In this paper using equi-statistical convergence, which is stronger than the usual uniform convergence and statistical uniform convergence, we obtain a general Korovkin type theorem. Then, we construct examples such that our new approximation result works but its classical and statistical cases do not work. | Turk J Math 33 (2009) , 159 – 168. ¨ ITAK ˙ c TUB doi: Equi-Statistical Extension of the Korovkin Type Approximation Theorem Sevda Karaku¸s, Kamil Demirci Abstract In this paper using equi-statistical convergence, which is stronger than the usual uniform convergence and statistical uniform convergence, we obtain a general Korovkin type theorem. Then, we construct examples such that our new approximation result works but its classical and statistical cases do not work. Key Words: Equi-statistical convergence, positive linear operator, Korovkin type theorem. 1. Introduction Throughout this paper I := [0, ∞). C (I) is the space of all real-valued continuous functions on I and CB (I) := {f ∈ C (I) : f is bounded on I} . The sup norm on CB (I) is given by f CB (I) := sup |f (x)| , x∈I (f ∈ CB (I)) . Also, let Hw be the space of all real valued functions f defined on I and satisfying x y , |f (x) − f (y)| ≤ w f; − 1 + x 1 + y () where w is the modulus of continuity given by, for any δ > 0 , w (f; δ) := sup |f (x) − f (y)| . x,y∈I |x−y| 0 , n ∈ N and the symbol |A| denotes the cardinality of the subset A. Definition 1 [12] (fn ) is said to be statistically pointwise convergent to f on I if st − limn→∞ fn (x) = f(x) for each x ∈ I, ., for every ε > 0 and for each x ∈ I, limn→∞ Ψn (x,ε) n = 0. Then, it is denoted by fn → f (stat) on I. Definition 2 [12] (fn ) is said to be equi-statistically convergent to f on I if for every ε > 0, limn→∞ 0 uniformly with respect to x ∈ I, which means that limn→∞ Ψn (.,ε) C n B (I) Ψn (x,ε) n = = 0 for every ε > 0. In this case, we denote this limit by fn → f (equi − stat) on I. Definition 3 [12] (fn ) is said to be statistically uniform convergent to f on I if st-limn→∞ fn − f CB (I) = 0, or lim n→∞ Φn (ε) n = 0. This limit is denoted by fn ⇒ f (stat) on I. Using the above definitions, we get the following result. Lemma 1 [12] fn ⇒ f on I (in the ordinary sense) implies fn ⇒
