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.
Approximation of functions of Lipschitz and zygmund classes have been considered by various researchers under different summability means. In the proposed paper, we have studied an estimation of the order of convergence of Fourier series in the weighted Zygmund class <em>W(Z<sub>r</sub><sup>(ω)</sup>)</em> by using Euler-Hausdorff product summability mean and accordingly established some (presumably new) results. Moreover, the results obtained here are the generalization of several known results.
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