Nebraska Academy of Sciences


Date of this Version



Published in Transactions of the Nebraska Academy of Sciences, Volume 2 (1973).


Copyright 1973 by the author(s).


1. Let A = (anm) and x = [Sm] (n, m = 0,1,2, ... ) be a matrix and a sequence of complex numbers, respectively. We write

(1) tn == An(x) = Ʃm=o anm Sm, and say that the sequence x (and the corresponding series Ʃm=o (Sm - Sm-1), with S-1 = 0 is summable A to the sum t if each of the series in (l) converges and limn tn exists and equals t. We say that the method A is regular provided it sums every convergent sequence to its limit. The method A is regular if and only if

(2) Ʃm=o │ anm │≤ k (n= 0,1,2,…),

(3) limn→∞ anm = 0 (m= 0,1,…),

(4) limn→∞ Ʃm=o anm = 1

where k is a constant independent of n. These are so called Silverman Toeplitz conditions.