The zeros of point evaluation extremals in Paley-Wiener spaces
Abstract
For 1≤ p < ∞, φp is the function of minimal Lp norm among all functions f of exponential type at most π for which f(0)=1. We start from the ansatz that the positive zeros τn of φp are close to n+1/p−1/2 and use a weighted convex correction to first place them in an ℓ2-neighborhood of this sequence. A bootstrap argument then improves this to a {Poincaré asymptotic} expansion of τn−(n+1/p−1/2) in terms of odd powers of (n+1/p−1/2)−1. We also get precise information about the Fourier transform φ̂p, a result that in particular implies that for all integers n≥ 2, φ̂2n is an integrable function which, when convolved with itself 2n−2 times, is constant on the interval [−π,π]. In particular, we identify the extremizer in an autoconvolution problem posed by B. Green in 2001.
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BibTeX
@misc{bondarenko2026zeros,
title = {{The zeros of point evaluation extremals in Paley-Wiener spaces}},
author = {Andriy Bondarenko and Joaquim Ortega-Cerdà and Kristian Seip},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19132},
eprint = {2610.19132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19132},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The zeros of point evaluation extremals in Paley-Wiener spaces
AU - Andriy Bondarenko
AU - Joaquim Ortega-Cerdà
AU - Kristian Seip
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19132
UR - https://arxiv.org/abs/2610.19132
AB - For 1≤ p < ∞, φ_p is the function of minimal L^p norm among all functions f of exponential type at most π for which f(0)=1. We start from the ansatz that the positive zeros τ_n of φ_p are close to n+1/p−1/2 and use a weighted convex correction to first place them in an ℓ^2-neighborhood of this sequence. A bootstrap argument then improves this to a {Poincaré asymptotic} expansion of τ_n−(n+1/p−1/2) in terms of odd powers of (n+1/p−1/2)^−1. We also get precise information about the Fourier transform φ̂_p, a result that in particular implies that for all integers n≥ 2, φ̂_2n is an integrable function which, when convolved with itself 2n−2 times, is constant on the interval [−π,π]. In particular, we identify the extremizer in an autoconvolution problem posed by B. Green in 2001.
ER -