Factor relations between some summability methods

Year 2021, Volume: 3 Issue: 2, 70 - 74, 30.12.2021

Mehmet Ali Sarıgöl

https://doi.org/10.47086/pims.1016103

Abstract

In the present paper, using the result of Bennett [1] on characterization of factorable matrices, we give necessary and sufficient conditions in order that Σλ_{n}x_{n} is summable |R,p_{n}|_{s} whenever Σμ_{n}x_{n} is summable |C,0|_{k}, and Σλ_{n}x_{n} is summable |C,0|_{s} whenever Σμ_{n}x_{n} is summable |R,p_{n}|_{r},. where 1

Keywords

Infinite series, Riesz and Cesàro mean, summability factors, matrix transformations, bounded linear operator

References

• 1 : G. Bennett, Some elemantery inequalities, Quart. J. Math. Oxford 38 (1987) 401-425.
• 2 : H. Bor, A new result on the high indices theorem, Analysis 29 (2009) 403-405.
• 3 : H. Bor, On two summability methods, Math. Proc. Camb. Philos. Soc. 97 (1985) 147-149.
• 4 : H. Bor, A note on |N,p_{n}|_{k} summability factors of infinite series, Indian J. Pure Appl. Math. 18 (1987) 330-336.
• 5 : H. Bor, Factors for absolute weighted arithmetic mean summability of infinite series, Int. J. Anal. Appl. 14 (2017) 175-179.
• 6 : L.S. Bosanquet, G. Das, Absolute summability factors for Nörlund means, Proc. London Math. Soc. 38 (1979) 1-52.
• 7 : T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 7 (1957) 113-141.
• 8 : F. Gökce, M.A. Sarıgöl, Generalization of the Absolute Cesáro Space and Some Matrix Transformations, Numer. Funct. Anal. Optim. 40 (2019) 1039-1052.
• 9 : F. Gökçe, M.A. Sarıgöl, Extension of Maddox's space l(µ) with Nörlund means, Asian Eur. J Math. 12 (2019) 1-12.
• 10 : G.C. Hazar, M.A. Sarıgöl, On factor relations between weighted and Nörlund means, Tamkang J. Math. 50 (2019) 61-69.
• 11 : S.M. Mazhar, On the absolute summability factors of infinite series, Tohoku Math. J. 23 (1971) 433-451.
• 12 : M.R. Mehdi, Summability factors for generalized absolute summability I, Proc. London Math. Soc. 10 (1960) 180-199.
• 13 : R.N. Mohapatra, On absolute Riesz summability factors, J. Indian Math. Soc. 32 (1968) 113-129.
• 14 : C. Orhan, M.A. Sarıgöl, On absolute weighted mean summability, Rocky Mount. J. Math. 23 (1993) 1091-1097.
• 15 : M.A. Sarıgöl, Extension of Mazhar's theorem on summability factors, Kuwait J. Sci. 42 (2015) 28-35
• 16 : M.A. Sarıgöl, Characterization of summability methods with high indices, Slovaca Math. 63 (2013) 1053-1058.
• 17 : M.A. Sarıgöl, On local property of summability of factored Fourier series, J. Math. Anal. Appl. 188 (1994) 118-127.
• 18 : M.A. Sarıgöl, A note on summability, Studia Sci. Math. Hungarica 28 (1993) 395-400.
• 19 : M.A. Sarıgöl, On two absolute Riesz summability factors of infinite series, Proc. Amer. Math. Soc. 118 (1993) 485-488.
• 20 : M.A. Sarıgöl, On absolute normal matrix summability methods, Glasnik Math. 28 (1993) 53-60.
• 21 : M.A. Sarıgöl, On absolute weighted mean summability methods, Proc. Amer. Math. Soc. 115 (1992) 157-160.
• 22 : M.A. Sarıgöl, and Bor, H., Characterization of absolute summability factors, J. Math. Anal. Appl. 195 (1995) 537-545.
• 23 : M.A. Sarıgöl, On |T|_{k} summability and absolute summability, Slovaca Math. 42 (1992) 325-329.
• 24 : W.T. Sulaiman, On some absolute summability factors of Infinite Series, Gen. Math. Notes 2 (2011) 7-11.