A A NEW THEOREM GENERALIZATION ON ABSOLUTE MATRIX SUMMABILITY OF FOURIER SERIES
Abstract
We generalize a main theorem dealing with weighted mean summability of Fourier series to the $|A,\varphi_n;\delta|_{k}$ summability factors of Fourier series. Also, some new and known results are obtained.
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References
\begin{thebibliography}{99}
%1
\bibitem{Bor}
H. Bor, On two summability methods, Math. Proc. Camb. Philos. Soc., 97 (1985), 147-149.
%2
\bibitem{Bor2}
H. Bor, On absolute summability factors, Proc. Am. Math. Soc., 118 (1993), 71-75.
\bibitem{Bor3}
H. Bor, On local property of $|\bar{N},p_n;\delta|_k$ summability of factored Fourier Series, J. Math. Anal. Appl., 179 (1993), 646-649.
\bibitem{Bor4}
H. Bor, Some new results on infinite series and Fourier series, Positivity, 19 (2015), 467-473.
\bibitem{Bor5}
H. Bor, On absolute weighted mean summability of infinite series and Fourier series, Filomat, 30(10) (2016), 2803–2807.
\bibitem{Yu}
H. Bor, D. S. Yu and P. Zhou, On local property of absolute summability of factored Fourier series, Filomat, 28(8) (2014), 1675-1686.
%5
\bibitem{Ce}
E. Ces{\`a}ro, Sur la multiplication des s\'{e}ries, Bull. Sci.
Math., 14 (1890), 114-120.
%6
\bibitem{Ch} K. K. Chen, Functions of bounded variation and the Ces\`aro means of Fourier series, Acad. Sinica Sci. Record, 1 (1945), 283-289.
\bibitem{Fl} 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
\bibitem{Ha} G. H. Hardy, Divergent Series, Oxford Univ. Press., Oxford, (1949).
\bibitem{Ko} E. Kogbetliantz, Sur l\'{e}s series absolument par la m\'{e}thode des moyannes arithm\'{e}tiques, Bull. Sci. Math., 49 (1925) 234-256.
\bibitem{ozarslan} H. S. Özarslan, A new study on generalised absolute matrix summability methods, Maejo Int. J. Sci. Technol. 12(3) (2018), 199–205.
\bibitem{o} H. S. Özarslan, H. N. Öğdük, Generalizations of two theorems on absolute summability methods, Aust. J. Math. Anal. Appl., 1 (2004), 7 pp.
\bibitem{o1}
H. S. Özarslan, Ş. Yıldız, A new study on the absolute summability factors of Fourier series, J. Math. Anal., 7 (2016), 31-36.
\bibitem{Sa}
M. A. Sarıgöl, On $|A|_k$ summability of factorable Fourier series, Trans. A. Razmadze Math. Inst., 178 (2024), 137-143.
\bibitem{zakawat} Z. U. Siddiqui, B. E. Basi, On absolute matrix summability factors using $\beta$-quasi power increasing sequences, Int. J. Phys. Appl. Sci., 2 (2015), 1-9.
\bibitem{Su} W. T. Sulaiman, Inclusion theorems for absolute matrix summability methods of an infinite series. IV, Indian J. Pure Appl. Math., 34 (11) (2003), 1547-1557.
\bibitem{Mil} N. Tanovic-Miller, On strong summability, Glas. Mat., III. Ser. 14 (34) (1979), 87-97.
\bibitem{yı}
Ş. Yıldız, A new theorem on absolute matrix summability of Fourier series, Publ. Inst. Math., 102 (116) (2017), 107-113.
\bibitem{yı1} Ş. Yıldız, On absolute matrix summability factors of infinite series and Fourier series, Bull. Sci. Math., 170 (2021), 10 pp.
\bibitem{yı2} Ş. Yıldız, An Application on an absolute matrix summability method, Trans. A. Razmadze Math. Inst., 176 (2022), 115-121.
\bibitem{cag} Ş. Yıldız Yar, Ç. Bülür, A Note on Absolute Matrix Summability of Fourier Series, (submitted to Maltepe J.M.)
\end{thebibliography}
%1
\bibitem{Bor}
H. Bor, On two summability methods, Math. Proc. Camb. Philos. Soc., 97 (1985), 147-149.
%2
\bibitem{Bor2}
H. Bor, On absolute summability factors, Proc. Am. Math. Soc., 118 (1993), 71-75.
\bibitem{Bor3}
H. Bor, On local property of $|\bar{N},p_n;\delta|_k$ summability of factored Fourier Series, J. Math. Anal. Appl., 179 (1993), 646-649.
\bibitem{Bor4}
H. Bor, Some new results on infinite series and Fourier series, Positivity, 19 (2015), 467-473.
\bibitem{Bor5}
H. Bor, On absolute weighted mean summability of infinite series and Fourier series, Filomat, 30(10) (2016), 2803–2807.
\bibitem{Yu}
H. Bor, D. S. Yu and P. Zhou, On local property of absolute summability of factored Fourier series, Filomat, 28(8) (2014), 1675-1686.
%5
\bibitem{Ce}
E. Ces{\`a}ro, Sur la multiplication des s\'{e}ries, Bull. Sci.
Math., 14 (1890), 114-120.
%6
\bibitem{Ch} K. K. Chen, Functions of bounded variation and the Ces\`aro means of Fourier series, Acad. Sinica Sci. Record, 1 (1945), 283-289.
\bibitem{Fl} 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
\bibitem{Ha} G. H. Hardy, Divergent Series, Oxford Univ. Press., Oxford, (1949).
\bibitem{Ko} E. Kogbetliantz, Sur l\'{e}s series absolument par la m\'{e}thode des moyannes arithm\'{e}tiques, Bull. Sci. Math., 49 (1925) 234-256.
\bibitem{ozarslan} H. S. Özarslan, A new study on generalised absolute matrix summability methods, Maejo Int. J. Sci. Technol. 12(3) (2018), 199–205.
\bibitem{o} H. S. Özarslan, H. N. Öğdük, Generalizations of two theorems on absolute summability methods, Aust. J. Math. Anal. Appl., 1 (2004), 7 pp.
\bibitem{o1}
H. S. Özarslan, Ş. Yıldız, A new study on the absolute summability factors of Fourier series, J. Math. Anal., 7 (2016), 31-36.
\bibitem{Sa}
M. A. Sarıgöl, On $|A|_k$ summability of factorable Fourier series, Trans. A. Razmadze Math. Inst., 178 (2024), 137-143.
\bibitem{zakawat} Z. U. Siddiqui, B. E. Basi, On absolute matrix summability factors using $\beta$-quasi power increasing sequences, Int. J. Phys. Appl. Sci., 2 (2015), 1-9.
\bibitem{Su} W. T. Sulaiman, Inclusion theorems for absolute matrix summability methods of an infinite series. IV, Indian J. Pure Appl. Math., 34 (11) (2003), 1547-1557.
\bibitem{Mil} N. Tanovic-Miller, On strong summability, Glas. Mat., III. Ser. 14 (34) (1979), 87-97.
\bibitem{yı}
Ş. Yıldız, A new theorem on absolute matrix summability of Fourier series, Publ. Inst. Math., 102 (116) (2017), 107-113.
\bibitem{yı1} Ş. Yıldız, On absolute matrix summability factors of infinite series and Fourier series, Bull. Sci. Math., 170 (2021), 10 pp.
\bibitem{yı2} Ş. Yıldız, An Application on an absolute matrix summability method, Trans. A. Razmadze Math. Inst., 176 (2022), 115-121.
\bibitem{cag} Ş. Yıldız Yar, Ç. Bülür, A Note on Absolute Matrix Summability of Fourier Series, (submitted to Maltepe J.M.)
\end{thebibliography}
Published
2025-12-20
Section
Mathematics and Computing - Innovations and Applications (ICMSC-2025)
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