The purpose of this paper is to generalize the concept of almost convergence for double sequence through the notion of de la Vallée-Poussin mean for double sequences. We also define and characterize the generalized regularly almost conservative and almost coercive four-dimensional matrices. Further, we characterize the infinite matrices which transform the sequence belonging to the space of absolutely convergent double series into the space of generalized almost convergence. 1. Introduction and Preliminaries Let be the Banach space of real bounded sequences with the usual norm . There exist continuous linear functionals on called Banach limits . It is well known that any Banach limit of lies between and . The idea of almost convergence of Lorentz  is narrowly connected with the limits of S. Banach; that is, a sequence is almost convergent to if all of its Banach limits are equal. As an application of almost convergence, Mohiuddine  obtained some approximation theorems for sequence of positive linear operator through this notion. For double sequence, the notion of almost convergence was first introduced by Móricz and Rhoades . The authors of  introduced the notion of Banach limit for double sequence and characterized the spaces of almost and strong almost convergence for double sequences through some sublinear functionals. For more details on these concepts, one can refer to [6–12]. We say that a double sequence of real or complex numbers is bounded if denoted by , the space of all bounded sequence . A double sequence of reals is called convergent to some number in Pringsheim’s sense (briefly, -convergent to )  if for every there exists such that whenever , where . If a double sequence in and is also -convergent to , then we say that it is boundedly -convergent to (briefly, -convergent to ). A double sequence is said to converge regularly to (briefly, -convergent to ) if is converges in Pringsheim’s sense, and the limits and exist. Note that in this case the limits and exist and are equal to the -limit of . Throughout this paper, by , , and , we denote the space of all -convergent, -convergent, and -convergent double sequences, respectively. Also, the linear space of all continuous linear functionals on is denoted by . Let be a four-dimensional infinite matrix of real numbers for all , and a space of double sequences. Let be a double sequences space, converging with respect to a convergence rule . Define Then, we say that a four-dimensional matrix maps the space into the space if and is denoted by . Móricz and Rhoades  extended the
F. Móricz and B. E. Rhoades, “Almost convergence of double sequences and strong regularity of summability matrices,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 2, pp. 283–294, 1988.