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 $${\int }_{\mathbb{R}}(M^{+}-M^{-})\mathrm{\mid }E(x){\mathrm{\mid }}^{-2}dx=\frac{1}{a^{2}K(0,0)}$$ 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 $(\pi a-\tanh (\pi a){)}^{-1}\pi a$.