Statistical \(\sigma \)-convergence of positive linear operators
Denote by \(C_M[a,b]\) the space of all continuous functions on \([a,b]\) and bounded on the entire line, that is \(| f(x)|\leq K_f\), \(-\infty<x<\infty\), where \(K_f\) is a constant depending on \(f\). By \(B[a,b]\) is denoted the space of all bounded functions on \([a,b]\). In this paper, the author obtains the following Korovkin-type approximation theorem by means of the concept of statistical \(\sigma\)-convergence. Theorem. Let \(\{ L_m\}\) be a sequence of positive linear operators acting from \(C_M[a,b]\) into \(B[a,b]\). Assume that the following condition hold: \[ \delta(\sigma)-\lim\|L_m(f_i)-f_i\|_{B[a,b]}=0,\quad \text{where}\quad f_i(x)=x^i,\quad i=0,1,2. \] Then, for all \(f\in C_M[a,b]\), we have \[ \delta(\sigma)-\lim\|L_m(f)-f\|_{B[a,b]}=0. \] The rates of statistical \(\sigma\)-convergence of a sequence of positive linear operators are computed.
scientific article; zbMATH DE number 5838267
scientific article