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.

Recently, the notion of positive linear operators by means of basic (or q-) Lagrange polynomials and {\mathcal{A}} -statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means of q-Lagrange polynomials and A-statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911–6918]. In our present investigation, we introduce a certain deferred weighted {\mathcal{A}} -statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1, t and {t^{2}} defined on a Banach space {C[0,1]} for a sequence of (presumably new) positive linear operators based upon {(p,q)} -Lagrange polynomials. Furthermore, we investigate the deferred weighted {\mathcal{A}} -statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.

The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. . The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

In the present work, we introduce and study the notion of statistical probability convergence for sequences of random variables as well as the concept of statistical convergence for sequences of real numbers, which are defined over a Banach space via product of deferred Cesàro and deferred Nörlund summability means. We first establish a theorem presenting a connection between them. Based upon our proposed method, we then prove a Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results (in classical as well as statistical versions). Finally, an illustrative example is presented here by means of the generalized Meyer-König and Zeller operators for a positive sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions.