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