Ideal convergence generated by double summability methods

Abstract

Abstract The main result of this note is that if I is an ideal generated by a regular double summability matrix summability method T that is the product of two nonnegative regular matrix methods for single sequences, then I-statistical convergence and convergence in I-density are equivalent. In particular, the method T generates a density µT with the additive property (AP) and hence, the additive property for null sets (APO). The densities used to generate statistical convergence, lacunary statistical convergence, and general de la Vallée-Poussin statistical convergence are generated by these types of double summability methods. If a matrix T generates a density with the additive property then T-statistical convergence, convergence in T-density and strong T-summabilty are equivalent for bounded sequences. An example is given to show that not every regular double summability matrix generates a density with additve property for null sets.