point $\alpha = ia$
on the imaginary axis. We show that the difference of majorant and minorant has
integral value $(\pi a - \tanh(\pi a))^{-1} \pi a$.
