Point evaluation in Paley--Wiener spaces
| Autor: | Brevig, Ole Fredrik, Chirre, Andrés, Ortega-Cerdà, Joaquim, Seip, Kristian |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | J. Anal. Math. 153 (2024), no. 2, 595--670 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s11854-024-0338-z |
| Popis: | We study the norm of point evaluation at the origin in the Paley--Wiener space $PW^p$ 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 $PW^p$. 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 $1 \leq p < 2$. We also estimate the asymptotic behavior of $\mathscr{C}_p$ as $p \to \infty$ and as $p \to 0^+$. Our approach is based on expressing $\mathscr{C}_p$ as the solution of an extremal problem. Extremal functions exist for all $0 |
| Databáze: | arXiv |
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