Extremal functions with vanishing condition

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 $(\pi a - \tanh(\pi a))^{-1} \pi a$.