Bernstein-Type Theorems and Uniqueness Theorems
Let \(f\) be an entire function of finite type with respect to finite order \(\rho {\text{ in }}\mathbb{C}^n \) and let \(\mathbb{E}\) be a subset of an open cone in a certain n-dimensional subspace \(\mathbb{R}^{2n} {\text{ ( = }}\mathbb{C}^n {\text{)}}\) (the smaller \(\rho \) ,...
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| Date: | 2004 |
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| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2004
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3743 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509876766113792 |
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| author | Logvinenko, V. Nazarova, N. Логвиненко, В. В. Назарова, Н. |
| author_facet | Logvinenko, V. Nazarova, N. Логвиненко, В. В. Назарова, Н. |
| author_sort | Logvinenko, V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:07:09Z |
| description | Let \(f\) be an entire function of finite type with respect to finite order \(\rho {\text{ in }}\mathbb{C}^n \) and let \(\mathbb{E}\) be a subset of an open cone in a certain n-dimensional subspace \(\mathbb{R}^{2n} {\text{ ( = }}\mathbb{C}^n {\text{)}}\) (the smaller \(\rho \) , the sparser \(\mathbb{E}\) ). We assume that this cone contains a ray \(\left\{ {z = tz^0 \in \mathbb{C}^n :t > 0} \right\}\) . It is shown that the radial indicator \(h_f (z^0 )\) of \(f\) at any point \(z^0 \in \mathbb{C}^n \backslash \{ 0\} \) may be evaluated in terms of function values at points of the discrete subset \(\mathbb{E}\) . Moreover, if \(f\) tends to zero fast enough as \(z \to \infty \) over \(\mathbb{E}\) , then this function vanishes identically. To prove these results, a special approximation technique is developed. In the last part of the paper, it is proved that, under certain conditions on \(\rho \) and \(\mathbb{E}\) , which are close to exact conditions, the function \(f\) bounded on \(\mathbb{E}\) is bounded on the ray. |
| first_indexed | 2026-03-24T02:48:04Z |
| format | Article |
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| id | umjimathkievua-article-3743 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:48:04Z |
| publishDate | 2004 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/bd/8568a614e6d9460b0de8a59ddba2bfbd.pdf |
| spelling | umjimathkievua-article-37432020-03-18T20:07:09Z Bernstein-Type Theorems and Uniqueness Theorems Теореми типу Бернштейна та теореми про єдність Logvinenko, V. Nazarova, N. Логвиненко, В. В. Назарова, Н. Let \(f\) be an entire function of finite type with respect to finite order \(\rho {\text{ in }}\mathbb{C}^n \) and let \(\mathbb{E}\) be a subset of an open cone in a certain n-dimensional subspace \(\mathbb{R}^{2n} {\text{ ( = }}\mathbb{C}^n {\text{)}}\) (the smaller \(\rho \) , the sparser \(\mathbb{E}\) ). We assume that this cone contains a ray \(\left\{ {z = tz^0 \in \mathbb{C}^n :t > 0} \right\}\) . It is shown that the radial indicator \(h_f (z^0 )\) of \(f\) at any point \(z^0 \in \mathbb{C}^n \backslash \{ 0\} \) may be evaluated in terms of function values at points of the discrete subset \(\mathbb{E}\) . Moreover, if \(f\) tends to zero fast enough as \(z \to \infty \) over \(\mathbb{E}\) , then this function vanishes identically. To prove these results, a special approximation technique is developed. In the last part of the paper, it is proved that, under certain conditions on \(\rho \) and \(\mathbb{E}\) , which are close to exact conditions, the function \(f\) bounded on \(\mathbb{E}\) is bounded on the ray. Нехай $f$ — ціла функція скінченного типу відносно порядку $\rho$ у $\mathbb{C}^n$, $\mathbb{E}$ — підмножииа відкритого конуса (чим менше $\rho$ , тим більш розрідженим є $\mathbb{E}$ у деякому $n$-вимірному підпросторі $\mathbb{R}^{2n} {\text{ ( = }}\mathbb{C}^n {\text{)}}$. Припускається, що даний конус містить промінь $\left\{ {z = tz^0 \in \mathbb{C}^n :t > 0} \right\}$. Показано, що радіальний індикатор $h_f (z^0 )$ функції $f$ у будь-якій точці $z^0 \in \mathbb{C}^n \backslash \{ 0\}$ можна оцінити через значення функції $f$ у точках дискретної множини $\mathbb{E}$. Крім того, якщо $f \to 0$ досить швидко при $z \to \infty$ на $\mathbb{E}$, то дана функція дорівнює нулю тотожно. Для доведення цих результатів розроблено спеціальну апроксимаційну техніку. В останній частині роботи доведено, що за деяких близьких до точних умов відносно $\rho$ і $\mathbb{E}$ функція /, обмежена на $\mathbb{E}$, буде обмеженою па всьому промені. Institute of Mathematics, NAS of Ukraine 2004-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3743 Ukrains’kyi Matematychnyi Zhurnal; Vol. 56 No. 2 (2004); 198-213 Український математичний журнал; Том 56 № 2 (2004); 198-213 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3743/4209 https://umj.imath.kiev.ua/index.php/umj/article/view/3743/4210 Copyright (c) 2004 Logvinenko V.; Nazarova N. |
| spellingShingle | Logvinenko, V. Nazarova, N. Логвиненко, В. В. Назарова, Н. Bernstein-Type Theorems and Uniqueness Theorems |
| title | Bernstein-Type Theorems and Uniqueness Theorems |
| title_alt | Теореми типу Бернштейна
та теореми про єдність |
| title_full | Bernstein-Type Theorems and Uniqueness Theorems |
| title_fullStr | Bernstein-Type Theorems and Uniqueness Theorems |
| title_full_unstemmed | Bernstein-Type Theorems and Uniqueness Theorems |
| title_short | Bernstein-Type Theorems and Uniqueness Theorems |
| title_sort | bernstein-type theorems and uniqueness theorems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3743 |
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