Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means
UDC 517.5 Let $(u_k)$ be a sequence of real or complex numbers. First, we consider a real sequence $(u_k)$ and formulate one-sided Tauberian conditions, which are necessary and sufficient for the  convergence of certain subsequences of $(u_k)$ to follow from its&nbs...
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| Datum: | 2024 |
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Institute of Mathematics, NAS of Ukraine
2024
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512678955450368 |
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| author | Sezer, Sefa Anıl Çanak, İbrahim Sezer, Sefa Anıl Çanak, İbrahim |
| author_facet | Sezer, Sefa Anıl Çanak, İbrahim Sezer, Sefa Anıl Çanak, İbrahim |
| author_sort | Sezer, Sefa Anıl |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-08-10T06:39:12Z |
| description | UDC 517.5
Let $(u_k)$ be a sequence of real or complex numbers. First, we consider a real sequence $(u_k)$ and formulate one-sided Tauberian conditions, which are necessary and sufficient for the  convergence of certain subsequences of $(u_k)$ to follow from its  deferred weighted summability. These conditions are satisfied if $(u_k)$ is deferred slowly decreasing or if $(u_k)$ obeys a Landau-type Tauberian condition. Second, we consider a complex sequence $(u_k)$ and present a two-sided Tauberian condition which is necessary and sufficient in order that the convergence of certain subsequences of $(u_k)$ follow from its deferred weighted summability.  This condition is satisfied either if $(u_k)$ is deferred slowly oscillating or if $(u_k)$ obeys a Hardy-type Tauberian condition. Finally, we extend these results to sequences in ordered linear spaces over the real numbers. |
| doi_str_mv | 10.3842/umzh.v76i7.7507 |
| first_indexed | 2026-03-24T03:32:37Z |
| format | Article |
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| id | umjimathkievua-article-7507 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:37Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-75072024-08-10T06:39:12Z Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means Sezer, Sefa Anıl Çanak, İbrahim Sezer, Sefa Anıl Çanak, İbrahim Summability by deferred weighted means Tauberian conditions deferred slow decrease and oscillation Landau and Hardy type conditions ordered linear spaces UDC 517.5 Let $(u_k)$ be a sequence of real or complex numbers. First, we consider a real sequence $(u_k)$ and formulate one-sided Tauberian conditions, which are necessary and sufficient for the  convergence of certain subsequences of $(u_k)$ to follow from its  deferred weighted summability. These conditions are satisfied if $(u_k)$ is deferred slowly decreasing or if $(u_k)$ obeys a Landau-type Tauberian condition. Second, we consider a complex sequence $(u_k)$ and present a two-sided Tauberian condition which is necessary and sufficient in order that the convergence of certain subsequences of $(u_k)$ follow from its deferred weighted summability.  This condition is satisfied either if $(u_k)$ is deferred slowly oscillating or if $(u_k)$ obeys a Hardy-type Tauberian condition. Finally, we extend these results to sequences in ordered linear spaces over the real numbers. УДК 517.5 Умови, за яких збіжність послідовності або деяких її  підпослідовностей випливає з сумовності за відкладеними ваговими середніми Нехай $(u_k)$ – послідовність дійсних або комплексних чисел. По-перше, ми розглядаємо дійсну послідовність $(u_k)$ і наводимо односторонні тауберові умови, що є необхідними і достатніми для того, щоб збіжність певних підпослідовностей послідовності $(u_k)$ випливала з її відкладеної вагової сумовності.  Ці умови виконуються, якщо $(u_k)$ є відкладено повільно спадною або якщо $(u_k)$ задовольняє тауберові умови типу Ландау. По-друге, ми розглядаємо комплексну послідовність $(u_k)$ і наводимо двосторонню тауберову умову, необхідну і достатню для того, щоб збіжність певних підпослідовностей послідовності $(u_k)$ випливала з її відкладеної вагової сумовності. Ця умова виконується, якщо $(u_k)$ є відкладено повільно осцилюючою  або якщо $(u_k)$ задовольняє тауберову умову типу Гарді. Насамкінець ми поширюємо ці результати на послідовності у впорядкованих лінійних просторах над дійсними числами.  Institute of Mathematics, NAS of Ukraine 2024-08-04 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7507 10.3842/umzh.v76i7.7507 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 7 (2024); 1041 - 1051 Український математичний журнал; Том 76 № 7 (2024); 1041 - 1051 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7507/10059 Copyright (c) 2024 Sefa Anıl Sezer, İbrahim Çanak |
| spellingShingle | Sezer, Sefa Anıl Çanak, İbrahim Sezer, Sefa Anıl Çanak, İbrahim Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means |
| title | Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means |
| title_alt | Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means |
| title_full | Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means |
| title_fullStr | Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means |
| title_full_unstemmed | Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means |
| title_short | Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means |
| title_sort | conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means |
| topic_facet | Summability by deferred weighted means Tauberian conditions deferred slow decrease and oscillation Landau and Hardy type conditions ordered linear spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7507 |
| work_keys_str_mv | AT sezersefaanıl conditionsunderwhichtheconvergenceofasequenceoritscertainsubsequencesfollowsfromthesummabilitybydeferredweightedmeans AT canakibrahim conditionsunderwhichtheconvergenceofasequenceoritscertainsubsequencesfollowsfromthesummabilitybydeferredweightedmeans AT sezersefaanıl conditionsunderwhichtheconvergenceofasequenceoritscertainsubsequencesfollowsfromthesummabilitybydeferredweightedmeans AT canakibrahim conditionsunderwhichtheconvergenceofasequenceoritscertainsubsequencesfollowsfromthesummabilitybydeferredweightedmeans |