Some inequalities for entire functions
Let $\mathcal{L}_p$ be the subspace of the space $L_p(\mathbb{R})$ 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 th...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2024-09-01
|
| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/542 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849416151599153152 |
|---|---|
| author | N. Sushchyk D. Lukivska |
| author_facet | N. Sushchyk D. Lukivska |
| author_sort | N. Sushchyk |
| collection | DOAJ |
| description | Let $\mathcal{L}_p$ be the subspace of the space $L_p(\mathbb{R})$ 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 $(f(n/2))_{n\in\mathbb{Z}},$ namely
$$
\frac12 \|f\|_{p,1}\le \|f\|_{\mathcal{L}_p}\le 2 \|f\|_{p,1},
$$
where $\|f\|_{p,1}:=\frac12(\|Jf\|_{\ell_p(\mathbb{Z})} +\|JT_{1/2}f\|_{\ell_p(\mathbb{Z})})$. Here $J:\mathcal{L}_p\to\ell_p(\mathbb{Z})$ is the linear operator given by the formula $$ (Jf)(n):=(-1)^nf(n), \quad n\in\mathbb{Z},
$$ and $T_\tau$ is the shift by $\tau\in\mathbb{R}$ of the function $f$,
$$ (T_\tau f)(z):=f(z+\tau), \quad z\in\mathbb{C}.
$$ |
| format | Article |
| id | doaj-art-69d03aa11c04471e8ac38724e91845bc |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2024-09-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-69d03aa11c04471e8ac38724e91845bc2025-08-20T03:33:17ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202024-09-0162110911210.30970/ms.62.1.109-112542Some inequalities for entire functionsN. Sushchyk0D. Lukivska1Ivan Franko National University of Lviv Lviv, UkraineIvan Franko National University of Lviv Lviv, UkraineLet $\mathcal{L}_p$ be the subspace of the space $L_p(\mathbb{R})$ 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 $(f(n/2))_{n\in\mathbb{Z}},$ namely $$ \frac12 \|f\|_{p,1}\le \|f\|_{\mathcal{L}_p}\le 2 \|f\|_{p,1}, $$ where $\|f\|_{p,1}:=\frac12(\|Jf\|_{\ell_p(\mathbb{Z})} +\|JT_{1/2}f\|_{\ell_p(\mathbb{Z})})$. Here $J:\mathcal{L}_p\to\ell_p(\mathbb{Z})$ is the linear operator given by the formula $$ (Jf)(n):=(-1)^nf(n), \quad n\in\mathbb{Z}, $$ and $T_\tau$ is the shift by $\tau\in\mathbb{R}$ of the function $f$, $$ (T_\tau f)(z):=f(z+\tau), \quad z\in\mathbb{C}. $$http://matstud.org.ua/ojs/index.php/matstud/article/view/542entire functionsbanach spaces |
| spellingShingle | N. Sushchyk D. Lukivska Some inequalities for entire functions Математичні Студії entire functions banach spaces |
| title | Some inequalities for entire functions |
| title_full | Some inequalities for entire functions |
| title_fullStr | Some inequalities for entire functions |
| title_full_unstemmed | Some inequalities for entire functions |
| title_short | Some inequalities for entire functions |
| title_sort | some inequalities for entire functions |
| topic | entire functions banach spaces |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/542 |
| work_keys_str_mv | AT nsushchyk someinequalitiesforentirefunctions AT dlukivska someinequalitiesforentirefunctions |