Abstract
We first review the characterization of the matrix maps between classical sequence spaces. Such a theory has been developed for a long time. We extend this theory to the concept of A-statistical convergence. We not only give the characterization of matrix maps of A-statistically convergent sequence spaces, but also obtain the corresponding limit formula. In addition, we evaluate the operator semi-norm of such matrix maps. Next, the converse problem is concerned. More precisely, we find conditions, called Tauberian conditions, under which the ordinary convergence of double sequences follows from the statistical convergence of its weighted means. To this aim, a new type of slow oscillation and slow decrease conditions are introduced. As a result, we conclude that Hardy’s two-sided condition, Landau’s condition and Schmidt’s condition are the desired Tauberian conditions. Finally, a parallel theory to the one given for double sequences is established for measurable functions of several variables. Our results in this thesis generalize many well-known recent works.