Volume 37, Issue 4
${[\ell_p]}_{e.r}$ Euler-Riesz Difference Sequence Spaces

Anal. Theory Appl., 37 (2021), pp. 557-571.

Published online: 2021-11

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• Abstract

Başar and Braha [1], introduced the sequence spaces $\breve{\ell}_\infty$, $\breve{c}$ and $\breve{c}_0$ of Euler-Cesáro bounded, convergent and null difference sequences and studied their some properties. Then, in [2], we introduced the sequence spaces ${[\ell_\infty]}_{e.r}, {[c]}_{e.r}$ and ${[c_0]}_{e.r}$ of Euler-Riesz bounded, convergent and null difference sequences by using the composition of the Euler mean $E_1$ and Riesz mean $R_q$ with backward difference operator $\Delta$. The main purpose of this study is to introduce the sequence space ${[\ell_p]}_{e.r}$ of Euler-Riesz $p-$absolutely convergent series, where $1 \leq p <\infty$, difference sequences by using the composition of the Euler mean $E_1$ and Riesz mean $R_q$ with backward difference operator $\Delta$. Furthermore, the inclusion $\ell_p\subset{[\ell_p]}_{e.r}$ hold, the basis of the sequence space ${[\ell_p]}_{e.r}$ is constructed and  $\alpha-$, $\beta-$ and $\gamma-$duals of the space are determined. Finally, the classes of matrix transformations from the ${[\ell_p]}_{e.r}$ Euler-Riesz difference sequence space to the spaces $\ell_\infty, c$ and $c_0$ are characterized. We devote the final section of the paper to examine some geometric properties of the space ${[\ell_p]}_{e.r}$.

• Keywords

Composition of summability methods, Riesz mean of order one, Euler mean of order one, backward difference operator, sequence space, BK space, Schauder basis, $\beta-$duals, matrix transformations.

40C05, 40A05, 46A45

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@Article{ATA-37-557, author = {Hacer Bilgin and Ellidokuzoğlu and and 21372 and and Hacer Bilgin Ellidokuzoğlu and Serkan and Demiriz and and 21373 and and Serkan Demiriz}, title = {${[\ell_p]}_{e.r}$ Euler-Riesz Difference Sequence Spaces}, journal = {Analysis in Theory and Applications}, year = {2021}, volume = {37}, number = {4}, pages = {557--571}, abstract = {

Başar and Braha [1], introduced the sequence spaces $\breve{\ell}_\infty$, $\breve{c}$ and $\breve{c}_0$ of Euler-Cesáro bounded, convergent and null difference sequences and studied their some properties. Then, in [2], we introduced the sequence spaces ${[\ell_\infty]}_{e.r}, {[c]}_{e.r}$ and ${[c_0]}_{e.r}$ of Euler-Riesz bounded, convergent and null difference sequences by using the composition of the Euler mean $E_1$ and Riesz mean $R_q$ with backward difference operator $\Delta$. The main purpose of this study is to introduce the sequence space ${[\ell_p]}_{e.r}$ of Euler-Riesz $p-$absolutely convergent series, where $1 \leq p <\infty$, difference sequences by using the composition of the Euler mean $E_1$ and Riesz mean $R_q$ with backward difference operator $\Delta$. Furthermore, the inclusion $\ell_p\subset{[\ell_p]}_{e.r}$ hold, the basis of the sequence space ${[\ell_p]}_{e.r}$ is constructed and  $\alpha-$, $\beta-$ and $\gamma-$duals of the space are determined. Finally, the classes of matrix transformations from the ${[\ell_p]}_{e.r}$ Euler-Riesz difference sequence space to the spaces $\ell_\infty, c$ and $c_0$ are characterized. We devote the final section of the paper to examine some geometric properties of the space ${[\ell_p]}_{e.r}$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2017-0068}, url = {http://global-sci.org/intro/article_detail/ata/19964.html} }
TY - JOUR T1 - ${[\ell_p]}_{e.r}$ Euler-Riesz Difference Sequence Spaces AU - Ellidokuzoğlu , Hacer Bilgin AU - Demiriz , Serkan JO - Analysis in Theory and Applications VL - 4 SP - 557 EP - 571 PY - 2021 DA - 2021/11 SN - 37 DO - http://doi.org/10.4208/ata.OA-2017-0068 UR - https://global-sci.org/intro/article_detail/ata/19964.html KW - Composition of summability methods, Riesz mean of order one, Euler mean of order one, backward difference operator, sequence space, BK space, Schauder basis, $\beta-$duals, matrix transformations. AB -

Başar and Braha [1], introduced the sequence spaces $\breve{\ell}_\infty$, $\breve{c}$ and $\breve{c}_0$ of Euler-Cesáro bounded, convergent and null difference sequences and studied their some properties. Then, in [2], we introduced the sequence spaces ${[\ell_\infty]}_{e.r}, {[c]}_{e.r}$ and ${[c_0]}_{e.r}$ of Euler-Riesz bounded, convergent and null difference sequences by using the composition of the Euler mean $E_1$ and Riesz mean $R_q$ with backward difference operator $\Delta$. The main purpose of this study is to introduce the sequence space ${[\ell_p]}_{e.r}$ of Euler-Riesz $p-$absolutely convergent series, where $1 \leq p <\infty$, difference sequences by using the composition of the Euler mean $E_1$ and Riesz mean $R_q$ with backward difference operator $\Delta$. Furthermore, the inclusion $\ell_p\subset{[\ell_p]}_{e.r}$ hold, the basis of the sequence space ${[\ell_p]}_{e.r}$ is constructed and  $\alpha-$, $\beta-$ and $\gamma-$duals of the space are determined. Finally, the classes of matrix transformations from the ${[\ell_p]}_{e.r}$ Euler-Riesz difference sequence space to the spaces $\ell_\infty, c$ and $c_0$ are characterized. We devote the final section of the paper to examine some geometric properties of the space ${[\ell_p]}_{e.r}$.

Hacer Bilgin Ellidokuzoğlu & Serkan Demiriz. (1970). ${[\ell_p]}_{e.r}$ Euler-Riesz Difference Sequence Spaces. Analysis in Theory and Applications. 37 (4). 557-571. doi:10.4208/ata.OA-2017-0068
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