International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 1, Pages 155-164
On the consistency of limitation methods for summable sequences
Department of Mathematics, Berhampur University, Berhampur 760007, Orissa, India
Received 8 November 1978; Revised 12 March 1979
Copyright © 1981 Nand Kishore and U. K. Misra. 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.
Two limitation methods, and , are said to be consistent for a class of sequences, iff, every sequence belonging to is limitable both by
and and that the -limit equals the -limit. Any two regular limitation methods are consistent for the class- of convergent sequences. However, this is not true in general and in fact, corresponding to every bounded non-convergent sequence it is possible to determine two -matrices such that they limit the sequence to two different values. In this paper, we establish the necessary and sufficient conditions for the consistency of two limitation methods, for summable sequences.