Equi-Statistical Relative Convergence and Korovkin-Type Approximation

Abstract

Classical approximation theory has started with the proof of Weierstrass approximation theorem and after that Korovkin [Linear operators and approximation theory, Hindustan Publ. Corp, Delhi, 1960] first established the necessary and sufficient conditions for uniform convergence of a sequence of positive linear operators to a function f . In classical Korovkin theorem, most of the classical operators tend to converge to the value of the function being approximated. Also, the attention of researchers has been attracted to the notion of statistical convergence because of the fact that it is stronger than the classical convergence method. Furthermore, the concept of equi-statistical convergence is more general than the statistical uniform convergence. In this work, we introduce our new convergence method named equi-statistical relative convergence to demonstrate a Korovkin type approximation theorems which were proven by earlier authors. Then, we present an example in support of our definition and result presented in this paper. Finally, we compute the rate of the convergence.