Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/215652
Title: Point evaluation in Paley-Wiener spaces
Author: Fredrik Brevig, Ole
Chirre, Andrés
Ortega Cerdà, Joaquim
Seip, Kristian
Keywords: Anàlisi harmònica
Espais funcionals
Harmonic analysis
Function spaces
Issue Date: 12-Sep-2024
Publisher: Springer
Abstract: We study the norm of point evaluation at the origin in the PaleyWiener space $P W^p$ for $0<p<\infty$, i.e., we search for the smallest positive constant $C$, called $\mathscr{C}_p$, such that the inequality $|f(0)|^p \leq C\|f\|_p^p$ holds for every $f$ in $P W^p$. We present evidence and prove several results supporting the following monotonicity conjecture: The function $p \mapsto \mathscr{C}_p / p$ is strictly decreasing on the half-line $(0, \infty)$. Our main result implies that $\mathscr{C}_p<p / 2$ for $2<p<\infty$, and we verify numerically that $\mathscr{C}_p>p / 2$ for $1 \leq p<2$. We also estimate the asymptotic behavior of $\mathscr{C}_p$ as $p \rightarrow \infty$ and as $p \rightarrow 0^{+}$. Our approach is based on expressing $\mathscr{C}_p$ as the solution of an extremal problem. Extremal functions exist for all $0<p<\infty$; they are real entire functions with only real zeros, and the extremal functions are known to be unique for $1 \leq p<\infty$. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau-Pollak-Slepian operator of time-frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range $1<p<\infty$, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to $\mathscr{C}_p$ and the extremal functions.
Note: Reproducció del document publicat a: https://doi.org/10.1007/s11854-024-0338-z
It is part of: Journal d'Analyse Mathematique, 2024, vol. 153, p. 595-670
URI: https://hdl.handle.net/2445/215652
Related resource: https://doi.org/10.1007/s11854-024-0338-z
ISSN: 0021-7670
Appears in Collections:Articles publicats en revistes (Matemàtiques i Informàtica)

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