Equivalence of methods for the summation of double sequences

In 1900 Pringsheim presented the following notion of convergence for double sequences: a double sequence [x] converges to L provided that given epsilon > 0 there exists K is an element of N such that vertical bar x(k.l) - L vertical bar < epsilon whenever k, l > K. Using this definition Robison and Hamilton, in 1926 and 1936, respectively, presented the following definition for the regularity of four dimensional matrices. A four dimensional matrix A is regular if it maps every bounded convergent sequence into a convergent sequence with the same limit. These notions shall be used to present simple conditions to ensure that T-m,T-n = Sigma(m,n)(k,l=0.0) a(m,n,k,l)x(k,l) and sigma(m,n) = Sigma(infinity.infinity)(k,l=0.0) a(m,n,k,l)x(k,l) are included by convergence. (C) 2010 Elsevier Ltd. All rights reserved.