Statistical Convergence with Respect to Power Series Methods and Applications to Approximation Theory

On statistical and strong convergence with respect to a modulus function and a power series method

“…Connor also established the relationship between statistical convergence and strong Cesàro convergence in his earlier paper [6]: A real sequence is strongly Cesàro convergent if and only if it is statistical convergent and bounded. Khan and Orhan [15] improved this result by replacing the boundedness condition with a strictly weaker condition called uniform integrability.Ünver and Orhan [27] has recently introduced the notions of statistical convergence, strong convergence and uniform integrability of a sequence defined by a power series method and established the similar relationship in the power series method setting.…”

“…, where E ε = {k ∈ N 0 : |x k − L| ≥ ε} and hereafter this set will always be denoted by E ε . The ideas of strong convergence, density and statistical convergence with respect to general power series methods, namely, in the case p (t) = ∞ k=0 p k t k has radius of convergence R ∈ (0, ∞], are introduced byÜnver and Orhan [27] and they called them as P p -strong convergence, P p -density and P p -statistical convergence, respectively. Note that if 0 < R < ∞ then it is sufficient to consider the case R = 1, since we may replace p k with p k R k (see [5], Remark 3.6.3).…”

“…Then p (t) = ∞ k=0 t 2k = 1/ 1 − t 2 for 0 < t < 1. Now if E = {2k + 1 : k ∈ N 0 }, then δ J p (E) = 1/2 but δ (E) = 0 (see [27]). Also note that in case p k = 1 for all k, J p -density is called Abel density introduced byÜnver in [26].…”

“…Any bounded sequence is J p -uniformly integrable but not conversely (see [27], Example 2). This notion and the following result will play a key role to obtain more general results in the second and third sections.…”

Statistical relative uniform convergence of a double sequence of functions at a point and applications to approximation theory

“…The statistical convergence in approximation theory was first used by Gadjiev and Orhan [3] to prove the Korovkin-type approximation theorem [4]. The studies and related results can be found in [5][6][7][8].…”