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We study the function \(\varphi _1\) of minimal \(L^1\) norm among all functions f of exponential type at most \(\pi \) for which \(f(0)=1\) . This function, first studied by Hörmander and Bernhardsson in 1993, has only real zeros \(\pm \tau _n\) , \(n=1,2, \ldots \) , and the sequence \((\tau _n-n-\frac 12)\) has \(\ell ^2\) norm bounded by \(0.13\) . The zeros \(\tau _n\) can be computed by means of a fixed point iteration.

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The Hörmander–Bernhardsson Extremal Function: A Preliminary Study

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  • 发表日期：

    2026年

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  • Andriy Bondarenko,
  • Joaquim Ortega-Cerdà,
  • Danylo Radchenko,
  • Kristian Seip

摘要

We study the function \(\varphi _1\) of minimal \(L^1\) norm among all functions f of exponential type at most \(\pi \) for which \(f(0)=1\) . This function, first studied by Hörmander and Bernhardsson in 1993, has only real zeros \(\pm \tau _n\) , \(n=1,2, \ldots \) , and the sequence \((\tau _n-n-\frac 12)\) has \(\ell ^2\) norm bounded by \(0.13\) . The zeros \(\tau _n\) can be computed by means of a fixed point iteration.