#### Vol. 24, No. 1, 1968

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Matrix summability over certain classes of sequences ordered with respect to rate of convergence

### David Fleming Dawson

Vol. 24 (1968), No. 1, 51–56
##### Abstract

Let C0 denote the set of all complex null sequences, and let S0 denote the set of all sequences in C0 which have at most a finite number of zero terms. If a = {ap}∈ S0 and b = {bp}∈ S0, we say that a converges faster than b,a < b, provided limap∕bp = 0. We say that a and b converge at the same rate, a b, provided 0 < lim liminf |ap∕bp| and limsup|ap∕bp| < . If a S0, let [a] = {x S0 : x a}. Let E0 = {[x] : x S0}. If [a],[b] E0, then we say that [a] is less than [b],[a] < [b], provided a < b. We note that E0 is partially ordered with respect to . In this paper we study matrix summability over subsets of S0 and over elements of E0. Open intervals in S0 will be denoted by (a,b),(a,), and (,b), where (a,) = {iχj S0 : a < x} and (,b) = {x S0 : x < b}. Some of our results characterize, for matrices, maximal summability intervals in S0. Such intervals are of the form (,b), never of the form (,b] = {x S0 : either x < b or x b}.

Primary: 40.31
##### Milestones
Received: 31 January 1967
Published: 1 January 1968
##### Authors
 David Fleming Dawson 