Euler-Riesz Difference Sequence Spaces

Year 2017, Volume: 7, 63 - 72, 19.12.2017

Hacer Bilgin Ellidokuzoğlu , Serkan Demiriz

Abstract

Ba\c{s}ar and Braha \cite{braha-basar-2016}, introduced the

sequence spaces $\ell_\infty$, $c$ and $c_0$  of Euler- Ces\'{a}ro

bounded, convergent and null difference sequences and studied

their some properties. The main purpose of this study is to

introduce 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

$\Delta$. Furthermore, the inclusions

$\ell_\infty\subset{[\ell_\infty]}_{e.r}, c\subset {[c]}_{e.r}$

and $c_0\subset{[c_0]}_{e.r}$ strictly hold, the basis of the

sequence spaces ${[c_0 ]}_(e.r)$ and ${[c]}_(e.r)$ is constucted

and alpha-, beta- and gamma-duals of these spaces are determined.

Finally, the classes of matrix transformations from the Euler-

Riesz difference sequence spaces to the spaces $\ell_\infty, c$

and $c_0$ are characterized. 

Keywords

Euler sequence spaces, Riesz sequence spaces, matrix transformations

References

• Altay, B., Başar, F., Some Euler sequence spaces of non-absolute type, Ukrainian Math. J., 57(2005), 1--17.
• Altay, B., Başar, F., Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 30(2006), 591--608.
• Altay, B., Başar, F., Mursaleen M., On the Euler sequence spaces which include the spaces $\ell_p$ and $\ell_\infty$ I, Inform. Sci., 176(2006), 1450--1462.
• Altay, B., Başar, F., Certain topological properties and duals of the domain of a triangle matrix in a sequence space, J. Math. Anal. Appl., 336(2007), 632--645.
• Altay, B., Başar, F., The fine spectrum and the matrix domain of the difference operator $\Delta$ on the sequence space $\ell_p, (0 < p < 1)$, Commun.Math. Anal., 2(2007), 1--11.
• Başar, F., Altay, B., On the space of sequences of $p-$bounded variation and related matrix mappings, Ukrainian Math. J., 55(2003), 136--147.
• Başar, F., Summability Theory and Its Applications, Bentham Science Publishers, e-books, Monographs, Istanbul, 2012.
• Başar, F., Domain of the composition of some triangles in the space of $p-$summable sequences, AIP Conference Proceedings, 1611(2014), 348--356.
• Başar, F., Braha, N. L., Euler- Ces\'{a}ro Difference Spaces of Bounded, Convergent and Null Sequences, Tamkang J. Math., 47(4)(2016), 405--420.
• Başarır, M., On the generalized Riesz $B-$difference sequence spaces, Filomat, 24.4(2010), 35--52.
• Choudhary, B., Mishra, S. K., A note on K\"{o}the-Toeplitz duals of certain sequence spaces and their matrix transformations, Internat. J.Math.Math. Sci., 18(1995), 681--688.
• Cooke, R. G., Infinite Matrices and Sequence Spaces, Macmillan and Co. Limited, London, 1950.
• Çolak, R., Et, M., Malkowsky, E., Some Topics of Sequence Spaces, in: Lecture Notes in Mathematics, F{\i}rat Univ. Press, (2004), 1--63, ISBN: 975-394-0386-6.
• Demiriz, S., Çakan, C., On some new paranormed Euler sequence spaces and Euler core, Acta Math. Sci., 26.7(2010), 1207--1222.
• Ercan, S., Bektaş, Ç. A., Some generalized difference sequence spaces of non-absolute type, Gen. Math. Notes., 27(2)(2015), 37--46.
• Grosse-Erdmann, K. G., On $\ell^1$-invariant sequence spaces, J. Math. Anal. Appl., 262(2001), 112--132.
• Kamthan, P. K., Gupta, M., Sequence Spaces and Series, Marcel Dekker Inc., New York and Basel, 1981.
• Kızmaz, H., On certain sequence spaces, Canad.Math. Bull., 24(1981), 169--176.
• Kirişçi, M., Başar, F., Some new sequence spaces derived by the domain of generalized difference matrix, Comput.Math. Appl., 60(2010), 1299--1309.
• Polat, H., Başar, F., Some Euler spaces of difference sequences of order $m$, Acta Math. Sci. Ser. B Engl. Ed., 27(2)(2007), 254--266.
• Sönmez, A., Some new sequence spaces derived by the domain of the triple bandmatrix, Comput.Math. Appl., 62(2011), 641--650.
• Stieglitz, M., Tietz, H., Matrix transformationen von folgenraumen eine ergebnisubersict, Math. Z., 154(1977), 1--16.
• Wilansky, A., Summability through Functional Analysis, North-HollandMathematics Studies 85, Amsterdam-Newyork-Oxford, 1984.