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 ϵ > 0 there exists K ∈ N such that | x k , l − L | < ϵ 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 , n = ∑ k , l = 0 , 0 m , n a m , n , k , l x k , l and σ m , n = ∑ k , l = 0 , 0 ∞ , ∞ a m , n , k , l x k , l are included by convergence.