A New Variation on Absolute Summability

Year 2021, Volume: 3 Issue: 1, 1 - 9, 15.06.2021

Şebnem Yıldız

Abstract

In [4], Bor has proved a main theorem dealing with absolute weighted arithmetic mean summability factors of infinite series by using a positive non-decreasing sequence. In this paper, we have extended this result to absolute matrix summability method by using an almost increasing sequence in place of a positive non-decreasing sequence. Also, some new and known results are also obtained.

Keywords

Minkowski, absolute matrix summability, summability factors, infinite series

References

• [1] N. K. Bari, S. B. Steckin, Best approximation and differential of two conjugate functions, Trudy. Moskov. Mat. Obsc. 5 (1956) 483-522 (in Russian).
• [2] H. Bor, On two summability methods, Math. Proc. Camb. Philos. Soc. 97 (1985) 147-149.
• [3] H. Bor, A note on $\left|\bar{N},p_n\right| _k$ summability factors of innite series, Indian J. Pure Appl. Math. 18 (1987) 330-336.
• [4] H. Bor, Factors for absolute weighted arithmetic mean summability of innite series, Int. J. Anal. Appl. 14 (2017) 175-179.
• [5] H. Bor, An application of quasi-monotone sequences to innite series and Fourier series, Anal. Math. Phys. 8 (2018) 7783.
• [6] H. Bor, On absolute summability of factored innite series and trigonometric Fourier series, Results Math. 73 (2018) 116.
• [7] H. Bor, On absolute Riesz summability factors of innite series and their application to Fourier series, Georgian Math. J. 26 (2019) 361366.
• [8] H. Bor, Certain new factor theorems for innite series and trigonometric Fourier series, Quaest. Math. 43 (2020) 441448
• [9] E. Cesaro, Sur la multiplication des series, Bull. Sci. Math. 14 (1890) 114-120.
• [10] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. Lond. Math. Soc. 7 (1957) 113-141.
• [11] G. H. Hardy, Divergent Series, Clarendon Press. Oxford, (1949).
• [12] K. N. Mishra, On the absolute Norlund summability factors of innite series, Indian J. Pure Appl. Math. 14 (1983) 40-43.
• [13] K. N. Mishra and R. S. L. Srivastava, On the absolute Cesaro summability factors of innite series, Portugal Math. 42 (1983/84) 53-61.
• [14] K. N. Mishra and R. S. L. Srivastava, On j N ; pnj summability factors of innite series, Indian J. Pure Appl. Math. 15 (1984) 651-656.
• [15] H. S.  Ozarslan, T. Kandefer, On the relative strength of two absolute summability methods, J. Comput. Anal. Appl. 11 (2009) 576{583.
• [16] B. E. Rhoades, Inclusion theorems for absolute matrix summability methods, J. Math. Anal. Appl. 238 (1999) 82-90.
• [17] W. T. Sulaiman, Inclusion theorems for absolute matrix summability methods of an innite series, IV. Indian J. Pure Appl. Math. 11 (2003) 1547-1557.
• [18] S. Yıldız, On application of matrix summability to Fourier series, Math. Methods Appl. Sci. (2018) 664{670.
• [19] S. Yıldız, On the absolute matrix summability factors of Fourier series, Math. Notes 103 (2018) 297-303.
• [20] S. Yıldız, A matrix application on absolute weighted arithmetic mean summability factors of innite series, Tibilisi Math. J. 11 (2018) 59-65.
• [21] S. Yıldız, On the generalizations of some factors theorems for innite series and Fourier series, Filomat 33 (2019) 4343-4351.
• [22] S. Yıldız, Matrix application of power increasing sequences to innite series and Fourier series, Ukranian Math. J. 72 (2020) 730-740.
• [23] S. Yıldız, A variation on absolute weighted mean summability factors of Fourier series and its conjugate series, Bol. Soc. Parana. Mat. 38 (2020) 105-113.
• [24] S. Yıldız, A recent extension of the weighted mean summability of innite series, J. Appl. Math. Inform. 39 (2021) 117-124.