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

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}.

Mathematical Subject Classification
Primary: 40.31
Received: 31 January 1967
Published: 1 January 1968
David Fleming Dawson