Some Tauberian theorems for the weighted mean method of summability of double sequence
UDC 517.5 Let $p=(p_j)$ and $q=(q_k)$ be real sequences of nonnegative numbers with the property that $P_m=\sum _{j=0}^{m} p_j \neq 0$ and $Q_n=\sum _{k=0}^{n} q_k \neq 0$ for all $m$ and $n.$ Let $(P_m)$ and $(Q_n)$ be regulary varying positive indices. Assume that $(u_{mn})$ is a doub...
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| Date: | 2023 |
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| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2023
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/509 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507021992787968 |
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| author | Totur, Ümit Çanak, İbrahim Totur, Ümit Çanak, İbrahim |
| author_facet | Totur, Ümit Çanak, İbrahim Totur, Ümit Çanak, İbrahim |
| author_sort | Totur, Ümit |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2024-06-19T00:34:36Z |
| description | UDC 517.5
Let $p=(p_j)$ and $q=(q_k)$ be real sequences of nonnegative numbers with the property that $P_m=\sum _{j=0}^{m} p_j \neq 0$ and $Q_n=\sum _{k=0}^{n} q_k \neq 0$ for all $m$ and $n.$ Let $(P_m)$ and $(Q_n)$ be regulary varying positive indices. Assume that $(u_{mn})$ is a double sequence of complex (real) numbers, which is $(\overline{N},p,q; \alpha,\beta)$ summable with a finite limit, where $(\alpha,\beta)=(1,1)$, $(1,0)$, or $(0,1)$.  We present some conditions imposed on the weights under which  $(u_{mn})$ converges in Pringsheim's sense.  These results generalize and extend the results obtained by authors in [Comput. Math. Appl., 62, No. 6, 2609–2615 (2011)]. |
| doi_str_mv | 10.3842/umzh.v75i9.509 |
| first_indexed | 2026-03-24T02:02:42Z |
| format | Article |
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| id | umjimathkievua-article-509 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:42Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-5092024-06-19T00:34:36Z Some Tauberian theorems for the weighted mean method of summability of double sequence Some Tauberian theorems for the weighted mean method of summability of double sequence Totur, Ümit Çanak, İbrahim Totur, Ümit Çanak, İbrahim TAUBERIAN THEOREMS UDC 517.5 Let $p=(p_j)$ and $q=(q_k)$ be real sequences of nonnegative numbers with the property that $P_m=\sum _{j=0}^{m} p_j \neq 0$ and $Q_n=\sum _{k=0}^{n} q_k \neq 0$ for all $m$ and $n.$ Let $(P_m)$ and $(Q_n)$ be regulary varying positive indices. Assume that $(u_{mn})$ is a double sequence of complex (real) numbers, which is $(\overline{N},p,q; \alpha,\beta)$ summable with a finite limit, where $(\alpha,\beta)=(1,1)$, $(1,0)$, or $(0,1)$.  We present some conditions imposed on the weights under which  $(u_{mn})$ converges in Pringsheim's sense.  These results generalize and extend the results obtained by authors in [Comput. Math. Appl., 62, No. 6, 2609–2615 (2011)]. УДК 517.5 Деякі тауберові теореми для методу зваженого  середнього підсумовування подвійних послідовностей Нехай $p=(p_j)$ і $q=(q_k)$ – дійсні послідовності невід’ємних чисел такі, що $P_m=\sum _{j=0}^{m} p_j \neq 0$ і $Q_n= \sum _{k=0}^{n} q_k \neq 0$ для всіх $m$ і $n.$ Нехай $(P_m)$ і $(Q_n)$ – регулярно змінні додатні індекси. Припустимо, що $(u_{mn})$ – подвійна послідовність комплексних (або дійсних) чисел, яка є $(\overline{N},p,q; \alpha,\beta)$ сумовною зі скінченною границею, де $ (\alpha,\beta)=(1,1)$, $(1,0)$ або $(0,1)$.  Наведено деякі умови, що накладені на ваги, за яких  $(u_{mn})$ збігається в розумінні Прінгсхейма.  Ці результати узагальнюють і розширюють результати, отримані авторами в [Comput. Math. Appl., 62, No. 6, 2609–2615 (2011)].  Institute of Mathematics, NAS of Ukraine 2023-09-26 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/509 10.3842/umzh.v75i9.509 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 9 (2023); 1276 - 1293 Український математичний журнал; Том 75 № 9 (2023); 1276 - 1293 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/509/9750 |
| spellingShingle | Totur, Ümit Çanak, İbrahim Totur, Ümit Çanak, İbrahim Some Tauberian theorems for the weighted mean method of summability of double sequence |
| title | Some Tauberian theorems for the weighted mean method of summability of double sequence |
| title_alt | Some Tauberian theorems for the weighted mean method of summability of double sequence |
| title_full | Some Tauberian theorems for the weighted mean method of summability of double sequence |
| title_fullStr | Some Tauberian theorems for the weighted mean method of summability of double sequence |
| title_full_unstemmed | Some Tauberian theorems for the weighted mean method of summability of double sequence |
| title_short | Some Tauberian theorems for the weighted mean method of summability of double sequence |
| title_sort | some tauberian theorems for the weighted mean method of summability of double sequence |
| topic_facet | TAUBERIAN THEOREMS |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/509 |
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