Extremal functions with vanishing condition

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

References in corpus (2)

• Extremal functions in de Branges and Euclidean spaces
• A Variation on Selberg's Approximation Problem

Cited by in corpus (5)

• Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function
• Extremal functions in de Branges and Euclidean spaces
• Bandlimited approximations and estimates for the Riemann zeta-function
• Extremal problems in de Branges spaces: the case of truncated and odd functions
• Extremal functions in de Branges and Euclidean spaces II