Abstract
Korovkin type approximation theorems are useful tools to check whether a given sequence ( L n ) n ≥ 1 of positive linear operators on C[0,1] of all continuous functions on the real interval [0,1] is an approximation process. That is, these theorems exhibit a variety of test functions, which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, x and x 2 in the space C[0,1] as well as for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line. In this paper, we use the notion of statistical summability (C,1) to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence.MSC:41A10, 41A25, 41A36, 40A30, 40G15.
Highlights
Introduction and preliminariesWe shall denote by N be the set of all natural numbers
The natural density of K is defined by δ(K) = limn n– |Kn| if the limit exists, where the vertical bars indicate the number of elements in the enclosed set
Korovkin second theorem is proved by Demirci and Dirik [ ] for statistical σ -convergence [ ]
Summary
Introduction and preliminariesWe shall denote by N be the set of all natural numbers. Móricz [ ] has defined the concept of statistical summability (C, ) as follows: For a sequence x In this case we write L = C (st)-lim x. Let x = (xk) be a sequence defined by
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