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

Abstract

In this paper, we define and study q-statistical limit point, q-statistical cluster point, q-statistically Cauchy, q-strongly Cesàro and statistically C1q-summable sequences. We establish relationships of q-statistical convergence with q-statistically Cauchy, q-strongly Cesàro and statistically C1q-summable sequences. Further, we apply q-statistical convergence to prove a Korovkin type approximation theorem.

A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter α

“…Mohiuddine et al [6], Acu et al [7], İçöz and Çekim [8,9], and Kajla and Micláus [10,11] constructed new sequences of linear positive operators to investigate the rapidity of convergence and order of approximation in diferent functional spaces in terms of several generating functions. Some other researchers developed many other useful operators [6,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] in the same feld. In the recent past, for g ∈ [0, 1], m ∈ N and α ∈ [−1, 1], Chen et al [31] constructed a sequence of new linear positive operators as…”

Approximation by q-Bernstein-Stancu-Kantorovich operators with shifted knots of real parameters

“…Since then several polynomials were generalized and studied by using q-calculus. For example related to our present theme, q-Bernstein shifted operators [20], q-Bernstein-Kantorovich operators [21], Bernstein-Kantorovich operators based on (p, q)-calculus [19], and other related operators [1], [4], [13], [25], etc.. The Bernstein operators have been extended in various forms for the purpose of approximating functions of different classes by replacing the point evaluation functionals by some integrals.…”

“…Recently, a variety of Bernstein-Kantorovich operators have been studied in [18], [23] and [17]. While the operators (1) with shifted knots were studied by Gadjiev et al [8] for the operators S ν 2 r,µ 2 :…”

On One- and Two-Dimensional α–Stancu–Schurer–Kantorovich Operators and Their Approximation Properties

“…In [13], the authors investigated various pointwise and uniform approximation results. Furthermore, many researchers, e.g., Kilicman et al [15], Acar et al [16], Aral et al [17], Cai et al [18,19], Çetin et al [20,21], Mohiuddine et al [22], Aslan et al [23,24], Acu et al [25], Agrawal [26], Nasiruzzaman et al [27], and Ayman-Mursaleen et al [28,29], have intensively studied α-Bernstein operators and their modifications for better approximation results. In [30][31][32] some interesting studies have been carried out.…”