By using modulus functions we introduce a new concept of density for sets of natural numbers. Consequently, we obtain a generalization of the notion of statistical convergence which is studied and characterized. As an application, we prove that the ordinary convergence is equivalent to the module statistical convergence for every unbounded modulus function. (2010): Primary 40A35, Secondary 46A45.
Mathematics Subject Classification
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.
We study f −statistical convergence, which is a generalization of the classical statistical convergence. In terms of it, we give a characterization of completeness in a normed space. We also introduce ' f −statistical cluster points', which is a richer concept than the classic one. Namely, each (usual) limit point of a sequence is an f −statistical cluster point for some f .
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