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Báo cáo: Size of Convergence Domains for Generalized Hausdorff Prime Matrices

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 131240, 14 pages doi:10.1155/2011/131240 Research Article Size of Convergence Domains for Generalized Hausdorff Prime Matrices T. Selmanogullari,1 E. Savas,2 and B. E. Rhoades3 1 2 Department of Mathematics, Mimar Sinan Fine Arts University, Besiktas, 34349 Istanbul, Turkey Department of Mathematics, Istanbul Commerce University, Uskudar, 34672 Istanbul, Turkey 3 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA Correspondence should be addressed to T. Selmanogullari, tugcenmat@yahoo.com Received 8 December 2010; Accepted 2 March 2011 Academic Editor: Q. Lan Copyright q 2011 T. Selmanogullari et al. This is an open access article distributed. | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 131240 14 pages doi 10.1155 2011 131240 Research Article Size of Convergence Domains for Generalized Hausdorff Prime Matrices T. Selmanogullari 1 E. Sava s 2 and B. E. Rhoades3 1 Department of Mathematics Mimar Sinan Fine Arts University Besiktas 34349 Istanbul Turkey 2 Department of Mathematics Istanbul Commerce University Uskudar 34672 Istanbul Turkey 3 Department of Mathematics Indiana University Bloomington IN 47405-7106 USA Correspondence should be addressed to T. Selmanogullari tugcenmat@yahoo.com Received 8 December 2010 Accepted 2 March 2011 Academic Editor Q. Lan Copyright 2011 T. Selmanogullari 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. We show that there exit E-J generalized Hausdorff matrices and unbounded sequences x such that each matrix has convergence domain c x. 1. Introduction The convergence domain of an infinite matrix A ank n k 0 1 . will be denoted by A and is defined by A x xn An x e c where c denotes the space of convergence sequences An x k ankxk. The necessary and sufficient conditions of Silverman and Toeplitz for a matrix to be conservative are limnank ak exists for each k limn k 0 ank t exists and A supn k 0 ankI TO. A conservative matrix A is called multiplicative if each ak 0 and regular if in addition t 1. The E-J generalized Hausdorff matrices under consideration were defined independently by Endl 1 2 and Jakimovski 3 . Each matrix H is a lower triangular matrix with nonzero entries fi hnk n a n-k. A k n - kỵ 1.1 where a is real number n is a real or complex sequence and A is forward difference operator defined by A k pik - pk 1 An 1 k A An k . We will consider here only nonnegative a. For a 0 one obtains an ordinary Hausdorff matrix. 2 Journal of Inequalities and .