Year: 2024
Type: article
Abstract: Let $\mathcal{L}_p$ be the subspace of the space $L_p(\bR)$ consisting of the restriction to the real axis of all entire functions of exponential type $\le \pi$. In this paper, for any function $f\in \mathcal{L}_p \,\,(1\le p\le \infty)$, we obtain estimates for the norm of $f$ in terms of the sequence $(f(n/2))_{n\in\bZ},$ namely$$\frac12 \|f\|_{p,1}\le \|f\|_{\mathcal{L}_p}\le 2 \|f\|_{p,1},$$where $\|f\|_{p,1}:=\frac12(\|Jf\|_{\ell_p(\bZ)} +\|JT_{1/2}f\|_{\ell_p(\bZ)})$. Here $J:\mathcal{L}_p\to\ell_p(\bZ)$ is the linear operator given by the formula $$ (Jf)(n):=(-1)^nf(n), \quad n\in\bZ,$$ and $T_\tau$ is the shift by $\tau\in\bR$ of the function $f$,$$ (T_\tau f)(z):=f(z+\tau), \quad z\in\bC.$$
Source: Matematychni Studii
Authors: N. Sushchyk, D. Lukivska
Institution: Lviv University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4402579379