PreprintCC BY 4.0arXiv
A Painlevé equation for the Hörmander-Bernhardsson constant
Abstract
The Hörmander-Bernhardsson constant 𝒞 is the sharp constant in |f(0)|≤ 𝒞 |f|1 for entire functions of exponential type ≤ π. We prove that 𝒞 = 2πθ*−2 where θ* is the least positive singularity of the regular solution v with v(0)=0 of the cosh-Gordon equation vθθ + vθ/θ= cosh(v). It is known that 𝒞 is a scaling limit in n from the analogous problem for polynomials of degree ≤ n. We reformulate the polynomial problem as a Padé approximation problem at infinity. The associated matrix Riemann-Hilbert problem is analyzed by a Deift-Zhou steepest descent whose local parametrix is built from a Painlevé transcendent.
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BibTeX
@misc{littmann2026painlevé,
title = {{A Painlevé equation for the Hörmander-Bernhardsson constant}},
author = {Friedrich Littmann},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30212},
eprint = {2608.30212},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30212},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Painlevé equation for the Hörmander-Bernhardsson constant
AU - Friedrich Littmann
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2608.30212
UR - https://arxiv.org/abs/2608.30212
AB - The Hörmander-Bernhardsson constant 𝒞 is the sharp constant in |f(0)|≤ 𝒞 |f|_1 for entire functions of exponential type ≤ π. We prove that 𝒞 = 2πθ_*^−2 where θ_* is the least positive singularity of the regular solution v with v(0)=0 of the cosh-Gordon equation v_θθ + v_θ/θ= cosh(v). It is known that 𝒞 is a scaling limit in n from the analogous problem for polynomials of degree ≤ n. We reformulate the polynomial problem as a Padé approximation problem at infinity. The associated matrix Riemann-Hilbert problem is analyzed by a Deift-Zhou steepest descent whose local parametrix is built from a Painlevé transcendent.
ER -