Necessary and sufficient Tauberian conditions under which convergence follows from $A^{r,\delta}$ summability
Let $x=(x_{mn})$ be a double sequence of real or complex numbers. The $A^{r,\delta}$-transform of a sequence $(x_{mn})$ is defined by $$ (A^{r,\delta}x)_{mn}={\sigma^{r,\delta}_{mn}(x)}=\frac{1}{(m+1)(n+1)}\sum_{j=0}^{m}\sum_{k=0}^{n}(1+r^j)(1+\delta^k)x_{jk}, \ \ \ \ \ 0<r, \delta<1 $$ The $...
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| Format: | Article |
| Language: | English |
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Sociedade Brasileira de Matemática
2022-12-01
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| Series: | Boletim da Sociedade Paranaense de Matemática |
| Online Access: | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/50823 |
| _version_ | 1797633608715862016 |
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| author | Çagla Kambak İbrahim Çanak |
| author_facet | Çagla Kambak İbrahim Çanak |
| author_sort | Çagla Kambak |
| collection | DOAJ |
| description |
Let $x=(x_{mn})$ be a double sequence of real or complex numbers. The $A^{r,\delta}$-transform of a sequence $(x_{mn})$ is defined by
$$
(A^{r,\delta}x)_{mn}={\sigma^{r,\delta}_{mn}(x)}=\frac{1}{(m+1)(n+1)}\sum_{j=0}^{m}\sum_{k=0}^{n}(1+r^j)(1+\delta^k)x_{jk}, \ \ \ \ \ 0<r, \delta<1
$$
The $A^{r,*}$ and $A^{*,\delta}$ transformations are defined respectively by
$$
(A^{r,*}x)_{mn}={\sigma^{r,*}_{mn}(x)}=\frac{1}{m+1}\sum_{j=0}^{m}(1+r^{j})x_{jn}, \ \ \ 0<r<1,
$$
and
$$
(A^{*,\delta}x)_{mn}={\sigma^{*,\delta}_{mn}(x)}=\frac{1}{n+1}\sum_{k=0}^{n}(1+\delta^{k})x_{mk},\ \ \ 0<\delta<1.
$$
We say that $(x_{mn})$ is ($A^{r,\delta}$,1,1) summable to $l$ if $({\sigma^{r,\delta}_{mn}}(x))$ has a finite limit $l$. It is known that if $\lim_{m,n \to \infty }x_{mn}=l$ and $(x_{mn})$ is bounded, then the limit $\lim _{m,n \to \infty} \sigma_{mn}^{r,\delta}(x)=l$ exists.
But the inverse of this implication is not true in general. Our aim is to obtain necessary and sufficient conditions for ($A^{r,\delta}$,1,1) summability method under which the inverse of this implication holds. Following Tauberian theorems for $(A^{r,\delta},1,1)$ summability method, we also introduce $A^{r,*}$ and $A^{*,\delta}$ transformations of double sequences and obtain Tauberian theorems for the $(A^{r,*},1,0)$ and $(A^{*,\delta},0,1)$ summability methods.
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| first_indexed | 2024-03-11T11:56:18Z |
| format | Article |
| id | doaj.art-e66d2ef4557f4022ad9fd94f895364d7 |
| institution | Directory Open Access Journal |
| issn | 0037-8712 2175-1188 |
| language | English |
| last_indexed | 2024-03-11T11:56:18Z |
| publishDate | 2022-12-01 |
| publisher | Sociedade Brasileira de Matemática |
| record_format | Article |
| series | Boletim da Sociedade Paranaense de Matemática |
| spelling | doaj.art-e66d2ef4557f4022ad9fd94f895364d72023-11-08T19:10:29ZengSociedade Brasileira de MatemáticaBoletim da Sociedade Paranaense de Matemática0037-87122175-11882022-12-014110.5269/bspm.50823Necessary and sufficient Tauberian conditions under which convergence follows from $A^{r,\delta}$ summabilityÇagla Kambak0İbrahim Çanak1Ege UniversityEge University Let $x=(x_{mn})$ be a double sequence of real or complex numbers. The $A^{r,\delta}$-transform of a sequence $(x_{mn})$ is defined by $$ (A^{r,\delta}x)_{mn}={\sigma^{r,\delta}_{mn}(x)}=\frac{1}{(m+1)(n+1)}\sum_{j=0}^{m}\sum_{k=0}^{n}(1+r^j)(1+\delta^k)x_{jk}, \ \ \ \ \ 0<r, \delta<1 $$ The $A^{r,*}$ and $A^{*,\delta}$ transformations are defined respectively by $$ (A^{r,*}x)_{mn}={\sigma^{r,*}_{mn}(x)}=\frac{1}{m+1}\sum_{j=0}^{m}(1+r^{j})x_{jn}, \ \ \ 0<r<1, $$ and $$ (A^{*,\delta}x)_{mn}={\sigma^{*,\delta}_{mn}(x)}=\frac{1}{n+1}\sum_{k=0}^{n}(1+\delta^{k})x_{mk},\ \ \ 0<\delta<1. $$ We say that $(x_{mn})$ is ($A^{r,\delta}$,1,1) summable to $l$ if $({\sigma^{r,\delta}_{mn}}(x))$ has a finite limit $l$. It is known that if $\lim_{m,n \to \infty }x_{mn}=l$ and $(x_{mn})$ is bounded, then the limit $\lim _{m,n \to \infty} \sigma_{mn}^{r,\delta}(x)=l$ exists. But the inverse of this implication is not true in general. Our aim is to obtain necessary and sufficient conditions for ($A^{r,\delta}$,1,1) summability method under which the inverse of this implication holds. Following Tauberian theorems for $(A^{r,\delta},1,1)$ summability method, we also introduce $A^{r,*}$ and $A^{*,\delta}$ transformations of double sequences and obtain Tauberian theorems for the $(A^{r,*},1,0)$ and $(A^{*,\delta},0,1)$ summability methods. https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/50823 |
| spellingShingle | Çagla Kambak İbrahim Çanak Necessary and sufficient Tauberian conditions under which convergence follows from $A^{r,\delta}$ summability Boletim da Sociedade Paranaense de Matemática |
| title | Necessary and sufficient Tauberian conditions under which convergence follows from $A^{r,\delta}$ summability |
| title_full | Necessary and sufficient Tauberian conditions under which convergence follows from $A^{r,\delta}$ summability |
| title_fullStr | Necessary and sufficient Tauberian conditions under which convergence follows from $A^{r,\delta}$ summability |
| title_full_unstemmed | Necessary and sufficient Tauberian conditions under which convergence follows from $A^{r,\delta}$ summability |
| title_short | Necessary and sufficient Tauberian conditions under which convergence follows from $A^{r,\delta}$ summability |
| title_sort | necessary and sufficient tauberian conditions under which convergence follows from a r delta summability |
| url | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/50823 |
| work_keys_str_mv | AT caglakambak necessaryandsufficienttauberianconditionsunderwhichconvergencefollowsfromardeltasummability AT ibrahimcanak necessaryandsufficienttauberianconditionsunderwhichconvergencefollowsfromardeltasummability |