No abstract

KEYWORDSBanach space, deferred weighted statistical convergence, Korovkin-type approximation theorems, periodic functions, positive linear operators, rate of convergence, statistical convergence INTRODUCTION, DEFINITIONS, AND MOTIVATIONIn the study of sequence spaces, classical convergence has got innumerable applications where the convergence of a sequence requires that almost all elements are to satisfy the convergence condition, that is, all of the elements of the sequence need to be in an arbitrarily small neighborhood of the limit. However, such restriction is relaxed in statistical convergence, where the validity of the convergence condition is achieved only for a majority of elements. The notion of statistical convergence was introduced and studied by Fast 1 and Steinhaus. 2 Recently, statistical convergence has been a dynamic research area due basically to the fact that it is more general than the classical convergence and such a theory as well as its various applications are discussed in the study in the areas of (for example) Fourier analysis, number theory, and approximation theory. For more details, see the recent works. [3][4][5][6][7][8][9][10][11][12][13][14] Let N be the set of natural numbers, and let K ⊆ N. Also, let K n = {k ∶ k ≦ n and k ∈ K}, Math Meth Appl Sci. 2018;41 671-683. wileyonlinelibrary.com/journal/mma

Statistical (C, 1) summability and a Korovkin type approximation theorem has been proved by Mohiuddine et al. [20] (see [S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical summability (C, 1) and a Korovkin type approximation theorem, J. Inequal. Appl. 2012 (2012), Article ID 172, 1-8). In this paper, we apply statistical deferred Cesàro summability method to prove a Korovkin type approximation theorem for the set of functions 1, e −x and e −2x defined on a Banach space C[0, ∞) and demonstrate that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. We also establish a result for the rate of statistical deferred Cesàro summability method. Some interesting examples are also discussed here in support of our definitions and results.

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.