IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 3 Ser. II (May – June 2020), PP 19-24 www.iosrjournals.org DOI: 10.9790/5728-1603021924 www.iosrjournals.org 19 | Page On [,] −Regular Four-Dimensional Matrices for [, ]-Almost Convergence of Double Sequences Zakawat U. Siddiqui,MuhammadA. Chamalwa Department of Mathematical Sciences, Faculty of Science, University of Maiduguri, Nigeria Abstract: Background: A double sequence A = ( ) is said to belong to the class (X, Y), where X and Y are two sequence spaces, if any sequence x = in X is transformed to a sequence ={ } in Y by the matrix transformation = ∞ = ∞ =0 such that the sequence exists and converges in the Pringsheim sense.A t sequence ={ } of real is said to be [, ]-almost convergent (briefly, ℱ [ , ] − ) to some number if ℱ [ , ] , where ℱ [ , ] ={ ={ }: − →∞ , ,, = , , ; = ℱ [, ] −}, , , ,, = 1 +, + ∈ ∈ . Materials and methods: For double sequences the Cauchy’s criterion of convergence has been modified by Pringsheim. Similarly, the necessary and sufficient conditions for the regularity of an infinite four dimensional matrix is a given by Robison. These concepts has been utilized to generalize the concept of [, ]-almost convergence double sequencesthrough de la Vallèe-Poussin mean and characterized some four-dimensional infinites matrices. We collect the relevant publications in this field and apply the same technique as applied in these papers to generalize the known results. Results: In this paper we characterizeinfinite four-dimensional matrices which transform the sequence belonging to the space of bounded double sequence into the space of generalized almost convergence double sequence (i.e. = ∈ (∁ , ℱ , )). We introduced the concept of [, ]-almost Cauchy double sequences. It has also been proved that the space generalized almost convergence double sequence is regular (i.e. ℱ [ , ] − ). Conclusion: The condition , ,, , , 1 +, + , , ∈ ∈ < ∞ has been found to be necessary and sufficient for a four-dimensional matrix = ∈ (∁ , ℱ , ) to be[, ]-almost convergent. Again the necessary and sufficient conditions have been established for amatrix =( ) to beℱ [ , ] − . Keywords: [, ]- almost convergence, ℱ [ , ] − matrix, [, ]-Cauchy double sequences, [, ]-almost coercive matrix. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 06-05-2020 Date of Acceptance: 19-05-2020 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction The definition of almost convergence of the sequences of real numbers ={ } was given by Lorentz (1948) 1 as follows: A sequence ={ } is said to be almost convergent to if for every >0, there exists N ∈ℕ, such that 1 + −1 =0 − < for all i > N. We write − = . Moricz and Rhoades (1988) 2 extended the concept of almost convergenceof a sequence ={ } to double sequences of real numbers ={ }. The sequence ={ } almost converges to , if for all >0, there exists∈ℕ, such that | 1 + , + − −1 =0 −1 =0 |< , for all p, q > N and for all (m, n)∈ℕ × ℕ. (1.1) Moricz and Rhoades also characterized some matrix classes involving this concept. As in the case of single sequences, every almost convergent double sequence is bounded. But a convergent double sequence need not be bounded. Thus, a convergent double sequence need not be almost convergent. However, every bounded convergent double sequence is almost convergent. The idea of almost convergence is narrowly connected with the Banach limits; that is, a sequence ∈ℓ ∞ is almost convergent to if all of its Banach limits are equal. As an application of almost convergence,