Publication: On statistical convergence and strong Cesaro convergence by moduli
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2019-11-14
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Leon-Saavedra, Fernando
Listan-Garcia, M. del Carmen
Perez Fernandez, Francisco Javier
Romero de la Rosa, Maria Pilar
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Springeropen
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Abstract
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 Cesaro convergence defined by a modulus function f. Namely, for every modulus function f, we will prove that a f-strongly Cesaro 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.
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Statistical convergence, Strong Cesaro convergence, Modulus function, Uniformly bounded sequence, Approximation theorems, Korovkin, Summability, Sequences, Spaces