Summability factors between the absolute Cesàro methods

Abstract

If Σε_{n}x_{n} is summable by the method Y whenever Σx_{n} is summable by the method X, then we say that the factor ε=(ε_{n}) is of type (X,Y) and denote by (X,Y). In this study we characterize the sets (|C,α|_{k},|C,-1|) , k>1 and (|C,-1|,|C,α|_{k}) , k≥1 for α>-1. Also, in the special case, we give some inclusion relations between methods, which completes some open problems in literature.

Keywords

• Sequence spaces,Absolute Cesàro summability,Summability Factors

References

1. [1] Bor, H., Some equivalence theorems on absolute summability methods, Acta Math. Hung.,149, (2016) 208-214.[2] Bor, H. and Thorpe, B., On some absolute summability methods, Analysis 7 (2) (1987),145-152.[3] Bor, H., On two summability methods, Math. Proc. Cambridge Philos Soc. 98, (1985),147-149.[4] Flett, T.M., On an extension of absolute summability and some theorems of Littlewoodand Paley, Proc. London Math. Soc. 7 (1957), 113-141.[5] Hardy, G. H., Divergent Series, Oxford, 1949.[6] Hazar, G. C. and Sar¬göl M. A., Compact and Matrix Operators on the Space jC;􀀀1jk,J. Comput. Anal. Appl., 25(6), (2018), 1014-1024.[7] Hazar, G. C. and Sar¬göl M. A., On factor relations between weighted and Nörlund means,Tamkang J. Math. (in press)[8] Kogbetliantz, E., Sur lesseries absolument sommables par la methods des moyannes arith-metiques, Bull. des Sci. Math. 49 (1925), 234-256.[9] Maddox, I.J., Elements of functinal analysis, Cambridge University Press, London, NewYork, (1970).[10] Mazhar, S.M., On the absolute summability factors of in…nite series, Tohoku Math. J.,23(1971), 433-451.[11] Mehdi, M.R., Summability factors for generalized absolute summability I, Proc. LondonMath. Soc.(3), 10 (1960), 180-199.[12] Mohapatra, R. N., On absolute Riesz summability factors, J. Indian Math. Soc. 32(1968),113-129.[13] Orhan, C. and Sar¬göl, On absolute weighted mean summability, Rocky Mount. J. Math.23 (1993), 1091-1097.[14] Sarıgöl, M.A., Spaces of series Summable by absolute Cesàro and Matrix Operators ,Comm. Math. Appl. 7(1), (2016), 11-22.[15] Sarıgöl, M.A., Extension of Mazhar’s theorem on summability factors, Kuwait J. Sci. 42(3) (2015), 28-35.[16] Sarıgöl, M.A., On the local properties of factored Fourier series, Appl. Math. Comp., 216(2010), 3386-3390.[17] Sarıgöl, M.A., and Bor, H., Characterization of absolute summability factors, J. Math.Anal. Appl. 195 (1995), 537-545.[18] Sarıgöl, M.A., On two absolute Riesz summability factors of in…nite series, Proc. Amer.Math. Soc. 118 (1993), 485-488.[19] Sarıgöl, M.A., A note on summability, Studia Sci. Math. Hungar. 28 (1993), 395-400.[20] Sulaiman, W.T., On summability factors of in…nite series, Proc. Amer. Math. Soc. 115(1992), 313-317.[21] Sulaiman, W.T., On some absolute summability factors of In…nite Series, Gen. Math.Notes, 2 (2) (2011), 7-13.[22] Thorpe, B., Matrix transformations of Cesàro summable Series, Acta Math. Hung., 48(3-4), (1986), 255-265.