Statistical Equal Convergence On Weighted Spaces

Abstract

The Korovkin theory has effective role in approximation theory. This theory is connected with the approximation to continuous functions by means of positive linear operators. Many mathematicians have investigated the Korovkin-type theorems by for a sequence of positive linear operators defined on different spaces by using various types of convergence. Firstly, A.D. Gadjiev has proved the weighted Korovkin type theorems, (Math. Zamet., 20 (1976) 781-786 (in Russian)). Later, these theorems are studied by many authors by means of different convergence methods. Recently, The definition of equal convergence for real functions was introduced by Császár and Laczkovich and they improved their investigations on this * convergence. Later Das et. al. introduced the ideas of I and I -equal convergence with the help of ideals by extending the equal convergence (Mat. Vesnik, vol:66, 2 (2014),165-177). In our work, we introduce a new type of statistical convergence on weighted spaces by using the notions of the equal convergence. We study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work.