Korovkin type approximation theorems via lacunary equistatistical convergence

Abstract

H. Aktuglu and H. Gezer [Central European J. Math. 7 (2009), 558-567] introduced the concepts of lacunary equistatistical convergence, lacunary statistical pointwise convergence and lacunary statistical uniform convergence for sequences of functions. In this paper, we apply the notion of lacunary equistatistical convergence to prove a Korovkin type approximation theorem by using test functions 1, x 1−x , ( x 1−x ) 2 .

On statistical convergence and strong Cesàro convergence by moduli

“…Among other results, Khan and Orhan show that a sequence is strongly Cesàro convergent if and only if it is statistically convergent and uniformly integrable. In this circle of ideas, a significant number of deep and beautiful results have been obtained by Connor, Fridy, Khan, Mursaleen, Orhan... and many others (see [2,9,13,26,27,[31][32][33]). Moreover, the convergence methods are an active research area with important applications (see the recent monograph by Mursaleen [24]).…”

Completeness of a normed space via strong p-Cesàro summability

“…Connor ([5]). Since then, in this circle of ideas, a significant number of deep and beautiful results have been obtained by Connor, Fridy, Mursaleen...and many others (see [1,4,8,11,12,[14][15][16]) There are also results that obtain characterizations of properties of Banach spaces through convergence types. For instance, Kolk [10] was one of the pionnering contributors.…”

Statistical Convergence via q-Calculus and a Korovkin’s Type Approximation Theorem

“…We prove a Korovkin type approximation theorem for q-statistical convergence analogous to that of given by Gadjiev and Orhan [30]. The Korovkin type approximation theorems have been proved by various authors through different summability methods; e.g., [38][39][40][41][42][43][44].…”