Statistical deferred weighted B $\mathcal{B}$ -summability and its applications to associated approximation theorems

Abstract

The notion of statistical weighted -summability was introduced very recently (Kadak et al. in Appl. Math. Comput. 302:80–96, 2017). In the paper, we study the concept of statistical deferred weighted -summability and deferred weighted -statistical convergence and then establish an inclusion relation between them. In particular, based on our proposed methods, we establish a new Korovkin-type approximation theorem for the functions of two variables defined on a Banach space and then present an illustrative example to show that our result is a non-trivial extension of some traditional and statistical versions of Korovkin-type approximation theorems which were demonstrated in the earlier works. Furthermore, we establish another result for the rate of deferred weighted -statistical convergence for the same set of functions via modulus of continuity. Finally, we consider a number of interesting special cases and illustrative examples in support of our findings of this paper.

Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

“…Moreover, since the sequence (x m ) is not deferred Nörlund statistically convergent, the finding of Srivastava et al [24] does not serve for our operator defined in (15). Thus, our Theorem 2 is certainly a non-trivial generalization of the findings of Srivastava et al [24] (see also [33,38]). Based upon the above outcomes, we conclude here that our chosen method has credibly worked under the operators defined in (15), and hence, it is stronger than the classical and statistical versions of the approximation of Korovkin-type theorems (see [24,33,38]) which were established earlier.…”

“…On the other hand, Dutta et al [28] studied another Korovkin type theorem over C[0, ∞) by considering the exponential test functions 1, e −x and e −2x on the basis of the deferred Cesàro mean. For more recent works in this direction, see [23,[29][30][31][32][33][34][35][36][37][38].…”

Strong Riesz summability of Fourier series

“…Also, the statistical convergene and statistical summability are more general than the ordinary convergene and ordinary summability. For recent works in this direction, see [1], [5], [6], [8], [9], [10], [11], [15], [16], [17], [18], [19] and [20]. Dealing with strong summability of Nörlund mean Mittal [13] has proved a theorem as follows.…”

Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems

“…and then by Ψ( f ; x, y) in (16), for a given κ > 0 there exists > 0, such that ω 5λ σ < κ. Then, by setting…”

“…As an extension of statistical versions of convergence, the idea of weighted statistical convergence of single sequences was presented by Karakaya and Chishti [7], and it has been further generalized by various authors (see [8][9][10][11][12]). Moreover, the concept of deferred weighted statistical convergence was studied and introduced by Srivastava et al [13] (see also [14][15][16][17][18][19]).…”