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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| Language: | deu |
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Ivan Franko National University of Lviv
2024-09-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/555 |
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| author | M. M. Sheremeta Yu. M. Gal' |
| author_facet | M. M. Sheremeta Yu. M. Gal' |
| author_sort | M. M. Sheremeta |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-92473e35814d46e8a57b9fcd19c24cca |
| institution | DOAJ |
| 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-92473e35814d46e8a57b9fcd19c24cca2025-08-20T02:40:18ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202024-09-01621465310.30970/ms.62.1.46-53555On some properties of the maximal term of series in systems of functionsM. M. Sheremeta0Yu. M. Gal'1Ivan Franko National University of Lviv, LvivDrogobych Ivan Franko Pedagogical State UniversityFor 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.http://matstud.org.ua/ojs/index.php/matstud/article/view/555entire function; regularly converging series; maximal term, lower generalized order |
| spellingShingle | M. M. Sheremeta Yu. M. Gal' On some properties of the maximal term of series in systems of functions Математичні Студії entire function; regularly converging series; maximal term, lower generalized order |
| title | On some properties of the maximal term of series in systems of functions |
| title_full | On some properties of the maximal term of series in systems of functions |
| title_fullStr | On some properties of the maximal term of series in systems of functions |
| title_full_unstemmed | On some properties of the maximal term of series in systems of functions |
| title_short | On some properties of the maximal term of series in systems of functions |
| title_sort | on some properties of the maximal term of series in systems of functions |
| topic | entire function; regularly converging series; maximal term, lower generalized order |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/555 |
| work_keys_str_mv | AT mmsheremeta onsomepropertiesofthemaximaltermofseriesinsystemsoffunctions AT yumgal onsomepropertiesofthemaximaltermofseriesinsystemsoffunctions |