On some properties of the maximal term of series in systems of functions
For an entire transcendental function $f$ and a sequence $(\lambda_n)$ of positive numbers increa\-sing to $+\infty$ a series $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ in the system $\{f(\lambda_nz)\}$ is said to be regularly convergent in ${\mathbb C}$ if $\mathfrak{M}(r,A)=\sum_{n=1}^{\infty}...
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| Main Authors: | , |
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| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2024-09-01
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| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/555 |
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| Summary: | For an entire transcendental function $f$ and a sequence $(\lambda_n)$ of positive numbers increa\-sing to $+\infty$ a series
$A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ in the system $\{f(\lambda_nz)\}$ is said to be regularly convergent in ${\mathbb C}$ if
$\mathfrak{M}(r,A)=\sum_{n=1}^{\infty} |a_n|M_f(r\lambda_n)<+\infty$ for all $r\in (0,+\infty)$, where $ M_f(r)=\max\{|f(z)|\colon |z|=r\}$.
Conditions are found on $(\lambda_n)$ and $f$, under which $\ln\mathfrak{M}(r,A)\sim \ln \mu(r,A)$ as $r\to+\infty$, where
$\mu(r,A)= \max\{|a_n|M_f(r\lambda_n)\colon n\ge 1\}$ is the maximal term of the series. A~formula for finding
the lower generalized order $$\lambda_{\alpha,\beta}[A]=\varliminf\limits_{r\to+\infty}\dfrac{\alpha(\ln \mathfrak{M}(r,A))}{\beta(r)}$$
is obtained, where the functions $\alpha$ and $\beta$ are positive, continuous and increasing to $+\infty$.
The open problems are formulated. |
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| ISSN: | 1027-4634 2411-0620 |