Analytic Continuation By Means Of The L(r,t)Summability Transform

Authors

Stanley D. Luke, University of Nebraska-Lincoln

Date of this Version

1973

Citation

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

Comments

Copyright 1973 by the author(s).

Abstract

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

(1) t_n == A_n(x) = ^∞Ʃ_m=o a_nm S_m, and say that the sequence x (and the corresponding series ^∞Ʃ_m=o (S_m - S_m-1), with S_-1 = 0 is summable A to the sum t if each of the series in (l) converges and lim_n t_n 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) ^∞Ʃ_m=o │ a_nm │≤ k (n= 0,1,2,…),

(3) lim_n→∞ a_nm = 0 (m= 0,1,…),

(4) lim_n→∞ ^∞Ʃ_m=o a_nm = 1

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