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Báo cáo hóa học: " Research Article ´ On the Cesaro Summability of Double Series"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article ´ On the Cesaro Summability of Double Series | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 257318 4 pages doi 10.1155 2008 257318 Research Article On the Cesaro Summability of Double Series E. Sava 1 H. Sevli 2 and B. E. Rhoades3 1 Department of Mathematics Istanbul Commerce University 34672 Uskudar Istanbul Turkey 2 Department of Mathematics Faculty of Arts Sciences Yuzuncu Yil University 65080 Van Turkey 3 Department of Mathematics Indiana University Bloomington IN 47405 USA Correspondence should be addressed to E. Savas ekremsavas@yahoo.com Received 18 July 2007 Accepted 19 August 2007 Recommended by Martin J. Bohner In a recent paper by Savas and S evli 2007 it was shown that each Cesaro matrix of order a for a -1 is absolutely kth power conservative for k 1. In this paper we extend this result to double Cesaro matrices. Copyright 2008 E. Savas et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. The concept of absolute summability of order k 1 was defined by Flett 1 as follows. Let y ak be a series with partial sums sn A an infinite matrix. Then y ak is said to be absolutely summable A of order k 1 if Ểnk-1 Tn- - Tn k O n 1 1 where O Tn ankSk. 2 k 0 Denote by Ak the sequence space defined by Ak CO sn n an O an sn - Sn-1 n 1 3 for k 1. A matrix T is said to be a bounded linear operator on Ak written T B Ak if T Ak Ak. In 1970 Das 2 defined such a matrix to be absolutely kth power conservative 2 Journal of Inequalities and Applications for k 1. In that paper he proved that every conservative Hausdorff matrix H G B Ak for k 1. In a recent paper 3 the first two authors proved every Cesaro matrix of order a for a -1 C a G Ak for k 1. Since the Cesaro matrices of order a for -1 a 0 are not conservative their result shows that being conservative is not a necessary condition for being absolutely kth .