General methods of convergence and summability

This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of N and study the space of convergence associated with the filter. We notice that c(X) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then l infinity (X) is a space of convergence associated with any free ultrafilter of N; and that if X is not complete, then l infinity (X) is never the space of convergence associated with any free filter of N. Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that l infinity (X) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then c(X) is a space of convergence through a certain class of such operators; and that if X is not complete, then c(X) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set HB(lim):={T is an element of B(l infinity (X),X):T|c(X)=lim and parallel to T parallel to =1} and prove that HB(lim) is a face of BLX0 if X has the Bade property, where LX0:={T is an element of B(l infinity (X),X):c0(X)subset of ker(T)}. Finally, we study the multipliers associated with series for the above methods of convergence

On statistical convergence and strong Cesaro convergence by moduli for double sequences

A remarkable result on summability states that the statistical convergence and the strong Cesàro convergence are closely connected. Given a modulus function f, we will establish that a double sequence that is f -strong Cesàro convergent is always f -statistically convergent. The converse, in general, is false even for bounded sequences. However, we will characterize analytically the modulus functions f for which the converse of this result remains true. The results of this paper adapt to several variables the results obtained in (León-Saavedra et al. in J. Inequal. Appl. 12:298, 2019)

On statistical convergence and strong Cesàro convergence by moduli

In this paper we will establish a result by Connor, Khan and Orhan (Analysis 8:47–63, 1988; Publ. Math. (Debr.) 76:77–88, 2010) in the framework of the statistical convergence and the strong Cesàro convergence defined by a modulus function f . Namely, for every modulus function f , we will prove that a f -strongly Cesàro convergent sequence is always f -statistically convergent and uniformly integrable. The converse of this result is not true even for bounded sequences. We will characterize analytically the modulus functions f for which the converse is true. We will prove that these modulus functions are those for which the statistically convergent sequences are f -statistically convergent, that is, we show that Connor–Khan–Orhan’s result is sharp in this sense

Influence of theta-Metric Spaces on the Diameter of Rough Weighted I-2-Limit Set

In this paper we continue our investigation of the recent summability notion introduced in [Math. Slovaca 69 (4) (2019) 871-890] (where rough weighted statistical convergence for double sequences is discussed over norm linear spaces) and introduce the notion of rough weighted I-2-convergence over theta-metric spaces. Also we exercise the behavior of weighted I-2-cluster points set over theta-metric spaces. Based on the new notion we vividly discuss some important results and perceive how the existing results are vacillating

Correction to: On statistical convergence and strong Cesàro convergence by moduli

Abstract We correct a logic mistake in our paper “On statistical convergence and strong Cesàro convergence by moduli” (León-Saavedra et al. in J. Inequal. Appl. 23:298, 2019)