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arXiv · 2608.22198

The H\"ormander--Bernhardsson function in higher dimensions

Felipe Gonçalves·Danylo Radchenko·Antonio Pedro Ramos
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Abstract

We study the problem of finding the norm of the point evaluation operator in the Paley--Wiener space $PW^{1}(\r^d)$, consisting of $d$-variable functions of spherical exponential type that are integrable on $\r^d$. The extremal functions can be taken radial, which naturally leads us to consider a related extremal problem in a weighted Paley--Wiener space of single-variable functions. We establish that the radial extremal function must satisfy a third-order linear ODE with polynomial coefficients for every $d \geq 1$, extending Gorbachev's recent odd-dimensional result. Along the way, we prove interpolation and reciprocal formulas involving the zeros of the extremizer.

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Felipe Gonçalves, Danylo Radchenko, Antonio Pedro Ramos. 2026-08-23. The H\"ormander--Bernhardsson function in higher dimensions. https://arxiv.org/abs/2608.22198

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