On Bornological Spaces of Series in Systems of Functions
Let $f$ be an entire transcendental function, $M_f(r) = \max\{| f(z)| : |z| = r\}$, $(\lambda_n)$ be a sequence of positive numbers increasing to $+\infty$ and suppose that the series \[ A(z) = \sum_{n=1}^{\infty} a_n f(\lambda_n z) \] regularly converges in $\mathbb{C}$, i.e., \[ \sum_{n=1}^{\infty...
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| Published in | Sarajevo journal of mathematics Vol. 20; no. 2; pp. 249 - 254 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
14.04.2025
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| Online Access | Get full text |
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| Summary: | Let $f$ be an entire transcendental function, $M_f(r) = \max\{| f(z)| : |z| = r\}$, $(\lambda_n)$ be a sequence of positive numbers increasing to $+\infty$ and suppose that the series \[ A(z) = \sum_{n=1}^{\infty} a_n f(\lambda_n z) \] regularly converges in $\mathbb{C}$, i.e., \[ \sum_{n=1}^{\infty} |a_n| M_f(r\lambda_n) < +\infty \quad \text{for all } r \in [0,+\infty). \] Bornology is introduced on a set of such series as a system of functions $f(\lambda_n z)$, and its connection with Fréchet spaces is studied. |
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| ISSN: | 1840-0655 2233-1964 |
| DOI: | 10.5644/SJM.20.02.07 |