Analogues of some fundamental theorems of summability theory
In 1911, Steinhaus presented the following theorem: if A is a regular matrix then there exists a sequence of 0's and 1's which is not A-summable. In 1943, R. C. Buck characterized convergent sequences as follows: a sequence x is convergent if and only if there exists a regular matrix A whi...
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| Format: | Article |
| Language: | English |
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Wiley
2000-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171200001782 |
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| author | Richard F. Patterson |
| author_facet | Richard F. Patterson |
| author_sort | Richard F. Patterson |
| collection | DOAJ |
| description | In 1911, Steinhaus presented the following theorem: if A is a regular matrix then there exists a sequence of 0's and 1's which is
not A-summable. In 1943, R. C. Buck characterized convergent
sequences as follows: a sequence x is convergent if and only if
there exists a regular matrix A which sums every subsequence of
x. In this paper, definitions for subsequences of a double
sequence and Pringsheim limit points of a double sequence are
introduced. In addition, multidimensional analogues of Steinhaus'
and Buck's theorems are proved. |
| format | Article |
| id | doaj-art-9dc9d3347c3b4a9a8428009ec1ba4739 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2000-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-9dc9d3347c3b4a9a8428009ec1ba47392025-02-03T05:53:37ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252000-01-012311910.1155/S0161171200001782Analogues of some fundamental theorems of summability theoryRichard F. Patterson0Department of Mathematics and Computer Science, Duquesne University, 440 College Hall, Pittsburgh 15282, PA, USAIn 1911, Steinhaus presented the following theorem: if A is a regular matrix then there exists a sequence of 0's and 1's which is not A-summable. In 1943, R. C. Buck characterized convergent sequences as follows: a sequence x is convergent if and only if there exists a regular matrix A which sums every subsequence of x. In this paper, definitions for subsequences of a double sequence and Pringsheim limit points of a double sequence are introduced. In addition, multidimensional analogues of Steinhaus' and Buck's theorems are proved.http://dx.doi.org/10.1155/S0161171200001782Subsequences of a double sequencePringsheim limit pointP-convergentP-divergentRH-regular. |
| spellingShingle | Richard F. Patterson Analogues of some fundamental theorems of summability theory International Journal of Mathematics and Mathematical Sciences Subsequences of a double sequence Pringsheim limit point P-convergent P-divergent RH-regular. |
| title | Analogues of some fundamental theorems of summability theory |
| title_full | Analogues of some fundamental theorems of summability theory |
| title_fullStr | Analogues of some fundamental theorems of summability theory |
| title_full_unstemmed | Analogues of some fundamental theorems of summability theory |
| title_short | Analogues of some fundamental theorems of summability theory |
| title_sort | analogues of some fundamental theorems of summability theory |
| topic | Subsequences of a double sequence Pringsheim limit point P-convergent P-divergent RH-regular. |
| url | http://dx.doi.org/10.1155/S0161171200001782 |
| work_keys_str_mv | AT richardfpatterson analoguesofsomefundamentaltheoremsofsummabilitytheory |