On Convergence with respect to an Ideal and a Family of Matrices

Abstract

P. Das et al. recently introduced and studied the notions of strong -summability with respect to an Orlicz function andstatistical convergence, where is a nonnegative regular matrix and is an ideal on the set of natural numbers. In this paper, we will generalise these notions by replacing with a family of matrices and with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' sup-limsup-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal has a countable base), continuing some of the author's previous work.