Discretization Theorems for Entire Functions of Exponential Type
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908373535948800 |
|---|---|
| author | Ganzburg, Michael I. |
| author_facet | Ganzburg, Michael I. |
| contents | We prove $L_q(\R^m)$--discretization inequalities
for entire functions $f$ of exponential type
in the form
\ba
C_2\|f\|_{L_q(\R^m)}
\le \left(\sum_{ν=1}^\iy \left\vert f\left(X_ν\right)
\right\vert^q\right)^{1/q}
\le C_1\|f\|_{L_q(\R^m)},\qquad q\in[1,\iy],
\ea
with estimates for $C_1$ and $C_2$.
We find a necessary and sufficient condition on
$Ω=\left\{X_ν\right\}_{ν=1}^\iy\subset\R^m$
for the right inequality to be valid and a
sufficient condition on $Ω$ for the left one to
hold true.
In addition, $L_\iy(Q^m_b)$-discretization
inequalities on an $m$-dimensional cube are proved for
entire functions of exponential type
and exponential polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15762 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discretization Theorems for Entire Functions of Exponential Type Ganzburg, Michael I. Classical Analysis and ODEs Primary 26D07, 26D15, Secondary 41A10, 41A63 We prove $L_q(\R^m)$--discretization inequalities for entire functions $f$ of exponential type in the form \ba C_2\|f\|_{L_q(\R^m)} \le \left(\sum_{ν=1}^\iy \left\vert f\left(X_ν\right) \right\vert^q\right)^{1/q} \le C_1\|f\|_{L_q(\R^m)},\qquad q\in[1,\iy], \ea with estimates for $C_1$ and $C_2$. We find a necessary and sufficient condition on $Ω=\left\{X_ν\right\}_{ν=1}^\iy\subset\R^m$ for the right inequality to be valid and a sufficient condition on $Ω$ for the left one to hold true. In addition, $L_\iy(Q^m_b)$-discretization inequalities on an $m$-dimensional cube are proved for entire functions of exponential type and exponential polynomials. |
| title | Discretization Theorems for Entire Functions of Exponential Type |
| topic | Classical Analysis and ODEs Primary 26D07, 26D15, Secondary 41A10, 41A63 |
| url | https://arxiv.org/abs/2505.15762 |