On factor relations between weighted and Nörlund means

G. Canan Hazar Güleç, Mehmet Ali Sarıgöl


By $\left( X,Y\right) ,$ we denote the set of all sequences $\epsilon =\left( \epsilon _{n}\right) $ such that $\Sigma \epsilon _{n}a_{n}$ is summable $Y$ whenever $\Sigma a_{n}$ is summable $X,$ where $X$ and $Y$ are two summability methods. In this study, we get necessary and sufficient conditions for $\epsilon \in \left( \left\vert N,q_{n},u_{n}\right\vert _{k},\left\vert \bar{N},p_{n}\right\vert \right) $ and $\epsilon \in \left( \left\vert \bar{N},p_{n}\right\vert ,\left\vert N,q_{n},u_{n}\right\vert _{k}\right) $, $k\geq 1,$ using functional analytic tecniques, where $% \left\vert \bar{N},p_{n}\right\vert $ and $\left\vert N,q_{n},u_{n}\right\vert _{k}$ are absolute weighted and N\"{o}rlund summability methods, respectively, \cite{1}, \cite{5}. Thus, in the special case, some well known results are also deduced.


Sequence spaces; Absolute Nörlund summability; Absolute weighted mean summability, Summability Factors

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H. Bor, On two summability methods}, Math. Proc. Cambridge Philos. Soc., 97(1985), No.1, 147--149.

D. Borwein and F. P. Cass, Strong Norlund summability, Math. Zeitschr., 103(1968), 94--111.

L. S. Bosanquet and G. Das, Absolute summability factors for Norlund means, Proc. London Math. Soc. (3), 38(1979), 1--52.

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

G. C. Hazar and M.A Sarigol, On absolute Norlund spaces and matrix operators, Acta Math. Sin. (Engl. Ser.), (2018),34(5), 812--826.

I. Kayashima, On relations between Norlund and Riesz means, Pac. J. Math., 49(2) (1973), 391--396.

I. J. Maddox, Elements of Functional 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.

M. F. Mears, Absolute Regularity and the Norlund Mean, Annals of Math., 38(3), (1937), 594--601.

M. R. Mehdi, Summability factors for generalized absolute summability I, Proc. London Math. Soc. (3), 10(1960), 180--199.

R. N. Mohapatra, On absolute Riesz summability factors, J. Indian Math. Soc., 32(1968), 113--129.

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

M. A. Sarigol, Extension of Mazhar's theorem on summability factors, Kuwait J. Sci., 42(3)(2015), 28--35.

M. A. Sarigol, Matrix operators on }$A_{k}$, Math. Comp. Model., 55(2012), 1763--1769.

M. A. Sarigol, Matrix transformations on fields of absolute weighted mean summability, Studia Sci.Math. Hungar., 48(3)(2011), 331--341.

M. A. Sarigol, On the local properties of factored Fourier series, Appl. Math. Comp., 216(2010), 3386--3390.

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

M. A. Sarigol, On two absolute Riesz summability Factors of infinite series, Proc. Amer. Math. Soc., 118(1993),485--488.

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

M. A. Sarigol, On absolute weighted mean summability methods, Proc. Amer. Math. Soc., 115(1)(1992), 157--160.

M. A. Sarigol, Necessary and sufficient conditions for the equivalence of the summability methods }$leftvert overline{N}%,p_{n}rightvert _{k} $textit{and }$leftvert C,1rightvert _{k}$,Indian J. Pure Appl. Math., 22(6) (1991), 483--489.

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

DOI: http://dx.doi.org/10.5556/j.tkjm.50.2019.2704

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