Fundamental theorems of summability theory for double sequences via stretching by blocks

Abstract

This paper extends fundamental theorems of summability theory to double sequences using stretching by blocks technique. We prove Steinhaus- and Buck-type theorems for double sequences, characterizing sequences that resist summability by RH-regular matrices and providing conditions for P-convergence. The preservation of P-divergence under block stretching is demonstrated, and a characterization of P-limit points via this method is established. We also show the impossibility of summing all block stretchings with a single RH-regular matrix. These results provide a more comprehensive framework for analyzing double sequences under matrix transformations.