Received: 22 September 2017 DOI: 10.1002/mma.4636 RESEARCH ARTICLE A certain class of weighted statistical convergence and associated Korovkin-type approximation theorems involving trigonometric functions H. M. Srivastava 1,2 Bidu Bhusan Jena 3 Susanta Kumar Paikray 3 U. K. Misra 4 1 Department of Mathematics and Statistics, University of Victoria, Victoria British Columbia V8W 3R4, Canada 2 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China 3 Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India 4 Department of Mathematics, National Institute of Science and Technology, Palur Hills Golanthara 761008, Odisha, India Correspondence H. M. Srivastava, Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada. Email: harimsri@math.uvic.ca Communicated by: W. Sprößig MOS Classification: Primary 40A05, 41A36; Secondary 40G15 The subject of statistical convergence has attracted a remarkably large num- ber of researchers due mainly to the fact that it is more general than the well-established theory of the ordinary (classical) convergence. In the year 2013, Edely et al 17 introduced and studied the notion of weighted statistical con- vergence. In our present investigation, we make use of the (presumably new) notion of the deferred weighted statistical convergence to present Korovkin-type approximation theorems associated with the periodic functions 1, cos x, and sin x defined on a Banach space C 2 (R). In particular, we apply our concept of the deferred weighted statistical convergence with a view to proving a Korovkin-type approximation theorem for periodic functions and also to demonstrate that our result is a nontrivial extension of several known Korovkin-type approximation theorems which were given in earlier works. Moreover, we establish another result for the rate of the deferred weighted statistical convergence for the same set of functions. Finally, we 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. KEYWORDS Banach space, deferred weighted statistical convergence, Korovkin-type approximation theorems, periodic functions, positive linear operators, rate of convergence, statistical convergence 1 INTRODUCTION, DEFINITIONS, AND MOTIVATION In 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. 3-14 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. 2017;1–13. wileyonlinelibrary.com/journal/mma Copyright © 2017 John Wiley & Sons, Ltd. 1