Lacunary equi-statistical convergence of positive linear operators

Abstract

In this paper, the concept of lacunary equi-statistical convergence is introduced and it is shown that lacunary equi-statistical convergence lies between lacunary statistical pointwise and lacunary statistical uniform convergence. Inclusion relations between equi-statistical and lacunary equi-statistical convergence are investigated and it is proved that, under some conditions, lacunary equi-statistical convergence and equi-statistical convergence are equivalent to each other. A Korovkin type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence by the help of modulus of continuity of positive linear operators are studied.

Korovkin type approximation theorems via lacunary equistatistical convergence

“…In [1], Aktuglu and Gezer proved the Korovkin theorem for lacunary eqistatistiacal convergence by using the test functions 1, x and x 2 ; while we use here the test functions 1,…”

“…[2], [15], [17] and [23]. The concept of equistatistical convergence was introduced by Balcerzak et al [3] and was subsequently applied for establishing approximation theorems in [1], [10], [11] and [12]. In [1], Aktuglu and Gezer [1] generalized the idea of statistical convergence to lacunary equistatistical convergence.…”

“…The concept of equistatistical convergence was introduced by Balcerzak et al [3] and was subsequently applied for establishing approximation theorems in [1], [10], [11] and [12]. In [1], Aktuglu and Gezer [1] generalized the idea of statistical convergence to lacunary equistatistical convergence. Recently, Y. Kaya and N. Gönül [13] established some analogues of the Korovkin approximation theorem via lacunary equistatistical convergence.…”

Lacunary statistical convergence of sequences of functions in intuitionistic fuzzy normed space

“…Also in [11], Mursaleen and Mohiuddine defined the concept of lacunary statistical convergence with respect to intuitionistic fuzzy normed space. Aktuglu and Gezer [5] defined lacunary statistical convergence of sequences of functions.…”

Weighted βγ-summability of fuzzy functions of order θ

“…If T βγ (n) satisfies(2) or(3), then the following conditionslim sup n→∞ T β([λγ])(n) T β([λγ])(n) − T βγ(n) < ∞ for every λ > 1(4)and lim sup n→∞ T β([λγ])(n) T βγ(n) − T β([λγ])(n)…”