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 $α= ia$ on the imaginary axis. We show that the difference of majorant and minorant has integral value $(πa - \tanh(πa))^{-1} πa$.

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