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arXiv · 1311.1157

Extremal functions with vanishing condition

Friedrich Littmann·Mark Spanier
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Abstract

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

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Friedrich Littmann, Mark Spanier. 2013-11-05. Extremal functions with vanishing condition. https://doi.org/10.1007/s00365-015-9304-4

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