The uniform convergence of a double sequence of functions at a point and Korovkin-type approximation theorems

Abstract

In this paper, we introduce an interesting kind of convergence for a double sequence called the uniform convergence at a point. We give an example and demonstrate a Korovkin-type approximation theorem for a double sequence of functions using the uniform convergence at a point. Then we show that our result is stronger than the Korovkin theorem given by Volkov and present several graphs. Finally, in the last section, we compute the rate of convergence.

Statistical relative uniform convergence of a double sequence of functions at a point and applications to approximation theory

“…Then ( ( )) ( = 0,1,2,3) converges statistically uniformly at ( 0 , 0 ) to iff for each ∈ ( 2 ), ( ( )) converges statistically uniformly at ( 0 , 0 ) to . Also, if one replaces the statistical limit with Pringsheim limit and scale function by a non-zero constant, then the next result which was obtained in [21] follows from our main Korovkin type approximation theorem.…”

“…More recently, Dirik et al [21] introduced the uniform convergence of a sequence of functions at a point for double sequences. Before this definition, we first give the following:…”

“…Another interesting type of convergence is the uniform convergence of a sequence of functions at a point [18]. More recently, Demirci et al [19] extended this type of convergence to relative uniform convergence of a sequence of functions at a point where the set of the neighborhoods of the point at which relative uniform convergence is considered (see, e.g., [20,21]).…”