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 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.
Highlights
A sequence on a normed space (X, · ) is said to be strongly Cesàro convergent to L if1n lim n→∞ n xk – L = 0. k=1The strong Cesàro convergence for real numbers was introduced by Hardy–Littlewood [14] and Fekete [12] in connection with the convergence of Fourier series.A sequence is statistically convergent to L if for any ε > 0 the subset {k : |xk – L| < ε} has density 1 on the natural numbers
The notion of f -strong Cesàro convergence that we introduce is very handy to use, and it fits as a glove to the f -statistical convergence
For which modulus functions do we find that all uniformly integrable and f -statistically convergent sequences are f -strongly Cesàro convergent?
Summary
A sequence (xk) on a normed space (X, · ) is said to be strongly Cesàro convergent to L if. Khan and Orhan show that a sequence is strongly Cesàro convergent if and only if it is statistically convergent and uniformly integrable. For which modulus functions do we find that all uniformly integrable and f -statistically convergent sequences are f -strongly Cesàro convergent?. (s) k=0 is uniformly integrable in L1μ[0, 1] (here χA(·) denotes the characteristic function of A) This measure theoretic approach was used by Khan and Orhan in [15], providing an answer to a problem posed by Connor [7] in the A-statistical-convergence setting and to another open question posed by Miller ([21]). A sequence (xn) is said to be f -strongly Cesàro convergent to L if f( lim n k=1 xk – L ) = 0.
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