Four-dimensional matrix transformation and A-statistical fuzzy Korovkin type approximation

Abstract

Abstract. In this paper, we prove a fuzzy Korovkin-type approximation theorem for fuzzy positive linear operators by using A-statistical convergence for four-dimensional summability matrices. Also, we obtain rates of A-statistical convergence of a double sequence of fuzzy positive linear operators for four-dimensional summability matrices. IntroductionAnastassiou . In this paper, we prove a fuzzy Korovkin-type approximation theorem for fuzzy positive linear operators by using A-statistical convergence for four-dimensional summability matrices. Then, we construct an example such that our new approximation result works but its classical case does not work. Also we obtain rates of A-statistical convergence of a double sequence of fuzzy positive linear operators for four-dimensional summability matrices.We now recall some basic definitions and notations used in the paper. A fuzzy number is a function µ : R → [0, 1], which is normal, convex, upper semi-continuous and the closure of the set supp(µ) is compact, where supp(µ) := {x ∈ R : µ(x) > 0}. The set of all fuzzy numbers are denoted by R F . Let

Generalized $A$-statistical convergence and a Korovkin type approximation theorem for double sequences

“…Then using the similar technique as in the proof of Theorem 4, one can also get the following result. If the ideal I in Theorem 5 is replaced by I 0 ; we immediately get the following result proved in [4].…”

“…Later, using different summability methods, some Korovkin type theorems have been obtained by many authors in several ways. For instance, see [4], [6][7][8][9], [10]. In recent years, the studies on double sequences has a rapid growth.…”

Approximation of Continuous Periodic Functions of Two Variables via Power Series Methods of Summability

“…Let f ∈ C * (R 2 ) and (x, y) ∈ [−π, π] × [−π, π]. Then in view of the proof of Theorem 9 in [6] we have…”

“…Besides, in connection with sequence transformations, weighted mean methods and power series methods of summability have been applied to Korovkin type theorems to recover the convergence of operators for which classical Korovkin theorems fail to work [5,8,11,12,14,15]. Furthermore, there are studies dealing with Korovkin type theorems via weighted mean summability methods for continuous functions of two variables [1,4,6,7]. In this paper, we prove a Korovkin type approximation theorem for 2π-periodic and real valued continuous functions on R 2 by using power series methods of summability and also give an approximation theorem by using modulus of continuity.…”

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

“…The statistical convergence in approximation theory was first used by Gadjiev and Orhan [3] to prove the Korovkin-type approximation theorem [4]. The studies and related results can be found in [5][6][7][8].…”