Necessary and sufficient Tauberian conditions for logarithmic summable sequences in two-normed spaces

Zerrin Ã–nder; Sefa AnÄ±l Sezer; Ä°brahim Çanak

doi:10.5269/bspm.64881

Autores/as

Zerrin Ã–nder
Sefa AnÄ±l Sezer
Ä°brahim Çanak Ege University http://orcid.org/0000-0002-1754-1685

DOI:

https://doi.org/10.5269/bspm.64881

Resumen

Our aim in this paper is to make a novel interpretation of the relation between the logarithmic summability method and convergence under the coverage of $2$-normed spaces. In line with this aim, we introduce a necessary and sufficient Tauberian condition of M\'{o}ricz-type for logarithmic summable sequences in these kinds of spaces. Following this, we investigate whether conditions designed as $O$-type such as the slow oscillation and Hardy-type conditions with respect to summability $(\ell,1)$ due to the non-existence of relation of ``order'' in $2$-normed spaces are the conditions needed for logarithmic summable sequences to be convergent.

Biografía del autor/a

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.