Spaces of Series Summable by Absolute Cesí ro and Matrix Operators

Authors

  • Mehmet Ali Sarıgöl Pamukkale University

DOI:

https://doi.org/10.26713/cma.v7i1.360

Keywords:

Summability factors, matrix transformations, sequence spaces, Cesí ro spaces

Abstract

In this paper giving some algebraic and topological properties of \(|C_\alpha|_k\), we characterize the classes of all infinite matrices \((|C_\alpha|,|C_\delta|_k)\) and \((|C_\alpha|_k,|C_\delta|)\) for \(\alpha,\delta>-1\) and \(k\ge 1\), show that each element of this classes correspond to a continuous linear mapping, which also enables us to extend some well known results of Flett [7], Orhan and Sarigol [15], Bosanquet [2], Mehdi [13], Mazhar [11], and Sarigol [18], where \(|C_\alpha|_k\) is the space of series summable by absolute Cesaro summability \(|C,\alpha|_k\) in Flett's notation.

Downloads

Download data is not yet available.

References

H. Bor and B. Thorpe, On some absolute summability methods, Analysis 7 (1987), 145–152.

L.S. Bosanquet, Note on convergence and summability factors I, J. London Math. Soc. 20 (1945), 39–48.

L.S. Bosanquet, Review on G. Sunouchi's paper "Notes on Fourier Analysis, 18, absolute summability of a series with constant terms”, Math. Rev. 11 (1950), 654.

L.S. Bosanquet and H.C. Chow, Some remarks on convergence and summability factors, J. London Math. Soc. 32 (1957), 73–82.

H.C. Chow, Note on convergence and summability factors, J. London Math. Soc. 29 (1954), 459–476.

G. Das, A Tauberian theorem for absolute summability, Proc. Cambridge Philos. 67 (1970), 321–326.

T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 7 (1957), 113–141.

E. Kogbetliantz, Sur lesseries absolument sommables par la methods des moyannes arithmetiques, Bull. des Sci. Math. 49 (1925), 234–256.

B. Kuttner, Some remarks on summability factors, Indian J. Pure Appl. Math. 16 (1985), 1017–1027.

I.J. Maddox, Elements of Functinal Analysis, Cambridge University Press, London, New York (1970).

S.M. Mazhar, On the absolute summability factors of infinite series, Tohoku Math. J. 23 (1971), 433–451.

L. McFadden, Absolute Norlund summability, Duke Math. J. 9 (1942), 168-207.

M.R. Mehdi, Summability factors for generalized absolute summability I, Proc. London Math. Soc. 10 (1960), 180–199.

R.N. Mohapatra and G. Das, Summability factors of lower-semi matrix transformations, Monatsh. Math. 79 (1975), 307–3015.

C. Orhan and M.A. Sarıgöl, On absolute weighted mean summability, Rocky Mount. J. Math. 23 (1993), 1091–1097.

A. Peyerimhoff, Summierbarkeitsfaktoren für absolut Cesí ro-summierbare Reiben, Math. Z. 59 (1954), 417–424.

M.A. Sarıgöl, An inequality on matrix operators and its applications, Journal of Classical Analysis 2 (2013), 145–150.

M.A. Sarıgöl, Matrix operators on Ak, Math. Comp. Model. 55 (2012), 1763–1769.

M.A. Sarıgöl, Matrix transformations on fields of absolute weighted mean summability, Studia Sci. Math. Hungar. 48 (2011), 331–341.

M.A. Sarıgöl and H. Bor, Characterization of absolute summability factors, J. Math. Anal. Appl. 195 (1995), 537–545.

M.A. Sarıgöl, A note on summability, Studia Sci. Math. Hungar. 28 (1993), 395–400.

M.A. Sarıgöl, On absolute weighted mean summability methods, Proc. Amer. Math. Soc. 115 (1992), 157–160.

M.A. Sarıgöl, Necessary and sufficient conditions for the equivalence of the summability methods

(|bar N,p_n |_k) and (|C,alpha |_k), Indian J. Pure Appl. Math. 22 (1991), 483–489.

M. Stieglitz and H. Tietz, Matrixtransformationen von Folgenraumen Eine Ergebnis"uberischt, Math Z. 154 (1977), 1–16.

W.T. Sulaiman, On summability factors of infinite series, Proc. Amer. Math. Soc. 115 (1992), 313–317.

G. Sunouchi, Notes on Fourier Analysis, 18, absolute summability of a series with constant terms, Tohoku Math. J. 1 (1949), 57–65.

Downloads

Published

15-06-2016
CITATION

How to Cite

Sarıgöl, M. A. (2016). Spaces of Series Summable by Absolute Cesí ro and Matrix Operators. Communications in Mathematics and Applications, 7(1), 11–22. https://doi.org/10.26713/cma.v7i1.360

Issue

Vol. 7 No. 1 (2016)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.