Tauberian theorems for the product of weighted and CesÃ ro summability methods for double sequences

GÃ¶kÅŸen FÄ±ndÄ±k; Ä°brahim Çanak

doi:10.5269/bspm.v38i7.44135

Autores

GÃ¶kÅŸen FÄ±ndÄ±k Ege University http://orcid.org/0000-0002-2356-3987
Ä°brahim Çanak Ege University http://orcid.org/0000-0002-1754-1685

DOI:

https://doi.org/10.5269/bspm.v38i7.44135

Resumo

In this paper, we obtain necessary and sufficient conditions, under which convergence of a double sequence in Pringsheim's sense follows from its weighted-Cesaro summability. These Tauberian conditions are one-sided or two-sided if it is a sequence of real or complex numbers, respectively.

Biografia do Autor

Ä°brahim Çanak, Ege University

Ibrahim Canak is Professor of Mathematics at Ege University, Izmir, Turkey. He was born in Tokat, Turkey on October 1, 1967. He received his primary and secondary education in Istanbul, Turkey. He received his B.S. degree in 1988 with the highest rank in Mathematics from University of Istanbul, Istanbul, Turkey. October 1988 through October 1992 he was a graduate student and teaching assistant. While he was a graduate teaching assistant at University of Istanbul in 1993, he won a scholarship from the Higher Education Council of Turkey to get Ph.D. degree in the USA. He received his doctoral degree in 1998 under the direction of Professor Dr. Caslav V. Stanojevic at the University of Missouri – Rolla, Missouri, USA. He joined Adnan Menderes University, Aydin, Turkey in the fall of 1993 as a research assistant. He is the author of over 60 research papers. His current area of research and interest is summability theory. He is a member of American Mathematical Society and Mathematical Association of Turkey. He has been on the faculty of Ege University since 2010. He has been a referee and/or reviewer for several journals, MathSciNet and Zentrallblatt-Math and is also on the editorial board of several mathematics journals.

Referências

Baron, S., Stadtmuller, U., Tauberian theorems for power series methods applied to double sequences, J. Math. Anal. Appl. 211, 574-589, (1997).
https://doi.org/10.1006/jmaa.1997.5473

Braha, N. L., Some weighted equi-statistical convergence and Korovkin type-theorem, Results Math. 70(3-4), 433-446, (2016).
https://doi.org/10.1007/s00025-016-0578-z

Chen, C., Hsu, J., Tauberian theorems for weighted means of double sequences, Anal. Math. 26(4), 243-262, (2000).

FÄ±ndÄ±k, G., Canak, I., Tauberian theorems for the product of weighted and Cesaro summability methods for double sequences, In Cakalli, H., Deger, O. (Eds.), Abstract book. Paper presented at the 2nd International Conference of Mathematical Sciences (ICMS 2018), Maltepe University, Istanbul, Turkey, July 31-August 6, 2018 (p. 89).

Knopp, K., Limitierungs-Umkehrsatze fur Doppelfolgen, Math. Z. 45, 573-589, (1939).
https://doi.org/10.1007/BF01580302

Loku, V., Braha, N. L., Tauberian theorems by weighted summability method, Armen. J. Math. 9(1), 35-42, (2017).

Moricz, F., Tauberian theorems for Ces'aro summable double sequences Stud. Math. 110(1), 83-96, (1994).
https://doi.org/10.4064/sm-110-1-83-96

Moricz, F., Rhoades, B. E., Necessary and sufficent Tauberian conditions for certain weighted mean methods of summability II., Acta Math. Hungar. 102(4), 279-285, (2004).
https://doi.org/10.1023/B:AMHU.0000024678.80514.94

Pringsheim A., Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53, 289-321, (1900).
https://doi.org/10.1007/BF01448977

Savas, R., Sezer, S. A., Tauberian theorems for sequences in 2-normed spaces, Results Math. 72(4), 1919-1931, (2017).
https://doi.org/10.1007/s00025-017-0747-8

Stadtmuller, U., Tauberian theorems for weighted means of double sequences, Anal. Math. 25(1), 57-68, (1999).
https://doi.org/10.1007/BF02908426